Geometric and Computational Spectral Theory [1 ed.] 9781470442583, 9781470426651

Citation preview

700

Geometric and Computational Spectral Theory Séminaire de Mathématiques Supérieures Geometric and Computational Spectral Theory June 15–26, 2015 Centre de Recherches Mathématiques, Université de Montréal, Montréal, Québec

Alexandre Girouard Dmitry Jakobson Michael Levitin Nilima Nigam Iosif Polterovich Frédéric Rochon Editors

American Mathematical Society Providence, Rhode Island Centre de Recherches Mathématiques Montréal, Québec, Canada

Geometric and Computational Spectral Theory Séminaire de Mathématiques Supérieures Geometric and Computational Spectral Theory June 15–26, 2015 Centre de Recherches Mathématiques, Université de Montréal, Montréal, Québec

Alexandre Girouard Dmitry Jakobson Michael Levitin Nilima Nigam Iosif Polterovich Frédéric Rochon Editors

700

Geometric and Computational Spectral Theory Séminaire de Mathématiques Supérieures Geometric and Computational Spectral Theory June 15–26, 2015 Centre de Recherches Mathématiques, Université de Montréal, Montréal, Québec

Alexandre Girouard Dmitry Jakobson Michael Levitin Nilima Nigam Iosif Polterovich Frédéric Rochon Editors

American Mathematical Society Providence, Rhode Island Centre de Recherches Mathématiques Montréal, Québec, Canada

Editorial Board of Contemporary Mathematics EDITORIAL COMMITTEE Dennis DeTurck, Managing Editor Michael Loss

Kailash Misra

Catherine Yan

Editorial Committee of the CRM Proceedings and Lecture Notes Jerry L. Bona Va˘sek Chvatal Galia Dafni Nicole Tomczak-Jaegermann

Lisa Jeffrey Ram Murty Christophe Reutenauer Pengfei Guan

Donald Dawson Nicolai Reshetikhin H´el´ene Esnault Luc Vinet

2010 Mathematics Subject Classification. Primary 58Jxx, 35Pxx, 65Nxx.

Library of Congress Cataloging-in-Publication Data Names: Girouard, Alexandre, 1976- editor. Title: Geometric and computational spectral theory : S´ eminaire de Math´ematiques Sup´ erieures, June 15-26, 2015, Centre de Recherches Math´ ematiques, Universit´ e de Montr´ eal, Montr´eal, Quebec, Canada / Alexandre Girouard, Dmitry Jakobson, Michael Levitin, Nilima Nigam, Iosif Polterovich, Fr´ ed´ eric Rochon, editors. Description: Providence, Rhode Island : American Mathematical Society ; Montr´eal, Quebec, Canada : Centre de Recherches Math´ematiques, [2017] | Series: Contemporary mathematics ; volume 700 | Includes bibliographical references. Identifiers: LCCN 2017030037 | ISBN 9781470426651 (alk. paper) Subjects: LCSH: Metric spaces–Congresses. | Spectral geometry–Congresses. | Geometry, Differential–Congresses. | AMS: Global analysis, analysis on manifolds – Partial differential equations on manifolds; differential operators – Partial differential equations on manifolds; differential operators. msc | Partial differential equations – Spectral theory and eigenvalue problems – Spectral theory and eigenvalue problems. msc | Numerical analysis – Partial differential equations, boundary value problems – Partial differential equations, boundary value problems. msc Classification: LCC QA611.28 .G46 2015 | DDC 515/.353–dc23 LC record available at https://lccn.loc.gov/2017030037 DOI: http://dx.doi.org/10.1090/conm/700

Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy select pages for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Permissions to reuse portions of AMS publication content are handled by Copyright Clearance Center’s RightsLink service. For more information, please visit: http://www.ams.org/rightslink. Send requests for translation rights and licensed reprints to [email protected]. Excluded from these provisions is material for which the author holds copyright. In such cases, requests for permission to reuse or reprint material should be addressed directly to the author(s). Copyright ownership is indicated on the copyright page, or on the lower right-hand corner of the first page of each article within proceedings volumes. c 2017 by the American Mathematical Society. All rights reserved. 

∞ The paper used in this book is acid-free and falls within the guidelines 

established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10 9 8 7 6 5 4 3 2 1

22 21 20 19 18 17

Contents

The spectrum of the Laplacian: A geometric approach Bruno Colbois

1

An elementary introduction to quantum graphs Gregory Berkolaiko

41

A free boundary approach to the Faber-Krahn inequality Dorin Bucur and Pedro Freitas

73

Some nodal properties of the quantum harmonic oscillator and other Schr¨odinger operators in R2 Pierre B´ erard and Bernard Helffer

87

Numerical solution of linear eigenvalue problems Jessica Bosch and Chen Greif

117

Finite element methods for variational eigenvalue problems Guido Kanschat

155

Computation of eigenvalues, spectral zeta functions and zeta-determinants on hyperbolic surfaces Alexander Strohmaier 177 Scales, blow-up and quasimode constructions Daniel Grieser

207

Scattering for the geodesic flow on surfaces with boundary Colin Guillarmou

267

v

Preface The 2015 edition of the S´eminaire de Math´ematiques Sup´erieures (SMS) took place on June 15-26, 2015, at the Universit´e de Montr´eal. The topic of the meeting was Geometric and computational spectral theory, and it was perhaps the largest school in spectral geometry since the late nineties. The event was sponsored by the Centre de recherche math´ematiques, as well as by the Fields Institute (Toronto), the Pacific Institute for the Mathematical Sciences (Vancouver), the Mathematical Sciences Research Institute (Berkeley), the Institut des sciences math´ematiques (Montr´eal) and the Canadian Mathematical Society. The summer school brought together students and internationally renowned experts in the geometric and computational aspects of spectral theory. The area of spectral theory has fascinated mathematicians and physicists for centuries, and recent years have seen remarkable progress in several branches of the field. The scientific program consisted of twelve minicourses in four main themes: geometry of eigenvalues, geometry of eigenfunctions, computational spectral theory, and spectral theory on singular spaces. A particular emphasis was made on the interplay between these topics, notably between the computational and the geometric part — which was one of the most novel aspects of the school. The minicourses were complemented by exercise sessions, computer labs and short presentations by selected junior participants who have already made important contributions to the subject. The school featured about 90 participants from 13 countries spanning five continents. The current volume contains the lecture notes and survey papers based on nine minicourses delivered at the summer school. The lecture notes by Bruno Colbois (Neuchˆatel) begin with an overview of some basic facts on the spectrum of the Laplace operator on Euclidean domains and Riemannian manifolds. The main part of the notes is concerned with the geometric eigenvalue inequalities, notably the Cheeger lower bound on the first eigenvalue and the Korevaar’s method for estimating from above the eigenvalues of the Laplacian on a Riemannian manifold in a fixed conformal class. The last two sections cover some related recent results obtained by the author and his collaborators. In particular, bounds on the eigenvalues of hypersurfaces are discussed, as well as the notions of the topological and conformal spectra. The lecture notes by Gregory Berkolaiko (Texas A & M) provide an introduction to spectral theory of Schr¨odinger operators on metric graphs, which in this setting are known as quantum graphs. A number of illuminating examples are treated in detail. A particular emphasis is made on the count of zeros of the eigenfunctions of quantum graphs. A survey paper by Dorin Bucur (Chamb´ery) and Pedro Freitas (Lisbon) focusses on the celebrated Faber-Krahn inequality for the fundamental tone of the vii

viii

PREFACE

Dirichlet Laplacian from the viewpoint of free boundary problems. The proof of the Faber-Krahn inequality which is presented is purely variational and, surprisingly, does not use rearrangement arguments. The advantage of this approach is its adaptability to extremal problems for higher eigenvalues and to more general Robin boundary conditions. The paper by Pierre B´erard (Institut Fourier) and Bernard Helffer (Nantes and Orsay) is concerned with the nodal properties of the quantum harmonic oscillator and other Schr¨ odinger operators on a Euclidean plane. In particular, the authors show that for the two-dimensional isotropic quantum harmonic oscillator there exists an infinite sequence of eigenfunctions with exactly two nodal domains. For the spherical Laplacian and the Dirichlet Laplacian on a square a similar result has been proved long time ago by A. Stern and H. Lewy. Other questions in nodal geometry are also discussed, such as bounds on the length of the zero set of a Schr¨ odinger eigenfunction in the classically permitted region. The next three contributions to the volume deal with various computational aspects of spectral theory. The paper by Jessica Bosch and Chen Greif (UBC) provides a review of the numerical methods for computing eigenvalues of matrices. A number of methods are discussed in detail, including the power method, the divideand-conquer algorithms for tridiagonal matrices, as well as several approaches to compute the eigenvalues of large and sparce matrices, such as Lanczos, Arnoldi and Jacobi-Davidson methods. The notes by Guido Kanschat (Heidelberg) give an introduction to the finite element methods for variational eigenvalue problems. In particular, Galerkin approximation is discussed in detail. Several eigenvalue problems are considered, including the standard boundary value problems for the Laplacian, as well as problems arising in the study of Maxwell and Stokes equations. In the appendix, some computer programs for experiments are presented. The lecture notes by Alexander Strohmaier (Leeds) provide a link between the “numerical” and the “geometric” themes of this volume. The notes focus on the method of particular solutions and its applications to the computation of eigenvalues of the Laplace operator on Riemannian manifolds, including hyperbolic surfaces. Several Mathematica programs are included for illustration, which is most helpful. Computations of various spectral quantities, such as the spectral zeta function and the zeta-regularized determinant of the Laplace operator, are also discussed. The remaining two contributions are based on minicourses requiring more advanced analytic background. The lecture notes by Daniel Grieser (Oldenburg) give an introduction to analysis on manifolds with corners in the spirit of R. Melrose. Its tools are used to construct quasimodes (i.e. approximate eigenfunctions) for degenerating families of domains, including adiabatic limit families and families exhibiting certain types of sclaing behaviour. The notes by Colin Guillarmou (ENS) are concerned with geometric inverse problems on surfaces with boundary, such as the lens rigidity problem and its special case, the boundary rigidity problem. These problems naturally arise when one needs to recover the Riemannian metric on a surface from the boundary measurements. A closely related notion of the X-ray transform is also discussed. The Proceedings of the 2015 S´eminaire de Math´ematiques Sup´erieures on Geometric and Computational Spectral Theory cover a large variety of topics and methods, combining geometric, analytic and numerical ideas. We hope that this volume

PREFACE

ix

will serve both as a useful reference for experts and as an inspiring reading for young mathematicians who would like to learn more about this fascinating and rapidly developing area of mathematics. The editors would like to express their gratitude to all the contributing authors, as well as to all the speakers at the summer school. Last but not least, it is our pleasure to thank the SMS coordinator, Sakina Benhima, for all her hard work and help with the organization of this event.

Contemporary Mathematics Volume 700, 2017 http://dx.doi.org/10.1090/conm/700/14181

The spectrum of the Laplacian: A geometric approach Bruno Colbois

Preamble: These notes correspond to a 4-hour lecture and one exercise session given in Montr´eal in June 2015 during the summer school Geometric and Computational Spectral Theory. The goal was to introduce the subject, that is to present various aspects of the spectrum of the Laplacian on a compact Riemannian manifold from a geometric viewpoint, and also to prepare the audience for some of the following lectures. In the first part, I recall without proof some classical facts, I try to illustrate the theory with examples and I discuss open questions. The aim is to give an intuition to the students discovering the subject, without being too formal. The exercise session was very useful, revealing that things which are obvious for the people of the field are far from being simple for those looking at the question for the first time. This is the reason why I decided to present in these notes really simple examples with a lot of details. In the second part, I focus mainly on a specific problem: the estimates of the spectrum in the conformal class of a given Riemannian metric. This is the opportunity to present recent developments on the subject and to explain some very interesting geometrical methods in order to get upper bounds for the spectrum. As already mentioned, I have tried not to be too formal, and the price for that is that things may sometimes lack precision. I have tried to give all needed references, and for the basics, I will mainly refer to the following books. The book of A. Henrot [He] constitutes a very good reference for the study of the spectrum of the Laplacian for Euclidean domains, and the books of I. Chavel [Ch1] and P. B´erard [Be] are very convenient for the spectrum of the Laplacian on Riemannian manifolds. They contain a lot of basic and important results. More recent references are the book of Rosenberg [Ro] (but rather far from the goal of this lecture) and the lecture notes about Spectral Theory and Geometry edited by B. Davies and Y. Safarov [ST]. Last but not least, it is always interesting to look at the famous book of BergerGauduchon-Mazet [BGM], who played an important role in the development of the Geometrical Spectral Theory. At the moment when the summer school took place, the book by O. Labl´ee [La] appeared. I did not use it to write these notes but it is an interesting introduction to the subject.

2010 Mathematics Subject Classification. Primary 35P15, 58J50. c 2017 American Mathematical Society

1

2

BRUNO COLBOIS

1. Introduction, basic results and examples In this section, I will describe without proof some classical results and examples concerning the spectrum of the Laplacian on domains and on Riemannian manifolds. Most of the time, I will introduce the different notions rather informally and refer to classical texts for more details. The aim is to give a very rough first idea of a part of the “landscape” of the geometrical spectral theory thanks to a couple of examples and results. It is as well an opportunity to introduce some of the other lectures of the summer school. 1.1. The Laplacian for Euclidean domains with Dirichlet boundary conditions. Let Ω ⊂ Rn be a bounded, open, connected domain with Lipschitz boundary. The Laplacian we will consider is given by n  ∂2f Δf = − ∂x2i i=1 where f ∈ C 2 (Ω). We investigate the spectrum of the Laplacian Δ on Ω with the Dirichlet boundary condition, that is we study the eigenvalue problem (1)

Δf = λf

under the Dirichlet condition (2)

f|∂Ω = 0. This problem has a discrete and real spectrum 0 < λ1 (Ω) < λ2 (Ω) ≤ λ3 (Ω) ≤ ... → ∞,

with the eigenvalues repeated according to their multiplicity. Example 1. If Ω is the interval ]0, π[⊂ R, the Laplacian is given by Δf = −f  and the spectrum, for the Dirichlet boundary condition, is given by λk = k2 , k = 1, 2, ... The eigenfunction corresponding to λk = k2 is fk (x) = sin kx. However, it is exceptional to be able to calculate explicitly the spectrum. In dimensions greater than one, it is generally not possible (apart from a few exceptions like the ball or a product of intervals) to calculate the spectrum of a domain. The calculation of the spectrum of a ball is classic, but not easy. It is done with enough details, for example, in the introduction of the Ph. D. thesis of A. Berger [Ber]. Given that, physically, the Dirichlet problem is a modelization of the vibration of a fixed membrane, it can be hoped to have connections between the geometry (or shape) of a domain Ω and its spectrum. By fixed membrane is understood a membrane whose boundary is fixed, like a drum. This leads to the following very classical image for two types of problems in the context of geometrical spectral theory: The direct problems. When we see a drum, we have an idea of how it would sound. The mathematical translation is: if we have some information about the geometry (or the shape) of a domain Ω, we can hope to get information (or estimates) about its spectrum. By information about the geometry can be understood for example the volume, the diameter of Ω, the mean curvature of its boundary, the inner radius. By estimates of the spectrum, we understand lower or upper bounds for some or all of the eigenvalues, with respect to the geometrical information we have.

THE SPECTRUM OF THE LAPLACIAN: A GEOMETRIC APPROACH

3

The inverse problems. If we hear a drum, even if we do not see it, we have an idea of its shape. The mathematical translation is: If we have information about the spectrum of a domain, we can hope to get information about its geometry. The most famous question is whether the shape of a domain is determined by its spectrum (question of M. Kac in 1966: can one hear the shape of a drum?). In other words: if we know all the spectrum of a domain, are we able to reconstruct the domain? The formal answer is no: There are a lot of domains having the same spectrum without being isometric. However, in some sense, this is often true, which makes things complicated. I will not go further in this direction in these notes, see [Go] for a survey of the subject. 1.1.1. The Faber-Krahn inequality. This is an old but very enlightening result found independently by Faber and Krahn in the 1920’s: More explanations and a sketch of the proof can be found in [As, p. 102–103]. The proof uses the isoperimetric inequality and is not easy. In his lecture, D. Bucur will offer an approach for a new proof of this inequality (see [BF]). Theorem 2. Let Ω ⊂ Rn be a bounded open domain in Rn and B ⊂ Rn , a ball with the same volume as Ω. If λ1 denotes the first eigenvalue for the Dirichlet boundary conditions, then λ1 (B) ≤ λ1 (Ω), with equality if and only if Ω is equal to B up to a displacement. Firstly, the theorem gives a lower bound for the first Dirichlet eigenvalue of a domain Ω which is a typical direct problem. If the volume of Ω is known, then one gets immediately information about the spectrum: a lower bound for the first eigenvalue. But the equality case can be understood as an inverse result. If Ω and B have the same volume and if λ1 (Ω) = λ1 (B), then Ω is a ball. This result is also very interesting because it allows very modern questions to be asked. We will mention a few of them, without going into much detail. The same type of questions could also be asked in other situations (other boundary conditions, Laplacian on Riemannian manifolds, other differential operators, etc.). (1) The stability. If a domain Ω ⊂ Rn has the same volume as a ball B and if, moreover, λ1 (Ω) is close to λ1 (B), can one say that Ω is close to the ball B? Of course, a difficulty is to state what is meant by “close”. There are a lot of results around this problem that I will not describe in detail, but see [FMP] for a recent and important contribution and [BP] for a very recent survey about this question. Note that it is not too difficult to see that one can do a small hole in any domain or add a “thin hair” without affecting the spectrum too much (easy part in Example 16 and more explanations in [BC], [CGI]). Therefore, it cannot be hoped for example, that Ω will be homeomorphic to a ball, or will be Hausdorff close to a ball. However, essentially, one can show that Ω is close to a ball in the sense of the measure, see Figure 1 and [FMP], [BP]. (2) The spectral stability. The next natural question is to see what this stability will imply: in [BC], we have shown that if Ω and B have the same volume, and if λ1 (Ω) is close to λ1 (B) then, for all k, λk (Ω) is close to λk (B) (but, of course, not uniformly in k). The difficulty is that the proximity of λ1 (Ω) and λ1 (B) implies only that the domains are “close

4

BRUNO COLBOIS

B

Figure 1. The domain Ω is close to the ball B in the sense of the measure in the sense of the measure”, and we have to show that this is enough to control all the eigenvalues. (3) The extremal domains. We have seen that, among all domains having the same volume, the ball is the domain having the minimal λ1 . We can of course ask for domain(s) having the minimal λk . The various aspects of this problem are very complicated and most are far from being solved. For λ2 , it is well known that the solution is the union of two disjoint balls (see [He, Thm. 4.1.1]). Note that this domain is not regular in the sense that was expected. But in general, it is already difficult to find a right context where it can be shown that an extremal domain exists. It will be one of the goals of D. Bucur’s lecture to explain this, and I do not develop the question more here. See also the contribution of D. Bucur in [BF]. It is interesting to sketch the proof for λ2 as it is done, for example, in [He]. It uses the Courant Nodal Theorem (see the lecture of B. Helffer, [BH] or [Ch1, p. 19]) which says the following: if {fi }∞ i=1 denotes an , then the number of orthonormal basis of eigenfunctions for {λi (Ω)}∞ i=1 nodal domains of fk is less than or equal to k, for every k ≥ 1, where a ¯ −1 (0). nodal domain of uk is a connected component of Ω/f k The first implication of the theorem of Courant is that any eigenfunction f1 associated to the first eigenvalue λ1 of a domain Ω has a constant sign (say it is positive) and has multiplicity one. The second eigenvalue λ2 of Ω could have multiplicity. If f2 is an eigenfuction associated to λ2 , it has to be orthogonal to f1 , and, as consequence, changes its sign. Let A1 = {x ∈ Ω : f2 (x) > 0}; A2 = {x ∈ Ω : f2 (x) < 0}. The Courant Nodal Theorem says that A1 and A2 are connected. Thanks to this information and to the theorem of Faber-Krahn, we can prove that the extremal domain for λ2 is the union of two balls of the same size. Let Ω be a domain, and f2 an eigenfunction for λ2 . The restriction of f2 to A1 and to A2 satisfies the Laplace equation, and f2 is an eigenfunction of eigenvalue λ2 . But on A1 and A2 , f does not change its sign. This means that f2 is an eigenfunction for the first eigenvalue of A1 and A2 , and it follows

THE SPECTRUM OF THE LAPLACIAN: A GEOMETRIC APPROACH

5

λ1 (A1 ) = λ1 (A2 ) = λ2 (Ω). Let B1 , B2 two balls with V ol(B1 ) = V ol(A1 ) and V ol(B2 ) = V ol(A2 ). By Faber-Krahn, we have λ1 (Ai ) ≥ λ1 (Bi ), i = 1, 2, with equality if Ai is congruent to Bi , so that λ2 (Ω) ≥ max(λ1 (B1 ), λ1 (B2 )). If we consider the disjoint union B ∗ of B1 and B2 , the spectrum of B ∗ is the union of both spectrum, and V ol(B ∗ ) = V ol(Ω), so that λ2 (B ∗ ) ≤ max(λ1 (B1 ), λ1 (B2 )). We deduce λ2 (B ∗ ) = max(λ1 (B1 ), λ1 (B2 )). In order to have the smallest possible λ2 , we have to choose two balls of the same radius, and the equality case in the Faber-Krahn inequality implies that Ω is the disjoint union of two identical balls. (4) Numerical estimates. Knowing about the existence of extremal domains from a theoretical viewpoint, one can try to investigate the shape of such domains. It is almost impossible to do this from a formal point of view, and this was investigated from a numerical point of view, in particular for domains of R2 . But this is also a difficult numerical problem, in particular due to the fact that there are a lot of local extrema. This approach was initiated in the thesis of E. Oudet. Note that it gives us a candidate to be extremal, but we have no proof that the domain appearing as extremal is the right one. It is important to apply different numerical approaches, and to see if they give the same results. See the thesis of A. Berger [Ber] for a description of results. Remark 3. A question could be whether a domain of given fixed volume could have a very large first eigenvalue λ1 : The answer is yes. For example, think of a long and thin domain, and more generally of “thin” domains, that is domains with small inradius. However, formally, a small inradius does not imply large eigenvalue λ1 . We have to take into account the connectivity of the domain, see [Cr], where the following is shown: if Ω is a bounded connected domain in R2 with inradius ρ and k boundary components, then the first eigenvalue λ1 (Ω) for the Dirichlet boundary considitions satisfies λ1 (Ω) ≥

1 4ρ2

λ1 (Ω) ≥

1 2kρ2

if k = 1, 2, and

for k ≥ 2. So, there is an intriguing relationship between k and ρ. For example, a ball with a lot of very small holes can have a first eigenvalue close to the first eigenvalue of the ball (the problem of the Crushed Ice described for example in [Ch1, p. 233]).

6

BRUNO COLBOIS

Figure 2. Examples with “large” Dirichlet eigenvalue 1.2. The Laplacian for Euclidean domains with Neumann boundary conditions. Let Ω ⊂ Rn be a bounded, connected domain with a Lipschitz boundary. The Laplacian is given by n  ∂2f Δf = − ∂x2i i=1 where f ∈ C 2 (Ω). What is investigated is the spectrum of the Laplacian Δ on Ω with the Neumann boundary condition, that is the eigenvalue problem (3)

Δf = λf

under the Neumann condition ∂f )|∂Ω = 0. ∂n where n denotes the exterior normal to the boundary. This problem has a discrete and real spectrum (4)

(

λ0 = 0 < λ1 (Ω) < λ2 (Ω) ≤ λ3 (Ω) ≤ ... → ∞, with the eigenvalues repeated according to their multiplicity. Example 4. If Ω is the interval ]0, π[⊂ R, the Laplacian is given by Δf = −f  and the spectrum, for the Neumann boundary condition, is given by λk = k2 , k = 0, 1, 2, ... The eigenfunction corresponding to λk = k2 is fk (x) = cos kx. We also refer to [Ber] for the calculation of the spectrum of the ball in dimensions 2 and 3. Some similarities between the spectrum for the Neumann boundary condition and Dirichlet boundary condition can be noticed. However, in this introduction, I will mainly focus on the differences. A first obvious difference is that zero is always an eigenvalue, corresponding to the constant eigenfunction. This means that the first interesting eigenvalue is the second, or the first nonzero eigenvalue. In Example 13, for the Dirichlet problem, we will see that if Ω1 ⊂ Ω2 , then we have for each k that λk (Ω1 ) ≥ λk (Ω2 ) (we say that we have monotonicity).

THE SPECTRUM OF THE LAPLACIAN: A GEOMETRIC APPROACH

7

On the contrary, in Example 18, we will see that we have no monotonicity for the Neumann problem. Indeed, in this case, the fact that Ω1 ⊂ Ω2 has no implication on the spectrum. We will also see that the spectrum for the Neumann conditions is very sensitive to perturbations of the boundary. At this stage, I must clarify that, because the goal of these lectures was to present a geometric approach, I have chosen bounded domains with a Lipschitz boundary. This guarantees the discreteness of the spectrum, but this hypothesis may be relaxed, in particular for the Dirichlet problem (see the lectures of D. Bucur). In Example 19, we will construct a bounded domain, with a smooth boundary up to one singular point, and a non-discrete spectrum. 1.2.1. The Szeg¨ o-Weinberger inequality. For further details, please refer to [As, p. 103]. It is similar to the Faber-Krahn inequality, but is more delicate to prove, precisely because it concerns the second eigenvalue λ1 . Theorem 5. Let Ω be an open bounded domain in Rn with Lipschitz boundary and B ⊂ Rn a ball with the same volume of Ω. If λ1 denotes the first eigenvalue for the Neumann boundary conditions, then λ1 (Ω) ≤ λ1 (B), with equality if and only if Ω is equal to B up to a displacement. Remark 6. (1) In contrast with the case of the Dirichlet boundary condition, λ1 cannot be very large for a domain of fixed volume when we take the Neumann boundary condition. (2) We will see that there are domains of fixed volume with as many eigenvalues close to 0 as we want. (3) There is no spectral stability in the Neumann problem. Indeed, we can perturb a ball by pasting a thin full cylinder of convenient length L: It does not affect the first nonzero eigenvalue but all the others. Using results in the spirit of [An1], the spectrum converges to the union of the spectrum of the ball and of the spectrum of [0, L], with the Dirichlet boundary condition on 0 and the Neumann condition on L. It suffices to choose L in such a way that the first eigenvalue of the interval lies between the two first eigenvalues (without multiplicity) of the ball. 1.3. The Laplacian on a Riemannian manifold. Let (M, g) be a Riemannian manifold. If f ∈ C 2 (M ), then let us define Δf = −div grad f, and look at the eigenvalue problem Δf = λ f. In local coordinates {xi }, the Laplacian reads  ∂  1 ∂ Δf = −  (g ij det(g) f ). ∂x ∂x det(g) i,j j i In particular, in the Euclidean case, we recover the usual expression  ∂ ∂ Δf = − f. ∂xj ∂xj j

8

BRUNO COLBOIS

We will not use very often the expression of the Laplacian in local coordinates. People who are not familiar with the Riemannian geometry could think of a Riemannian manifold as a submanifold of the Euclidean space. Most of the operations we will make consist in “deforming” the Riemannian metric g in order to construct examples or counter-examples. These deformations can be viewed as deformations of the submanifold. However, this intuitive way of thinking has its limits: Some deformations are much easier to realize abstractly as deformations of the Riemannian metric, than to be viewed directly as deformations of a submanifold. This is the case of the conformal deformation which will be considered in the second part of this lecture. When M is compact and connected, we have a discrete spectrum 0 = λ0 (M, g) < λ1 (M, g) ≤ λ2 (M, g) ≤ ... → ∞. The eigenvalue 0 corresponds to the constant eigenfunction. Example 7. Let us take for M the circle of radius 1 that we can identify with the interval [0, 2π] with periodic condition. The spectrum is given by λ0 = 0 and λ2k−1 = λ2k = k2 for k ≥ 1 corresponding to the eigenfuctions f2k−1 (x) = sin(kx), f2k (x) = cos(kx). Here as well, there are very few situations where the spectrum can be explicitly calculated. However, there are more examples in the context of Riemannian manifolds than for domains. In the book [BGM], the case of the n-dimensional round sphere, and of the flat tori in dimension 2 are studied. Note that, in dimension greater than 1, the eigenvalues are generically of multiplicity 1 [Uh]. However, in the very few examples where the spectrum can be calculated, large multiplicity often arises. This comes from the fact that the examples where we can calculate explicitly the spectrum have a lot of symmetries, and this implies very often the emergence of multiplicity. 1.3.1. Large and small eigenvalues. We can address a similar question as that asked about domains. Given a compact connected differential manifold M without boundary, is it possible to find a metric g on M with λ1 (M, g) arbitrarily large or small? First, we have to observe that some constraint is needed: for any Riemannian metric g, we have the relation 1 λk (M, g). t2 This means that, thanks to a homothety of the metric, arbitrarily large or small eigenvalues can be produced. This is also true for Euclidean domains. A possible constraint is to fix the volume, or, equivalently, to estimate λk (M, g)V ol(M, g)2/n rather than λk (M, g) , where n = dim M , because this expression is invariant by homothety. λk (M, t2 g) =

Small eigenvalues. We will soon see that it is easy to produce a metric of given volume with as many small eigenvalues as we want. This leads to the question of finding lower bounds depending on the geometry. Large eigenvalues. For large eigenvalues, the result is surprising and it leads to a lot of different types of questions.

THE SPECTRUM OF THE LAPLACIAN: A GEOMETRIC APPROACH

9

The case dimM ≥ 3. If M is a given compact differential manifold with dim M ≥ 3, then it was proved in [CD] that we can choose g of given volume with λ1 as large as we want. To prove this result: - first, prove this for the sphere of odd dimension (see for example [Bl]) using that the sphere of dimension (2n + 1) is a S 1 -bundle over CP n . - extend the result to the even dimension sphere [Mu]. - prove it for every manifold M of dimension ≥ 3 [CD], using classical, but rather difficult results from spectral theory. The idea is to deform the metric of the manifold to make it “close” to the metric of a sphere with large eigenvalue, and then to conclude using results about the behavior of the spectrum under surgery. .

The case of surfaces. For surfaces, there is an upper bound for λk which depends on the genus and on k. This will be proven in the second part of this lecture. But the case of the first nonzero eigenvalue λ1 is of special interest, and I will conclude this first section with some consideration about this question. 1.3.2. The case of the first nonzero eigenvalue for surfaces. For this paragraph, please refer to the recent survey of R. Schoen [Sc] paragraph 3, where one can find precise references and explanations, and to [CE1]. A first result was proved by Hersch for the sphere in 1970 [He]: for any Riemannian metric g on the sphere S 2 (5)

λ1 (S 2 , g)V ol(S 2 , g) ≤ 8π

with equality if and only if g is the canonical round metric. In 1980, Yang and Yau extended the result to every genus: if S is an orientable surface of genus γ and g a Riemannian metric on S γ+3 λ1 (S, g)V ol(S, g) ≤ 8πE( (6) ) 2 where E denotes the integer part. This result is good, in the following sense: There exists a universal constant C > 0 such that, for each γ, there exists a surface (S, g) of genus γ with λ1 (S, g)V ol(S, g) ≥ Cγ (and this is not easy at all to construct). However, it is not optimal, in the sense that there is no equality case. There are only a few surfaces where the maximum/extremum of λ1 (S, g)V ol(S, g) is known: - the projective space, where the maximum is 12π, reached for the metric with constant curvature; 2 √ , reached by the flat equilateral torus; - the torus, where the maximum is 8π 3 √

- the Klein bottle, where the maximum is 12πE( 2 3 2 ) with an equality case. - the surface of genus 2. In this situation, the maximum is 16π and is reached by a family of singular metrics. At some point in the proof, there is a numerical estimate. For surfaces with genus greater than 2, almost nothing is known about maximum/supremum for λ1 . 2. Variational characterization of the spectrum and simple applications In this section, I give a variational characterization of the spectrum: Roughly speaking, this allows us to investigate the spectrum of the Laplacian without looking at the equation Δu = λu itself, but rather by considering some test functions.

10

BRUNO COLBOIS

Moreover, we have only to take into account the gradient of these test functions, and not their second derivative, which is much easier. One can say that this variational characterization is one of the fundamental tools allowing us to use the geometry in order to investigate the spectrum. To illustrate this, I will explain with some detail very elementary constructions of small eigenvalues: These constructions are well known by the specialists of the topic, and, in general, they are mentioned in one sentence. Although these are used here at an elementary level, they allow us to understand principles that will be used later in a more difficult context. Please, refer to the book of P. B´erard [Be] for more details about the theory. Let (M, g) be a compact manifold with boundary ∂M . Let f, h ∈ C 2 (M ). Then we have the well-known Green’s Formula    df Δf hdvolg = df, dh dvolg − h dA dn M M ∂M df where dn denotes the derivative of f in the direction of the outward unit normal vector field n on ∂M . dvolg the volume form on (M, g) (associated to g) and dA the volume form on ∂M . df )|∂M = In particular, if one of the following conditions ∂M = ∅, h|∂M = 0 or ( dn 0 is satisfied, then we have the relation

(Δf, h) = (df, dh) = (f, Δh) where (, ) denotes the L2 -scalar product. In the sequel, we will study the following eigenvalue problems when M is compact: • Closed Problem: Δf = λf in M ; ∂M = ∅; • Dirichlet Problem Δf = λf in M ; f|∂M =0 ; • Neumann Problem: df = 0. ) dn |∂M The following standard result about the spectrum can be seen in [Be, p. 53]. Δf = λf in M ; (

Theorem 8. Let M be a compact connected manifold with boundary ∂M (which may be empty), and consider one of the above mentioned eigenvalue problems. Then: (1) The set of eigenvalue consists of an infinite sequence 0 < λ1 ≤ λ2 ≤ λ3 ≤ ... → ∞, where 0 is not an eigenvalue in the Dirichlet problem; (2) Each eigenvalue has finite multiplicity and the eigenspaces corresponding to distinct eigenvalues are L2 (M )-orthogonal; (3) The direct sum of the eigenspaces E(λi ) is dense in L2 (M ) for the L2 norm. Furthermore, each eigenfunction is C ∞ -smooth. Remark 9. The Laplace operator depends only on the given Riemannian metric. If F : (M, g) → (N, h)

THE SPECTRUM OF THE LAPLACIAN: A GEOMETRIC APPROACH

11

is an isometry, then (M, g) and (N, h) have the same spectrum, and if f is an eigenfunction on (N, h), then f ◦ F is an eigenfunction on (M, g) for the same eigenvalue. As already mentioned, the spectrum cannot be computed explicitly, with a few exceptions. In general, it is only possible to get estimates of the spectrum, and these estimations are related to the geometry of the considered manifold (M, g). However, asymptotically, we know how the spectrum behaves. This is the Weyl law. Weyl law: If (M, g) is a compact Riemannian manifold of dimension n (with or without boundary), then, for each of the above eigenvalue problems, λk (M, g) ∼

(7)

(2π)2 2/n



k V ol(M, g)

2/n

ωn as k → ∞, where ωn denotes the volume of the unit ball of Rn . This also means  2/n 1 λk (M, g) (2π)2 lim (8) = 2/n k→∞ V ol(M, g) k2/n ωn In particular, the spectrum of (M, g) determines the volume of (M, g). It is important to stress that the result is asymptotic: We do not know in general for which k the asymptotic estimate becomes good! However, this formula is a guide when trying to get upper bounds. To introduce some geometry on the study of the Laplacian, it is very relevant to use the variational characterization of the spectrum. To this aim, let us introduce the Rayleigh quotient. In order to do this, we have to consider functions in the Sobolev space H 1 (M ) or H01 (M ). It is possible to find the definition of H 1 (M ) or H01 (M ) in every book on PDE, but what we will really consider are functions which are of class C 1 by part and glue them continuously along a submanifold. Moreover, for H01 (M ), we consider only functions taking the value 0 on ∂M . If a function f lies in H 1 (M ) in the closed and Neumann problems, and in for the Dirichlet problem, the Rayleigh quotient of f is  |df |2 dvolg (df, df ) . = R(f ) = M 2 (f, f ) f dvolg M

H01 (M )

Note that in the case where f is an eigenfunction for the eigenvalue λk , then   |df |2 dvolg Δf f dvolg M  = M = λk . R(f ) =  2 f dvolg f 2 dvolg M M Theorem 10. (Variational characterization of the spectrum, [Be, pp. 60–61]) Let us consider one of the 3 eigenvalues problems. Min-Max formula: we have λk = inf sup{R(u) : u = 0, u ∈ Vk } Vk

where Vk runs through k + 1-dimensional subspaces of H 1 (M ) (k-dimensional subspaces of H01 (M ) for the Dirichlet eigenvalue problem).

12

BRUNO COLBOIS

This min-max formula is useless to calculate λk , but it is very useful to find an upper bound because of the following fact: For any given (k + 1) dimensional vector subspace V of H 1 (M ), we have λk (M, g) ≤ sup{R(u) : u = 0, u ∈ V }. This gives immediately an upper bound for λk (M, g) if it is possible to estimate the Rayleigh quotient R(u) of all the functions u ∈ Vk . Note that there is no need to calculate the Rayleigh quotient, it suffices to estimate it from above. Of course, this upper bound is useful if the vector space V is conveniently chosen. It would be a delicate problem to estimate the Rayleigh quotient of all functions u ∈ V . A special situation is when V is generated by k + 1 disjointly supported functions f1 , ..., fk+1 . In this case, (9)

sup{R(u) : u = 0, u ∈ V } = sup{R(fi ) : i = 1, ..., k + 1},

which makes the estimation easier to do, because we have only to estimate from above the Rayleigh quotient of (k + 1) functions. We will use this fact in the sequel. To see it, let f = α1 f1 + ... + αk+1 fk+1 . Then 2 (f, f ) = α12 (f1 , f1 ) + ... + αk+1 (fk+1 , fk+1 ).

and 2 (df, df ) = α12 (df1 , df1 ) + ... + αk+1 (dfk+1 , dfk+1 ).

so that R(f ) is bounded from above by the maximum of the Rayleigh quotients R(fi ), i = 1, ..., k + 1. Remark 11. We have indeed shown a little more: If the test functions {fi }k+1 i=1 are orthonormal and if the {dfi }k+1 are orthogonal, then (9) is also true. i=1 Remark 12. We can see already two advantages to this variational characterization of the spectrum. First, there is no need to work with solutions of the Laplace equation, but only with ”test functions”, which is easier. Then, only one derivative of the test function has to be controlled, and not two, as in the case of the Laplace equation. To see this concretely, let us give with some details a couple of simple examples. Example 13. Monotonicity in the Dirichlet problem. Let Ω1 ⊂ Ω2 ⊂ (M, g), two domains of the same dimension n of a Riemannian manifold (M, g). Let us suppose that Ω1 and Ω2 are both compact connected manifolds with a smooth boundary. If we consider the Dirichlet eigenvalue problem for Ω1 and Ω2 with the induced metric, then for each k λk (Ω2 ) ≤ λk (Ω1 ). The proof is very simple: Each eigenfunction of Ω1 may be continuously extended by 0 on Ω2 and may be used as a test function for the Dirichlet problem on Ω2 . Let us construct an upper bound for λk (Ω2 ): for Vk , we choose the vector subspace of H01 (Ω2 ) generated by an orthonormal basis f1 , ..., fk of eigenfunctions of Ω1 extended by 0 on Ω2 . Clearly, these functions vanish on ∂Ω2 and they are c C ∞ on Ω1 and Ω2 ∩ Ω¯1 . They are continuous on ∂Ω1 .

THE SPECTRUM OF THE LAPLACIAN: A GEOMETRIC APPROACH

13

Let f = α1 f1 + ... + αk fk . We have (f, f ) = α12 + ... + αk2 . We have



 dfi , dfj dvolg =

(dfi , dfj ) = Ω2

 =

dfi , dfj dvolg  Δfi , fj dvolg = λi fi fj dvolg , Ω1

Ω1

Ω1

and it follows that (dfi , dfi ) = λi (Ω1 )(fi , fi ) and (dfi , dfj ) = 0 if i = j. We have (df, df ) = α12 λ1 (Ω1 ) + ... + αk2 λk (Ω1 ). We conclude that R(f ) ≤ λk (Ω1 ), and we have λk (Ω2 ) ≤ λk (Ω1 ). As this is true for each k, we have the result. Example 14. As a consequence of (9), if M is a compact Riemannian manifold without boundary, and if Ω1 ,...,Ωk+1 are domains in M with disjoint interiors, then λk (M, g) ≤ max(μ1 (Ω1 ), ..., μ1 (Ωk+1 )), where μ1 (Ω) denotes the first eigenvalue of Ω for the Dirichlet problem. Let us choose (k +1) as test functions the eigenfunctions u1 , ...uk+1 corresponding to μ1 (Ω1 ), ..., μ1 (Ωk+1 ) respectively, extended by 0 on M . They have μ1 (Ω1 ), ..., μ1 (Ωk+1 ) as Rayleigh quotients, and this allows us to show the inequality. Example 15. Let us see how to deform a Riemannian manifold (keeping the volume fixed), in order to produce as many small eigenvalues as we want. We first consider the cylinder CL = [0, L] × S n−1 , where the first Dirichlet eigenvalue satisfies π2 λ1 (CL ) ≤ 2 L To verify that, it suffices to consider the function πr f (r, x) = f (r) = sin L which is an eigenfunction for

π2 L2 .

Then, on each manifold M , we can construct a family of Riemannian metrics of fixed volume, with as many small eigenvalues as we want. We just deform the metric without changing the topology as follows: we deform locally a small ball into a long and thin “nose”, that is a cylinder [0, L] × Sn−1 closed by a cap. We L , and, can divide this cylindrical part into (k + 1) disjoint cylinders of length k+1 considering the Dirichlet problem on each of these cylinders, we get π 2 (k + 1)2 L2 which is arbitrarily small for k fixed and L → ∞. λk ≤

In the sequel, we will explain how to produce arbitrarily small eigenvalues for a Riemannian manifold with fixed volume and bounded diameter (Cheeger’s dumbbell construction and its applications). But let us first apply the min-max formula to an elementary situation of surgery.

14

BRUNO COLBOIS

Example 16. Elementary surgery. We will explain in detail the already mentioned fact: A small hole in a domain does not affect the spectrum “too much” (see ( 10) below). Let Ω ⊂ Rn a domain with smooth boundary. Let x ∈ Ω, and B(x, ) the ball of radius centered at the point x ( is chosen small enough such that B(x, ) ⊂ Ω). We denote by Ω the subset Ω − B(x, ), that is we make a hole of radius on Ω. We consider the two domains Ω and Ω with the Dirichlet boundary conditions, ∞ and denote by {λk }∞ k=1 and by {λk ( )}k=1 their respective spectrums. Then, for each k, we have (10)

lim λk ( ) = λk

→0

Before showing this in detail, let us make a few remarks. Remark 17. (1) The convergence is not uniform in k. (2) This result is a very specific part of a much more general facts. We get this type of results on manifolds, with subset much more general than balls, for example tubular neighborhood of submanifolds of codimension greater than 1. For the interested reader, we refer to the paper of Courtois [Cou] and to the book of Chavel [Ch1]. (3) We can get very precise asymptotic estimates of λk ( ) in terms of . Again, we refer to [Cou] and references therein. (4) We have also the convergence of the eigenspace associated to λk ( ) to the eigenspace associated to λk , but this has to be defined precisely: a problem occurs if the multiplicity of λk is not equal to the multiplicity of λk ( ), see [Cou]. In order to see the convergence, we first observe that by monotonicity, for all k λk ≤ λk ( ). Our goal is to show that λk ( ) ≤ λk + c( ), with c( ) → 0 as → 0. In order to use the min-max construction, we need to consider a family of testfunctions. This family is constructed using a perturbation of the eigenfunctions of Ω. Note that we do the proof for n ≥ 3, but the strategy is the same if n = 2. ∞ Let {fi }∞ i=1 be an orthonormal basis of eigenfuctions on Ω. We fix a C -plateau function χ : R → R defined by ⎧ if r ≤ 32 ⎨ 0 1 if r ≥ 2 χ(r) = ⎩ 0 ≤ χ(r) ≤ 1 if 32 ≤ r ≤ 2

The test function fi, associated to fi is defined by d(p, x) )fi . By construction, the function fi, (p) takes the value 0 on the ball B(x, r) and satisfies the Dirichlet boundary condition. So, it can be used as a test function in order to estimate λk, . fi, (p) = χ(

THE SPECTRUM OF THE LAPLACIAN: A GEOMETRIC APPROACH

15

Let f be a function on the vector space generated by {fi, }ki=1 . In order to estimate its Rayleigh quotient, it makes a great simplification if the basis is orthonormal and the (dfi ) are orthogonal. This is not the case here. However, the basis is almost orthonormal and the (dfi ) are almost orthogonal: this is what we will first show and use. Note that this is a very common situation. The basis {fi, }ki=1 is almost orthonormal. Let us denote by ai (resp. bi ) the maximum of the function |fi | (resp. |dfi |) on Ω. Then   2 2 n n (11) fi ≤ ai 2 ωn , |dfi |2 ≤ b2i 2n ωn n B(x,2)

B(x,2)

where ωn is the volume of the unit ball in Rn . Let us show that δij − Ck n/2 ≤ fi, , fj, ≤ δij + Ck n/2 , where Ck depends on a1 , ..., ak . fi, , fj, = χ fi , χ fj = (χ − 1)fi + fi , (χ − 1)fj + fj = = fi , fj + (χ − 1)fi , fj + (χ − 1)fj , fi + (χ − 1)fi , (χ − 1)fj . By (11), (χ − 1)fi 2 ≤ a2i 2n ωn n , and by Cauchy-Schwarz δij − Ck n/2 ≤ fi, , fj, ≤ δij + Ck n/2 , where Ck depends on a1 , .., ak . The set {dfi, }ki=1 is almost orthogonal. Let us obverse that dfi, = dχ fi + χ dfi , and dfi, , dfj, = dχ fi , dχ fj + dχ fi , χ dfj + χ dfi , dχ fj + χ dfi , χ dfj . This implies dχ fi 2 ≤ C

1 2

 fi2 ≤ Ca2i 2n ωn n−2 B(x,2)

where C depends only on the derivative χ of χ. This is the place where n ≥ 3 is used in order to have a positive exponent to . With the same considerations as before | dfi, , dfj, | ≤ dfi , dfj + Ci n/2 = λi δij + Ck n/2 . where Ck depends on a1 , ..., ak , b1 , ..., bk , C. Now, we can estimate the Rayleigh quotient of a function f ∈ [f1, , ..., fk, ]. This is done exactly in the same spirit as the proof of the monotonicity, but we have to deal with the fact that the basis of test functions is only almost-orthonormal. Let f = α1 f1, + ... + αk fk, . Thanks to the above considerations, we have f 2 = α12 + ... + αk2 + O( n/2 ), df 2 = α12 λ1 + ... + αk2 λk + O( This implies R(f ) ≤ λk + O( and gives the result.

n−2 2

)

n−2 2

).

16

BRUNO COLBOIS

The next example explains how to produce arbitrarily small eigenvalues for Riemannian manifold with fixed volume and bounded diameter. The Cheeger dumbbell construction for Riemannian manifolds. The idea is to consider two n-spheres of fixed volume A connected by a small cylinder C of length 2L and radius . We denote by M this manifold. The first nonzero eigenvalue of M converges to 0 as goes to 0. It is even possible to estimate very precisely the asymptotic of λ1 in term of (see [An2]), but here, let us just shows that it converges to 0.

Figure 3. Cheeger Dumbell Let f be a function with value 1 on the first sphere, −1 on the second and decreasing linearly along the cylinder.

f=1

f=1 f=x/L

Cylinder Lenth 2L

Sphere Volume V

Sphere Volume V

The maximum norm of its gradient is L1 . By construction (and  because we can suppose for simplicity that the manifold is symmetric), we have M f dvolg = 0. Let V be the vector space generated by f and by the constant function 1. If h ∈ V , we can write h = a + bf , a, b ∈ R and   h2 dvolg = a2 V ol(M ) + b2 f 2 dvolg , M



 |dh|2 dvolg = b2

M

M

|df |2 dvolg . M

So, as λ1 = inf {R(h) : h ∈ V }, we get λ1 (M ) ≤ R(f ). The function f varies only on the cylinder C and its gradient has norm implies  1 |df |2 dvolg = 2 V ol(C). L M

1 L.

This

THE SPECTRUM OF THE LAPLACIAN: A GEOMETRIC APPROACH

17

Moreover, because f 2 takes the value 1 on both spheres of volume A, we have  f 2 dvolg ≥ 2A. M

This implies that the Rayleigh quotient of f is bounded above by V olC/L2 2A which goes to 0 as does. A similar construction with k spheres connected by thin cylinders shows that there exist examples with k arbitrarily small eigenvalues.

Figure 4. Examples with 3 “small” nonzero eigenvalues Observe that we can easily fix the volume and the diameter in all these constructions: Thus to fix the volume and diameter is not enough to have a lower bound on the spectrum. Cheeger dumbbell for the Neumann problem. The same construction as above may be done in the case of the Neumann problem for a domain in Rn . A Cheeger dumbbell is simply given by two disjointed balls related by a thin full cylinder. This gives two interesting examples. The first shows that we have no monotonicity for the Neumann problem. Example 18. Let Ω ⊂ Rn a domain. There exists a domain Ω1 ⊂ Ω with a smooth boundary such that the k first eigenvalues of Ω1 are arbitrarily small. To do this, choose Ω1 to be a like a Cheeger dumbbell, with k + 1 balls related by very thin cylinders. We see immediately that the k first eigenvalues can be made as small as we wish, with the same calculations as above. Note that we have no monotonicity for the Neumann problem even when we restrict ourself to convex domains. For example, in dimension 2, we can consider a square [0, π] × [0, π]. The first nonzero eigenvalue is 1. But we can construct inside this √ square a thin rectangle around 1the diagonal of the square of length almost 2π: the first eigenvalue is close to 2 . Note, however, that we cannot find convex subsets of [0, π] × [0, π] with arbitrarily small first nonzero eigenvalue. By a result of Payne and Weinberger [PW], we have for Ω ⊂ Rn , convex, the inequality λ1 (Ω) ≥

π2 . diam(Ω)2

Example 19. This example shows that a small, local perturbation of a domain may strongly affect the spectrum of the Laplacian for the Neumann conditions. The text gives only qualitative arguements, the calculations are left to the reader if he needs them to understand the examples.

18

BRUNO COLBOIS

We begin with a domain Ω (a ball in the picture) and show that a very small perturbation of the boundary may change drastically the spectrum for the Neumann boundary condition: A nonzero eigenvalue arbitrarily close to zero may appear as a consequence of a very small perturbation.

Figure 5. We want to perturb locally a domain Ω We add a small ball close to Ω (but at a positive distance) and link it to Ω using a thin cylinder. The important point is not the size of the ball or of the cylinder, but the ratio. As on the picture, the radius of the cylinder must be much smaller than the radius of the ball. This creates a Cheeger dumbbell: the only difference with the above calculations is that the two “thick” parts do not have the same size, but it is easy to adapt the calculations and show that if the radius of the cylinder goes to zero, then the same is true for the first nonzero eigenvalue. The ball can be very small and very close to Ω, so the perturbation may be confined in an arbitrarily small region.

Figure 6. Add a small ball and link it to Ω with a thin cylinder We can iterate this first deformation, constructing a new deformation on the first, a new one on the second, etc.

Figure 7. Add a small ball and link it to Ω with a thin cylinder This can be understood as a family of balls Bk of radius rk = 212k , with center xk . All the centers are on the same line, and the distance between xk and xk+1 is 1 . 2k

THE SPECTRUM OF THE LAPLACIAN: A GEOMETRIC APPROACH

19

The balls Bk and Bk+1 are joined using a cylinder of radius 21k . This radius 2 is very thin in comparison with the radius of the ball. The result of this construction is a domain with an infinite number of Cheeger dumbbells. The domain is not compact, but taking its adherence add only one point, which is of course a singular point. In conclusion, we get a domain with a smooth boundary up to one singular point, and infinitely many small eigenvalues: This implies that the spectrum is not discrete and shows that we must be careful with the smoothness of the boundary. Conclusion: More generally, by a Cheeger dumbbell, we understand a manifold where two (or more)“thick parts” are separated by one (or more) “thin parts”. In such situations, the presence of small eigenvalue(s) may be suspected.

Figure 8. “Generalized” Cheeger dumbbell A very general question is to understand under which conditions we can get lower bounds for the spectrum, and in particular for the first nonzero eigenvalue λ1 . 3. Lower bounds for the first nonzero eigenvalue As mentioned in the preamble, the second part of this lecture will be concerned with the subject upper bounds for the spectrum. One of the participants to the school asked me if it is more important or difficult to find upper bounds than lower bounds. The answer is no. The question about lower bounds is a world by itself, and the goal of this short section is to present the tip of the iceberg: Two different types of results are briefly explained. In some sense, there are two ways to bound from below the first nonzero eigenvalue of a compact Riemannian manifold. After understanding which deformations

20

BRUNO COLBOIS

of the Riemannian metric can produce small eigenvalues, we have the choice between two strategies. - A comparison with a geometric constant taking account of such local or global deformations of the Riemannian metric or - To impose geometric constraints in order to avoid such local or global deformations of the Riemannian metric. It has already been explained which deformations of the Riemannian metric can produce small eigenvalues:

Figure 9. A local deformation of the Riemannian metric can produce small eigenvalues

Figure 10. “Small mushroom” or a local Cheeger dumbbell

Figure 11. Presence of small eigenvalues coming from a global deformation of the metric It can be local deformations, typically a deformation of the metric around a point producing a dumbbell (one says also sometimes that we construct a small mushroom on the manifold). It can be a global deformation, adding a long nose to the manifold. Of course, it is not clear that these types of deformations are the only ones producing small eigenvalues, and the difficulty is to find constraints which allow us to control the spectrum from below.

THE SPECTRUM OF THE LAPLACIAN: A GEOMETRIC APPROACH

21

3.1. A first method: Cheeger’s inequality. The Cheeger inequality is in some sense the counter-part of the dumbbell example. It is presented here in the case of a compact Riemannian manifold without boundary, but it may be generalized to compact manifolds with boundary (for both Neumann or Dirichlet boundary conditions) or to non-compact, complete, Riemannian manifolds. Definition 20. Let (M, g) be an n-dimensional compact Riemannian manifold without boundary. The Cheeger’s isoperimetric constant h = h(M ) is h(M ) = inf {J(C); J(C) = C

V oln−1 C }, min(V oln M1 , V oln M2 )

where C runs through all compact codimension one submanifolds which divide M into two disjoint connected open submanifolds M1 , M2 with common boundary C = ∂M1 = ∂M2 . Theorem 21. ( Cheeger’s inequality, 1978). We have the inequality λ1 (M, g) ≥

h2 (M, g) . 4

A proof may be found in Chavel’s book [Ch1] and developments and other statement in Buser’s paper [Bu2]. In particular, Buser shows that Cheeger’s inequality is sharp ([Bu2, Thm. 1.19]). This inequality is remarkable, because it relates an analytic quantity (λ1 ) to a geometric quantity (h) without any other assumption on the geometry of the manifold. Note that the Cheeger constant becomes small in the presence of a Cheeger dumbbell or of a long cylinder. The Cheeger constant takes account of the deformations producing small eigenvalues. Note that it is not always easy to estimate the Cheeger constant h of a Riemannian manifold. Sometimes, it is easier to estimate the first nonzero eigenvalue λ1 in order to get an estimate of h. Under some geometrical assumptions there is an upper bound of λ1 in terms of the Cheeger constant (see [Bu2]). Theorem 22. (Buser, 1982) Let (M n , g) be a compact Riemannian manifold with Ricci curvature bounded below Ric(M, g) ≥ −δ 2 (n − 1), δ ≥ 0. Then we have λ1 (M, g) ≤ C(δh + h2 ), where C is a constant depending only on the dimension and h is the Cheeger constant. The condition about the Ricci curvature is necessary. In [Bu2], Buser constructs a surface with an arbitrarily small Cheeger constant, but with λ1 uniformly bounded from below. It is easy to generalize it to any dimension. This example is very interesting, and we will describe the original construction of Buser in detail. Example 23. Let S 1 × S 1 be a torus with its product metric g and coordinates (x, y), −π ≤ x, y ≤ π and a conformal metric g = χ2 g. The function χ is an even function depending only on x, takes the value at 0, π, 1 outside an -neighborhood of 0 and π. It follows immediately that the Cheeger constant h(g ) → 0 as → 0 and it remains to show that λ1 (g ) is uniformly bounded from below.

22

BRUNO COLBOIS

Note that the example seems to be very close to the Cheeger dumbbell. The difference is that in the case of the dumbbell, the length of the cylinder joining the two “big” parts may be small but is fixed. Here, the length of the cylinder goes to 0 as → 0. Let f be an eigenfunction for λ1 (g ). We have  |df |2 dvolg λ1 (g ) = R(f ) = T 2  . f dvolg T Let S1 = {p : f (p) ≥ 0} and S2 = {p : f (p) ≤ 0} and let  f (p) if p ∈ S1 F (p) = af (p) if p ∈ S2  where a is chosen such that T F dvolg = 0. Let V be the vector space generated by the constant function 1 and the function F of integral 0. The situation is exactly the same as for the calculation around the Cheeger dumbbell and it follows λ1 (g) ≤ Rg (F ). So, it is enough to show that Rg (F ) ≤ λ1 (g ) :   |df |2g dvolg + a2 S2 |df |2g dvolg S1  Rg (F ) =  . f 2 dvolg + a2 S2 f 2 dvolg S1 By the conformal invariance in dimension 2 (see Prop. 29),   |df |2g dVg dvolg = |df |2g dvolg , Si

and by construction

Si



 f 2 dvolg ≤

Si

f 2 dvolg . Si

This implies λ1 (g ) = Rg (f ) ≥ Rg (F ) ≥ λ1 (g). The geometrical idea behind this construction is the following: the geometry changes very quickly. This allows to get a small Cheeger constant, but the first eigenfunction has not enough place to vary, and the first eigenvalue remains essentially unchanged. 3.2. A second method: control of the curvature and of the diameter. An upper bound on the diameter is clearly a necessary condition to bound the spectrum from below. A lower bound on the (Ricci) curvature allows avoiding local deformation such as “mushrooms”. More precisely, the Ricci curvature allows us to control the growth of the volume of the balls: This is the Theorem of Bishop-Gromov that will explained in section 4, in particular Theorem 34.

THE SPECTRUM OF THE LAPLACIAN: A GEOMETRIC APPROACH

23

Here is a lower bound obtained by Li and Yau in 1979: Theorem 24 (See [LY]). . Let (M, g) be a compact n-dimensional Riemannian manifold without boundary. Suppose that the Ricci curvature satisfies Ric(M, g) ≥ (n − 1)K and that d denotes the diameter of (M, g). Then, if K < 0, λ1 (M, g) ≥

exp − (1 + (1 − 4(n − 1)2 d2 K)1/2 ) , 2(n − 1)2 d2

and if K = 0, then π2 . 4d2 This first inequality is difficult to read. It means essentially that, for large d, λ1 (M, g) ≥

√

e−(2(n−1)d −K) λ1 (M, g) ≥ C(n) . d2 This type of results was generalized in different directions, see for example [BBG]. Note that a control of the diameter and of the curvature also allows us to get upper bounds for the spectrum. Theorem 25 (Cheng Comparison Theorem, 1975, [Che]). Let (M n , g) be a compact n-dimensional Riemannian manifold without boundary. Suppose that the Ricci curvature satisfies Ric(M, g) ≥ (n − 1)K and that d denotes the diameter of (M, g). Then C(n)k (n − 1)2 K 2 + λk (M, g) ≤ 4 d2 where C(n) is a constant depending only on the dimension. 4. Estimates on the conformal class This section is the more technical part of the text. I will present with some details the proof given by Korevaar of the existence of upper bounds for all the eigenvalues in a given conformal class of Riemannian metric. This is based on a text by Grigor’yan, Netrusov and Yau [GNY] that I find particularly enlightening. 4.1. Introduction. Let us begin by recalling the following result from [CD] already mentioned in the first part: Theorem 26. Let M be any compact manifold of dimension n ≥ 3 and λ > 0. Then there exists a Riemannian metric g on M with V ol(M, g) = 1 and λ1 (M, g) ≥ λ. However, if we stay on the conformal class of a given Riemannian metric g0 , then, we get upper bounds for the spectrum on volume one metrics, and it is the goal of this section to explain this. Definition 27. If g0 is a Riemannian metric on M , a metric g is said to be conformal to the metric g0 if there is a differentiable function f on M , f > 0, such that g(p) = f 2 (p)g0 (p) for all p ∈ M . We denote by [g0 ] the conformal class of g0 , that is the family of all Riemannian metrics g conformal to g0

24

BRUNO COLBOIS

Remark 28. Geometrically, if two metrics are conformal, the angle between two tangent vectors is preserved. However, it is difficult to see “intuitively” that two Riemannian metrics (for example two submanifolds) are conformal. For the proof, only one specific property is needed: Proposition 29. Let M be a manifold of dimension n and g a Riemannian metric. Let h be a differentiable function on M . Then, the expression |∇g h(p)|ng dvolg is a conformal invariant, where dvolg denotes the volume form. This means that, if g(p) = f 2 (p)g0 (p), then, for each p ∈ M , we have |∇g h(p)|ng dvolg = |∇g0 h(p)|ng0 dvolg0 Proof. This is a purely linear algebra arguement. If dimM = n and if g(p) = f 2 (p)g0 (p), then dvolg (p) = f n (p)dvolg0 ; If h is a function, |∇g h(p)|2g = f 21(p) |∇g0 h(p)|2g0 . This implies that |∇g h(p)|ng dvolg = |∇g0 h(p)|ng0 dvolg0 , and the expression |∇g h(p)|ng dvolg is a conformal invariant. Remark 30. (1) If a metric g0 is conformally deformed, even keeping the volume equal to 1, we have neither control of the diameter nor of the curvature. (2) It is easy to produce arbitrarily small eigenvalues on a conformal class keeping the volume fixed: the Cheeger dumbbell type construction may be done via a conformal deformation of the metric around a point (see [CE1]). (3) For a more complete story of the question about ”upper bounds on the conformal class”, one can read the introduction of [CE1]. Our goal is to prove the following theorem from Korevaar [Ko]: Theorem 31. Let (M n , g0 ) be a compact Riemannian manifold. Then, there exists a constant C(g0 ) depending on g0 such that for any Riemannian metric g ∈ [g0 ] λk (M, g)V ol(M, g)2/n ≤ C(g0 )k2/n . Moreover, if the Ricci curvature of g0 is non-negative, we can replace the constant C(g0 ) by a constant depending only on the dimension n. In the special case of surfaces, the bound depends only on the topology. Theorem 32. Let S be an oriented surface of genus γ. Then, there exists a universal constant C such that for any Riemannian metric g on S λk (S, g)V ol(S) ≤ C(γ + 1)k. The approach of Korevaar for the proof of this theorem is very original and not easy to understand. Around 10 years after the publication of [Ko], Grigor’yan, Netrusov and Yau wrote a paper [GNY] where they explained in particular the main ideas of [Ko] from a more general and more conceptual viewpoint. This paper was used recently in a lot of different situations, in particular in order to find upper

THE SPECTRUM OF THE LAPLACIAN: A GEOMETRIC APPROACH

25

bounds for the spectrum of the Steklov problem [CEG1]. In this lecture, I will explain how to use one of the main results of [GNY]. Remark 33. (1) Recall that λk (M, g)V ol(M, g)2/n is invariant through homothety of the metric, and this control is equivalent to fixing the volume. (2) The estimate is compatible with the Weyl law: λk (M, g)V ol(M, g)2/n ≤ C(g0 ). k2/n But this leads to a natural question: Can we get a similar estimate depending asymptotically only on the dimension and on k, but not on g0 or on the genus? This is part of the Ph.D. thesis of A. Hassannezhad (see Theorem 41 and Corollary 43 in this paper). (3) These estimates are not sharp in general. (4) These results were already known for k = 1, with different kinds of proofs and different authors (see for example the introduction of [CE1]). However, in order to make a proof for all k, Korevaar used a completely new approach.

λk (M, g)V ol(M, g)2/n ≤ C(g0 )k2/n implies

4.2. The Bishop-Gromov estimate. Before going into the description of the proof, let us recall a classical result of Riemannian geometry which appears very often in such situation: the theorem of Bishop-Gromov. The Bishop-Gromov inequality allows us to control the growth of balls. Let us begin with 3 simple examples: The volume of a ball of radius r in Rn is cn r n and we have V ol(B(2r)) = 2n . V ol(B(r)) The volume of a ball of radius r in the hyperbolic space Hn is cn e(n−1)r asymptotically as r → ∞, and we have V ol(B(2r)) ∼ e(n−1)r , V ol(B(r)) and the ratio grows to ∞ as r → ∞. In the hyperbolic space with curvature −a2 , the volume of a ball of radius r is a(n−1)r asymptotically as a → ∞ and we have cn e V ol(B(2r)) ∼ ea(n−1)r . V ol(B(r)) The ratio growths to ∞ for a fixed radius as a → ∞. More generally, on a given manifold (M, g), what can be said about the ratio How is it possible to control it, even for manifold with “non-simple” topology? The result of Bishop-Gromov says that the situations we have described in the hyperbolic spaces are the worst case. V ol(B(2r)) V ol(B(r)) ?

Theorem 34 (Bishop-Gromov, see [Sa] for a proof). Let (M, g) be a complete Riemannian manifold (without boundary): if Ricci(M, g) ≥ −(n − 1)a2 g, with a ≥ 0, then for x ∈ M and 0 < r < R, V olB a (x, R) V olB(x, R) ≤ V olB(x, r) V olB a (x, r)

26

BRUNO COLBOIS

B(r)

B(2r)

where B a denotes the ball on the model space of constant curvature −a2 . i ,2r) If a = 0, that is if Ricci(M, g) ≥ 0, the ratio VVolB(x olB(xi ,r) is controlled by a similar ratio as in the Euclidean space, and this depends only on the dimension! i ,2r) If a > 0, the control of the ratio VVolB(x olB(xi ,r) is exponential in r. However, if M is compact, this ratio is controlled in function of the diameter of (M, g) and of a.

Another way to read this inequality is that V ol(B(x, 2R)) ≤

V olB a (x, 2R) V ol(B(x, R)). V olB a (x, R)

This means that the volume of a ball of radius 2R is controlled by the volume of a ball of radius R. If the manifold is compact, this control depends only on a and on the diameter. In particular, if (M, g) is compact, it is easy to see, using the Bishop-Gromov inequality, that the number of disjoint balls of radius r/2 contained in a ball of radius r is controlled by the lower bound a of the Ricci curvature and the diameter of (M, g) (see [Zu, Lemma 3.6, p. 230]). Definition 35. This number (maximal number of disjoint balls of radius R contained in a ball of radius 2R) is called the packing constant. Strategy of the proof of Korevaar’s theorem. During the proof, the way to get upper bounds is to construct test functions, and, as said at point (9) of Theorem 10, it is nice to have disjointly supported functions. These test functions are associated to a convenient family of disjoint domains. We will see that the construction of Korevaar ([Ko], [GNY]) gives us a family of disjoint annuli. Let us first see how we can associate a test function to an annulus. 4.3. Test functions associated to an annulus. This is word-for-word the method of [GNY] as it is described in section 2 of [CES, Lemma 2.1]. Let us fix a reference metric g0 ∈ [g] and denote by d0 the distance associated to g0 . An annulus A ⊂ M is a subset of M of the form {x ∈ M : r < d0 (x, a) < R} where a ∈ M and 0 ≤ r < R (if necessary, we will denote it A(a, r, R)). The annulus 2A is by definition the annulus {x ∈ M : r/2 < d0 (x, a) < 2R}. To such an annulus we associate the function uA supported in 2A and such that ⎧ if r2 ≤ d0 (x, a) ≤ r ⎨ 1 − 2r d0 (x, A) 1 if x ∈ A uA (x) = ⎩ 1 − R1 d0 (x, A) if R ≤ d0 (x, a) ≤ 2R

THE SPECTRUM OF THE LAPLACIAN: A GEOMETRIC APPROACH

27

U=0 2A

A

U=1

U=0

Figure 12. Test function associated to an annulus

Remark 36. On a Riemannian manifold (M, g), we use frequently the distance d associated to g in order to construct new functions. It could be, like here, the distance to a subset A of M . Note that, even if the boundary of A is not smooth, the function d(∂A, ·) ”distance to the boundary of A” is well known to be 1-Lipschitz on M . According to Rademacher’s theorem (see Section 3.1.2, page 81–84 in [EG]), d(∂A, ·) is differentiable almost everywhere (since dvolg is absolutely continuous with respect to Lebesgue’s measure), and its g-gradient satisfies |∇d(∂A, ·)|g ≤ 1, almost everywhere. We introduce the following constant: Γ(g0 ) =

Vg0 (B(x, r)) rn x∈M,r>0 sup

where B(x, r) stands for the ball of radius r centered at x in (M, d0 ). Notice that since M is compact, the constant Γ(g0 ) is finite and depends only on g0 . This constant can be bounded from above in terms of a lower bound of the Ricci curvature Ricg0 and an upper bound of the diameter diam(M, g0 ) (Bishop-Gromov inequality). In particular, if the Ricci curvature of g0 is non-negative, then Γ(g0 ) is bounded above by a constant depending only on the dimension n. Lemma 37. For every annulus A ⊂ (M, d0 ) one has 

 |∇

g

uA |2g dvolg

≤ 8 Γ(g0 )

M

1− n2

2 n

dvolg

2

2

= 8 Γ(g0 ) n V olg (2A)1− n .

2A

Proof. Let A = A(a, r, R) be an annulus of (M, d0 ). Since uA is supported in 2A, we get, using H¨ older inequality, |∇

g

M





 uA |2g dvolg

|∇

g

= 2A

uA |2g dvolg

≤

|∇

g

2A

uA |ng dvolg

 n2 

1− n2 dvolg

2A

.

28

BRUNO COLBOIS

 From the conformal invariance of 2A |∇g uA |ng dvolg we have   g n |∇ uA |g dvolg = |∇g0 uA |ng0 dvolg0 2A

2A

with a.e.

|∇ uA | = g0

⎧ ⎨ ⎩

2 r

0 1 R

if r2 ≤ d0 (x, a) ≤ r if r ≤ d0 (x, a) ≤ R if R ≤ d0 (x, a) ≤ 2R.

Hence,  n  n  2 1 g0 n |∇ uA |g0 dvolg0 ≤ Vg0 (B(a, r)) + Vg0 (B(a, 2R)) ≤ 2n+1 Γ(g0 ) r R 2A where the last inequality follows from the definition of Γ(g0 ). Putting together all the previous inequalities, we obtain the result of the Lemma.  In order to prove Theorem 31 and 32, we need to consider the Rayleigh quotient of the test-functions. It is given by  |∇g uA |2g dvolg R(uA ) = M , u2 dvolg M A and we have 2

R(uA ) ≤

2

8 Γ(g0 ) n V olg (2A)1− n  . u2 dvolg 2A A 2

1− n from above So, in order  to2conclude, we need to control the term V ol(2A) u dvol from below. The first point is easy, but for the second, we and the term 2A A g need to know about the existence of annuli of rather large measure, which is difficult, because we do not have a good geometrical understanding of the geometry under conformal deformations. This is precisely the essential contribution of Korevaar, revisited by Grigor’yan, Netrusov and Yau, to do this.

4.4. The construction of Korevaar and Grigor’yan-Netrusov-Yau, [Ko], [GNY]. The construction is metric and it is explained in the context of metric measured spaces. Definition 38. Let (X, d) be a metric space. A(a, r, R), (with a ∈ X and 0 ≤ r < R) is the set

The annulus, denoted by

A(a, r, R) = {x ∈ X : r ≤ d(x, a) ≤ R}. Moreover, if λ ≥ 1, we will denote by λA the annulus A(a, λr , λR). Let (X, d) be a metric space with a finite measure ν. We make the following hypothesis about this space: (1) The balls are precompact (the closed balls are compact); (2) The measure ν is non-atomic (Recall that an atom is a measurable set which has positive measure and contains no set of smaller but positive measure. A measure which has no atoms is called non-atomic; in particular, the points have measure 0);

THE SPECTRUM OF THE LAPLACIAN: A GEOMETRIC APPROACH

29

(3) There exists N > 0 such that, for each r, a ball of radius r may be covered by at most N balls of radius r/2. The last hypothesis plays, in some sense, the role of a control of the curvature, but, as we will see, it is much weaker. Note that it is purely metric, and has nothing to do with the measure, even if, in order to show its validity, we use the Bishop-Gromov estimate about the volume growth of balls. If these hypotheses are satisfied, we have the following result Theorem 39. For each positive integer k, there exists a family of annuli {Ai }ki=1 such that (1) We have ν(Ai ) ≥ C(N ) ν(X) k , where C(N ) is an explicit constant depending only on N ; (2) The annuli 2Ai are disjoint from each other. 4.5. Application. Proof of Theorem 31. The metric space X will be the manifold M with the Riemannian distance associated to g0 and the measure ν will be the measure associated to the volume form dvolg . Note that this is one of the crucial ideas of the proof : we endowe M with the distance associated to the model Riemannian metric g0 , but with a measure associated to the Riemannian metric g to be studied. As M is compact, the theorem of Bishop-Gromov gives us a constant C(N ) = C(g0 ) > 0, such that, for each r > 0 and x ∈ M , V olg0 B(x, r) ≤ C(g0 ); V olg0 B(x, r/2) The constant C(g0 ) depends on the lower bound of Ricci(g0 ) and of the diameter of (M, g0 ), and only on the dimension if the Ricci curvature is non-negative. Remark 40. In the sequel, different constants depending on the lower bound of Ricci(g0 ) and of the diameter of (M, g0 ) will appear: they are denoted by C(g0 ). It would be an interesting exercise to make these constants really explicit. The goal is to find an upper bound for λk (g). To this aim, let us take a family of V ol (M ) disjoint 2k +2 annuli given by Theorem 39 and satisfying V olg (Ai ) ≥ C(g0 ) gk (note that the same constant C(g0 ) as in the previous step could be used without lost of generality). Using for each of the annuli A the test function uA defined above, we recall that 2 2 2 2 8 Γ(g0 ) n V ol(2A)1− n Γ(g0 ) n V olg (2A)1− n  R(uA ) ≤ , R(u ) ≤ A V olg (A) u2 dv 2A A g Moreover, by Theorem 39, V olg (M ) . k As we have 2k + 2 disjoint annuli, at least k + 1 of them have a measure less than V olg (M ) . So, k   1− n2 2/n 2 k V olg (M ) k n = C(g0 ) R(uA ) ≤ Γ(g0 ) , k C(g0 )V olg (M ) V olg (M ) V olg (A) ≥ C(g0 )

30

BRUNO COLBOIS

and this allows us to prove Theorem (31) Moreover, if the Ricci curvature of g0 is bounded below by 0, Γ(g0 ) depends only on the dimension n of M and it is the same for C(g0 ). Proof of Theorem 32. Let (S, g) be a compact orientable surface of genus γ. It is known as a corollary of the Riemann-Roch theorem (see [YY]) that (S, g) admits a conformal branched cover ψ over (S2 , g0 ) with degree deg(ψ) ≤ γ + 1. Let us endow S2 with the usual spherical distance d0 associated to the canonical metric g0 , and with a measure associated to g, the push-forward measure ν = ψ∗ (dvolg ). For any open set O ⊂ S2 ,  (12) dvolg . ν(O) = ψ −1 (O)

We apply Theorem 39 to the metric measure space (S2 , d0 , ν): Note that the con 2 stant c = c N (S , d0 ) arising from Theorem 39 depends now only on the canonical distance on the sphere S 2 . It can be seen as a universal constant.

We deduce that there exist an absolute constant c = c N (S2 , d0 ) and k + 1 annuli A1 , . . . , Ak+1 ⊂ S2 such that the annuli 2A1 , . . . , 2Ak+1 are mutually disjoint and, ∀i ≤ k, ν(S2 ) . k For each i ≤ k, set vi = uAi ◦ ψ. From the conformal invariance of the energy and Lemma 37, one deduces that, for every i ≤ k,   |∇g vi |2g dvolg , = deg(ψ) |∇g0 uAi |2g0 dvolg0 ≤ 8Γ(S2 , g0 )(γ + 1), ν(Ai ) ≥ c

(13)

S2

S

while, since uAi is equal to 1 on Ai , 

V olg (S) ν(S2 ) =c . vi2 dvolg ≥ V olg ψ −1 (Ai ) = ν(Ai ) ≥ c k k S Therefore, 8Γ(S2 , g0 ) C(γ + 1) (γ + 1)k ≤ k cV olg (S) V olg (S) where C is an absolute constant. Noticing that the k + 1 functions v1 , . . . vk+1 are disjointly supported in S, we deduce the desired inequality for λk (S, g). R(vi ) ≤

4.6. A recent development: the result of A. Hassannezhad. The upper bounds obtained by N. Korevaar are good in the sense where they are compatible with the Weyl law: Asymptotically, they are of the type c(g0 )k2/n in dimension greater than 2 and C(genus)k in dimension 2. By [CD],[CE1], some dependance on the geometry or in the topology is needed. On the other side, the Weyl law tells us that for a given compact Riemannian manifold of dimension n, the k-th eigenvalue is asymptotically of the type C(n)k2/n . So, it is natural to ask if one can get uniform upper bounds, asymptotically of the type C(n)k2/n . This is the main result of [Ha]. Theorem 41. For each n ≥ 2, there exist two positive constants An , Bn such that, for every compact Riemannian manifold (M, g) of dimension n λk (M, g)V olg (M )2/n ≤ An V ([g])2/n + Bn k2/n

THE SPECTRUM OF THE LAPLACIAN: A GEOMETRIC APPROACH

31

The only dependence on the metric g is the term V ([g]) called the min-conformal volume: Definition 42. Let (M, g) be a compact Riemannian manifold of dimension n. The min-conformal volume of (M, g) depends only on the conformal class [g] of g and is defined by V ([g]) = inf {V olg0 (M ) : g0 ∈ [g]; Ric(g0 ) ≥ −(n − 1)}. In particular, if there is g0 ∈ [g] with Ric(g0 ) ≥ 0, then V ([g]) = 0. Theorem (41) shows that the asymptotic in k is controlled by B(n)k2/n : λk (M, g)V olg (M )2/n V ([g0 ])2/n ≤ A(n) + Bn , 2/n k k2/n and V ([g0 ])2/n →0 k2/n as k → ∞. In the case of surfaces, A. Hassannezhad got Corollary 43. There exist two constants A, B such that for every compact surface (S, g) of genus γ, we have λk (S, g)V olg (S) ≤ Aγ + Bk, and, in particular, the asymptotic in k is Bk. Remark 44. Note that Corollary 43 applies for both orientable and nonorientable surfaces. For the proof these theorems, the method consists in constructing disjointly supported test functions associated to disjoint subsets of the manifold. The new ingredient is to mix the method of Korevaar with a construction due to Colbois and Maerten [CM]. Roughly speaking, the method of Korevaar gives us annuli, and the estimates become bad when these annuli are too large. The main idea is to use the method of [CM] in such situations. Remark 45. Each time we have a result of the type  λk ≤ C(geometry)

k V ol

2/n

one can ask if one can get a result of the type λk V ol2/n ≤ A(n)C(geometry) + B(n)k2/n . Most of the questions of this type are open: An example will be given in section 5, Remark 49.

32

BRUNO COLBOIS

5. Another geometric method to construct upper bounds and applications. In this section, I will present (with much less detail) another method to find upper bounds, which was established by D. Maerten and myself [CM], with one application to the spectrum of hypersurfaces. Roughly speaking, the method of [Ko], [GNY] is adequate for conformal metrics, but needs a comparison with one Riemannian metric. In the previous results, all the statements depend on metric g0 . The method of [CM] does not require such a comparison, but it supposes that there is no concentration of the metric. Let us state the version of [CEG1]: Theorem 46. Let (X, d, μ) be a complete, locally compact metric measured space, where μ is a finite measure. We assume that for all r > 0, there exists an integer N (r) such that each ball of radius r can be covered by N (r) balls of radius r/2. If there exist an integer K > 0 and a radius r > 0 such that, for each x ∈ X μ(B(x, r)) ≤

μ(X) , 4N 2 (r)K

then, there exist K μ-measurable subsets A1 , ..., AK of X such that, ∀i ≤ K, μ(X) μ(Ai ) ≥ 2N (r)K and, for i = j, d(Ai , Aj ) ≥ 3r.

Figure 13. A surface with 3 subsets as in Theorem 46 Remark 47. (1) The first hypothesis is essentially the same as in the construction of Korevaar and suppose a control of the packing constant. (2) In the version given in [CM], there is the hypothesis that the volume of the r-balls tends to 0 uniformly on X as r → 0. This is not necessary; it is enough that the measure μ(B(x, r)) of the ball is small enough. (3) In order to get a bound for λk comparable with the Weyl law, r has to be )1/n , and K = k + 1. of the size of ( Cμ(X) k (4) If r is chosen too small, the upper bounds we get are far from being optimal. As in the previous construction we associate to each subset Ai the test function uAi which is 1 inside Ai and 0 outside the r-neighborhood Ari of Ai . If A is one of

THE SPECTRUM OF THE LAPLACIAN: A GEOMETRIC APPROACH

33

the subsets Ai and Ar = {p ∈ X : d(p, A) ≤ r} ⎧ if x ∈ Ar ⎨ 1 − 1r d(x, A) 1 if x ∈ A uA (x) = ⎩ 0 if d(x, A) ≥ r An application. Let us describe a situation where this method can be used. Let Σ ⊂ Rm+1 be a compact hypersurface. As a consequence of the result of [CM] and of a classical theorem of Nash-Kuiper, it is possible to deform Σ in Rm+1 in order to have λ1 (Σ)V ol(Σ)2/m arbitrarily large. See [CDE, paragraph 5], for a detailed explanation. The upper bounds for Σ will depend of the isoperimetric ratio of the hypersurface. Let Ω ⊂ Rm+1 be a bounded domain with smooth boundary Σ = ∂Ω. We denote by V ol(Σ) I(Ω) = V ol(Ω)m/(m+1) the isoperimetric ratio of Ω. Theorem 48 (C-El Soufi-Girouard, [CEG2]). For any bounded domain Ω ⊂ Rm+1 with smooth boundary Σ = ∂Ω, and all k ≥ 1, (14)

λk (Σ)|Σ|2/m ≤ Cm I(Ω)1+2/m k2/m

with Cm an explicit constant depending on m, and where I(Ω) denotes the isoperimetric ratio. The proof, given in [CEG1], uses the construction of [CM]: First, we need to introduce (X, d, μ) be a complete, locally compact metric measure space, where μ is a finite measure satisfying the first hypothesis: For all r > 0, there exists an integer N (r) such that each ball of radius r can be covered by N (r) balls of radius r/2. In our situation, X = Rm+1 , d is the Euclidean distance. This implies that the packing constant N depends only on the dimension m. The measure μ is the measure associated to the hypersurface Σ: μ(U ) = V ol(U ∩ Σ). The main difficulty for the proof is that the second hypothesis is not satisfied. It is not always true that there exists an integer K > 0 and a radius r > 0 such that, for each x ∈ X μ(B(x, r)) ≤ 4Nμ(X) 2 (r)K .. The reason is that the measure can concentrate some place.  1/m Recall that for rk of the type Cμ(Σ) , we hope to have k μ(B(x, rk )) ≤

μ(Σ) . 4C(m)k

The idea of the proof of Theorem 48 is as follows. The difficult part of the proof (that I will not explain here) is to show that the hypothesis about the concentration is correct, but only for a part of Σ, proportional to the measure μ(Σ), with factor of proportionality C(m) I(Ω) .

34

BRUNO COLBOIS

This leads us to choose

 rk =

C1 (m)μ(Σ) kI(Ω)

1/m .

The Rayleigh quotient of a test function uA can be estimated as follows. If dvΣ denotes the volume form on Σ,  1 |∇uA |2 dvΣ ≤ 2 μ(Ark ) rk Σ and

 u2A dvΣ ≥ Σ

μ(Σ) C2 (n)I(Ω)k

we get R(uA ) ≤

1 μ(Ark ) C2 (n)I(Ω)k rk2 μ(Σ)

and we have a problem with the term μ(Ark )... As in the proof of the theorem of Korevaar, we can begin with 2k + 2 subsets, so that, for at least (k + 1) of them, μ(Ark ) ≤ μ(Σ) k+1 . This implies  2/m 2 k 1 μ(Σ) C2 (n)I(Ω)k ≤ C(n)I(Ω)1+ m R(uA ) ≤ 2 . rk k + 1 μ(Σ) V ol(Σ) Remark 49. This is a typical situation where the open question occurs: is it possible to establish an inequality of the type 2

λk (Σ)V ol(Σ)2/m ≤ Am I(Ω)1+ m + Bm k2/m for two constant Am , Bm depending only on the dimension m? For this specific question the answer is no (thesis of Luc P´etiard, article in preparation for 2017). 6. The conformal spectrum Having upper (or lower) bounds leads to investigate the spectrum from a qualitative or quantitative viewpoint. Let us give the example of the conformal spectrum and of the topological spectrum we developed in [CE1] with A. El Soufi (see also [Co] for a short survey). Definition 50. For any natural integer k and any conformal class of metrics [g0 ] on M , we define the conformal k-th eigenvalue of (M, [g0 ]) to be   λck (M, [g0 ]) = sup λk (M, g)V ol(M, g)2/n | g is conformal to g0 . The sequence {λck (M, [g0 ])} constitutes the conformal spectrum of (M, [g0 ]). In dimension 2, one can also define a topological spectrum by setting, for any genus γ and any integer k ≥ 0, λtop k (γ) = sup {λk (M, g)V ol(M, g)} , where g describes the set of Riemannian metrics on the orientable compact surface M of genus γ.

THE SPECTRUM OF THE LAPLACIAN: A GEOMETRIC APPROACH

35

Regarding the quantitative aspects, in some very special situations, the hope is to detect the maximum or the supremum in a conformal class. This concerns mainly the first nonzero eigenvalue. Note that for the topological spectrum on surfaces, it was already explained in (1.3.2) that it was a very difficult question. It is as well difficult to decide if the supremum is a maximum: Does it a Riemannian metric g ∈ [g0 ] exists such that λk V ol(M, g)2/n is maximum? Regarding the conformal first eigenvalue, El Soufi and Ilias [EI1] gave a sufficlass [g]: cient condition for a Riemannian metric g to maximize λ1 in its conformal  if there exists a family f1 , f2 , ..., fp of first eigenfunctions satisfying i dfi ⊗dfi = g, then λc1 (M, [g]) = λ1 (g). This condition is fulfilled in particular by the metric of any homogeneous Riemannian space with irreducible isotropy representation. For instance, the first conformal eigenvalues of the rank one symmetric spaces endowed with their standard conformal classes [gs ], are given by 2/n

• λc1 (Sn , [gs ]) = nωn , where ωn is the volume of the n-dimensional Euclidean sphere of radius one, n−2 2/n • λc1 (RP n , [gs ]) = 2 n (n + 1)ωn , c d −1/d • λ1 (CP , [gs ]) = 4π(d + 1)d! , −1/2d • λc1 (HP d , [gs ]) = 8π(d + 1)(2d + 1)!√ , 6 1/8 9 1/8 c 2 = 8π 6( 385 ) . • λ1 (CaP , [gs ]) = 48π( 11! ) Remark 51. Note that this is a strong hypothesis to suppose that the maximum is reached by a regular Riemannian metric. This cannot be expected in general, and even in dimension 2, it is a fundamental question to understand what kinds of singularities may occurs for the maximal metrics (with respect to λ1 or λk ). I will mainly describe the qualitative aspects of the problem and the first qualitative question is the following: On a compact manifold M , does a Riemannian metric g exists such that λc1 [g] is arbitrarily small? The answer is no: Among all the possible conformal classes of metrics on manifolds, the standard conformal class of the sphere is the one having the lowest conformal spectrum. Theorem 52. For any conformal class [g] on M and any integer k ≥ 0, λck (M, [g]) ≥ λck (Sn , [gs ]). Although the eigenvalues of a given Riemannian metric may have nontrivial multiplicities, the conformal eigenvalues are all simple: The conformal spectrum consists of a strictly increasing sequence, and, moreover, the gap between two consecutive conformal eigenvalues is uniformly bounded. Precisely, we have the following theorem: Theorem 53. For any conformal class [g] on M and any integer k ≥ 0, λck+1 (M, [g])n/2 − λck (M, [g])n/2 ≥ λc1 (Sn , [gs ]) = nn/2 ωn , where ωn is the volume of the n-dimensional Euclidean sphere of radius one. An immediate consequence of these two theorems is the following explicit estimate of λck (M, [g]): Corollary 54. For any conformal class [g] on M and any integer k ≥ 0, λck (M, [g]) ≥ nωn2/n k2/n .

36

BRUNO COLBOIS

Combined with the Hassannezhad estimates [Ha, Corollary 54] gives nωn2/n k2/n ≤ λck (M, [g]) ≤ An V ([g])2/n + Bn k2/n for some constants An , Bn depending only on n, and V ([g]) depending on a lower bound of Ric d2 , where Ric is the Ricci curvature and d is the diameter of g or of another representative of [g]). This implies the relation  2/n V ([g]) λck (M, [g]) 2/n ≤ An + Bn . nωn ≤ k k2/n Asymptotically, as k → ∞, this shows that

λck (M,[g]) k2/n

2/n

is between nωn

and Bn .

Corollary 54 implies also that, if the k-th eigenvalue λk (g) of a metric g is 2/n less than nωn k2/n , then g does not maximize λk on its conformal class [g]. For instance, the standard metric gs of S2 , which maximizes λ1 , does not maximize λk on [gs ] for any k ≥ 2. This fact answers a question of Yau (see [Y, p. 686]). El Soufi and Ilias have also shown that the property for a Riemannian metric to be critical (in a generalized sense) implies multiplicity. As a corollary of this and of the previous results, we get Corollary 55. If a Riemannian metric g maximizes λ1 on its conformal class [g], then it does not maximize λ2 on [g]. More generally, a Riemannian metric g cannot maximize simultaneously three consecutive eigenvalues λk , λk+1 , λk+2 on [g]. Note however that it is a very difficult question to decide if there exists a Riemannian metric g1 ∈ [g] such that λck ([g]) = λk (g1 ). Regarding the topological spectrum of surfaces, we have similar results: Theorem 56. For any fixed genus γ and any integer k ≥ 0, we have top λtop k+1 (γ) − λk (γ) ≥ 8π,

and top λtop k (γ) ≥ 8(k − 1)π + λ1 (γ) ≥ 8kπ

However, we have good lower bounds for λtop 1 (γ) (see for example [BM]) so that we can say that, 4 λtop 1 (γ) ≥ π(γ − 1), 5 and it follows 4 λtop k (γ) ≥ π(γ − 1) + 8(k − 1)π. 5 Combined with Hassannezhad estimates [Ha], we get for an oriented surface of genus γ 4 π(γ − 1) + 8(k − 1)π ≤ λtop k (γ) ≤ A(γ − 1) + Bk. 5 for A, B constant. We can also look at the behavior of λck (γ) in terms of the genus γ. Theorem 57. We have top λtop k (γ + 1) ≥ λk (γ).

THE SPECTRUM OF THE LAPLACIAN: A GEOMETRIC APPROACH

37

Note that it is an open question to know if the inequality is strict or not for each γ. A few words about the proof. Locally, a Riemannian manifold is almost Euclidean and one can deform a small neighbourhood of a point into a (almost) round sphere (blow up). Then, we can use these deformations to get our results, using classical facts about the behavior of the spectrum under surgery. This explains the presence of the first eigenvalue of the round sphere in all these estimates. top In order to show λtop k (γ + 1) ≥ λk (γ), we have to compare two surfaces with different topology and this needs another idea: to add a thin and short handle in order to pass from a surface of genus γ to a surface of genus γ + 1. By classical surgery results, this neither affects the eigenvalues between λ1 and λk (if the handle is short enough) nor the area. However, it cannot be proved with this method that the inequality is strict.

Remark 58. Each time there are lower or upper bounds, it is possible to study the spectrum from a qualitative viewpoint. However, this is not always easy! Conside for example the following question in the case of hypersurfaces. For a positive number I (large enough), define λk (I) = sup{λk (Σ)V ol(Σ)2/m , } where Σ = ∂Ω, Ω domain in Rm+1 such that I(Ω) ≤ I. What can be said about {λk (I)}? For example, do we have λk+1 (I) > λk (I)? What about the gap λk+1 (I) − λk (I)? Recent results about this topic. (1) For the second conformal eigenvalue of the canonical metric on the sphere, asymptotically sharp upper bound were found in [GNP] and [Pe1]. (2) In [Pe2], Petrides proved the following result: let (M, g) be a compact Riemannian manifold of dimension n ≥ 3. The first conformal eigenvalue 2/n is always greater than nωn (the value it takes for the round sphere) except if (M, g) is conformally diffeomorphic to the standard sphere. (3) In the specific case of the first eigenvalue for compact surfaces, there have been several works recently published about the existence and the regularity of maximal metrics: Roughly speaking, the maximal metrics are not expected to be regular, but singular with very few singularities consisting only in conical points [Kok], [Pe3].

38

BRUNO COLBOIS

Acknowledgments The author thanks the organizers of the summer school (especially Alexandre Girouard and Iosif Polterovich) for inviting me to Montr´eal and for the opportunity to write this article. The author also thank the referee for his very useful comments. References Colette Ann´ e, Spectre du laplacien et ´ ecrasement d’anses (French, with English sum´ mary), Ann. Sci. Ecole Norm. Sup. (4) 20 (1987), no. 2, 271–280. MR911759 [An2] Colette Ann´ e, A note on the generalized dumbbell problem, Proc. Amer. Math. Soc. 123 (1995), no. 8, 2595–2599, DOI 10.2307/2161294. MR1257096 [As] Mark S. Ashbaugh, Isoperimetric and universal inequalities for eigenvalues, Spectral theory and geometry (Edinburgh, 1998), London Math. Soc. Lecture Note Ser., vol. 273, Cambridge Univ. Press, Cambridge, 1999, pp. 95–139, DOI 10.1017/CBO9780511566165.007. MR1736867 [BBG] P. B´ erard, G. Besson, and S. Gallot, Sur une in´ egalit´ e isop´ erim´ etrique qui g´ en´ eralise celle de Paul L´ evy-Gromov (French), Invent. Math. 80 (1985), no. 2, 295–308, DOI 10.1007/BF01388608. MR788412 [BC] J. Bertrand and B. Colbois, Capacit´ e et in´ egalit´ e de Faber-Krahn dans Rn (French, with English and French summaries), J. Funct. Anal. 232 (2006), no. 1, 1–28, DOI 10.1016/j.jfa.2005.04.015. MR2200165 [Be] Pierre H. B´ erard, Spectral geometry: direct and inverse problems, Lecture Notes in Mathematics, vol. 1207, Springer-Verlag, Berlin, 1986. With appendixes by G´ erard Besson, and by B´ erard and Marcel Berger. MR861271 [Ber] Berger, A.; Optimisation du spectre du Laplacien avec conditions de Dirichlet et Neumann atel, (2015). dans R2 et R3 , Thesis, Grenoble and Neuchˆ [BF] Bucur, D., Freitas P.; A free boundary approach to the Faber-Krahn inequality, to appear in this Volume. [BGM] Marcel Berger, Paul Gauduchon, and Edmond Mazet, Le spectre d’une vari´ et´ e riemannienne (French), Lecture Notes in Mathematics, Vol. 194, Springer-Verlag, Berlin-New York, 1971. MR0282313 [BH] B´ erard, P., Helffer B.; On the nodal patterns of the 2D isotropic quantum harmonic oscillator, arXiv:1506.02374, to appear in this Volume. [Bl] David D. Bleecker, The spectrum of a Riemannian manifold with a unit Killing vector field, Trans. Amer. Math. Soc. 275 (1983), no. 1, 409–416, DOI 10.2307/1999029. MR678360 [BM] Robert Brooks and Eran Makover, Riemann surfaces with large first eigenvalue, J. Anal. Math. 83 (2001), 243–258, DOI 10.1007/BF02790263. MR1828493 [BP] Brasco, L., De Philippis, G.; Spectral inequalities in quantitative form, arXiv:1604.05072. This is a chapter of the forthcoming book ”Shape Optimization and Spectral Theory”, edited by Antoine Henrot and published by De Gruyter. ´ [Bu2] Peter Buser, A note on the isoperimetric constant, Ann. Sci. Ecole Norm. Sup. (4) 15 (1982), no. 2, 213–230. MR683635 [CD] B. Colbois and J. Dodziuk, Riemannian metrics with large λ1 , Proc. Amer. Math. Soc. 122 (1994), no. 3, 905–906, DOI 10.2307/2160770. MR1213857 [CDE] Bruno Colbois, Emily B. Dryden, and Ahmad El Soufi, Bounding the eigenvalues of the Laplace-Beltrami operator on compact submanifolds, Bull. Lond. Math. Soc. 42 (2010), no. 1, 96–108, DOI 10.1112/blms/bdp100. MR2586970 [CE1] Bruno Colbois and Ahmad El Soufi, Extremal eigenvalues of the Laplacian in a conformal class of metrics: the ‘conformal spectrum’, Ann. Global Anal. Geom. 24 (2003), no. 4, 337–349, DOI 10.1023/A:1026257431539. MR2015867 [CEG1] Bruno Colbois, Ahmad El Soufi, and Alexandre Girouard, Isoperimetric control of the Steklov spectrum, J. Funct. Anal. 261 (2011), no. 5, 1384–1399, DOI 10.1016/j.jfa.2011.05.006. MR2807105 [CEG2] Bruno Colbois, Ahmad El Soufi, and Alexandre Girouard, Isoperimetric control of the spectrum of a compact hypersurface, J. Reine Angew. Math. 683 (2013), 49–65. MR3181547 [An1]

THE SPECTRUM OF THE LAPLACIAN: A GEOMETRIC APPROACH

39

Bruno Colbois, Ahmad El Soufi, and Alessandro Savo, Eigenvalues of the Laplacian on a compact manifold with density, Comm. Anal. Geom. 23 (2015), no. 3, 639–670, DOI 10.4310/CAG.2015.v23.n3.a6. MR3310527 [CGI] Bruno Colbois, Alexandre Girouard, and Mette Iversen, Uniform stability of the Dirichlet spectrum for rough outer perturbations, J. Spectr. Theory 3 (2013), no. 4, 575–599, DOI 10.4171/JST/57. MR3122224 [Ch1] Isaac Chavel, Eigenvalues in Riemannian geometry, Pure and Applied Mathematics, vol. 115, Academic Press, Inc., Orlando, FL, 1984. Including a chapter by Burton Randol; With an appendix by Jozef Dodziuk. MR768584 [Ch2] Isaac Chavel, On A. Hurwitz’ method in isoperimetric inequalities, Proc. Amer. Math. Soc. 71 (1978), no. 2, 275–279, DOI 10.2307/2042848. MR0493885 [Che] Shiu Yuen Cheng, Eigenvalue comparison theorems and its geometric applications, Math. Z. 143 (1975), no. 3, 289–297, DOI 10.1007/BF01214381. MR0378001 [CM] Bruno Colbois and Daniel Maerten, Eigenvalues estimate for the Neumann problem of a bounded domain, J. Geom. Anal. 18 (2008), no. 4, 1022–1032, DOI 10.1007/s12220-0089041-z. MR2438909 [Co] Bruno Colbois, Spectre conforme et m´ etriques extr´ emales (French, with French summary), S´eminaire de Th´eorie Spectrale et G´eom´ etrie. Vol. 22. Ann´ ee 2003–2004, S´emin. Th´ eor. Spectr. G´eom., vol. 22, Univ. Grenoble I, Saint-Martin-d’H` eres, 2004, pp. 93–101. MR2136138 [CoHi] R. Courant and D. Hilbert, Methods of mathematical physics. Vol. I, Interscience Publishers, Inc., New York, N.Y., 1953. MR0065391 [Cou] Gilles Courtois, Spectrum of manifolds with holes, J. Funct. Anal. 134 (1995), no. 1, 194–221, DOI 10.1006/jfan.1995.1142. MR1359926 [Cr] Christopher B. Croke, The first eigenvalue of the Laplacian for plane domains, Proc. Amer. Math. Soc. 81 (1981), no. 2, 304–305, DOI 10.2307/2044213. MR593476 [EG] Lawrence C. Evans and Ronald F. Gariepy, Measure theory and fine properties of functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992. MR1158660 [EHI] Ahmad El Soufi, Evans M. Harrell II, and Said Ilias, Universal inequalities for the eigenvalues of Laplace and Schr¨ odinger operators on submanifolds, Trans. Amer. Math. Soc. 361 (2009), no. 5, 2337–2350, DOI 10.1090/S0002-9947-08-04780-6. MR2471921 [EI1] A. El Soufi and S. Ilias, Immersions minimales, premi` ere valeur propre du laplacien et volume conforme (French), Math. Ann. 275 (1986), no. 2, 257–267, DOI 10.1007/BF01458460. MR854009 [EI2] A. El Soufi and S. Ilias, Majoration de la seconde valeur propre d’un op´ erateur de Schr¨ odinger sur une vari´ et´ e compacte et applications (French, with English summary), J. Funct. Anal. 103 (1992), no. 2, 294–316, DOI 10.1016/0022-1236(92)90123-Z. MR1151550 [FMP] Nicola Fusco, Francesco Maggi, and Aldo Pratelli, Stability estimates for certain FaberKrahn, isocapacitary and Cheeger inequalities, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 8 (2009), no. 1, 51–71. MR2512200 [GNP] Alexandre Girouard, Nikolai Nadirashvili, and Iosif Polterovich, Maximization of the second positive Neumann eigenvalue for planar domains, J. Differential Geom. 83 (2009), no. 3, 637–661. MR2581359 [GNY] Alexander Grigoryan, Yuri Netrusov, and Shing-Tung Yau, Eigenvalues of elliptic operators and geometric applications, Surveys in differential geometry. Vol. IX, Surv. Differ. Geom., vol. 9, Int. Press, Somerville, MA, 2004, pp. 147–217, DOI 10.4310/SDG.2004.v9.n1.a5. MR2195408 [Go] Carolyn S. Gordon, Survey of isospectral manifolds, Handbook of differential geometry, Vol. I, North-Holland, Amsterdam, 2000, pp. 747–778, DOI 10.1016/S18745741(00)80009-6. MR1736857 [Ha] Asma Hassannezhad, Conformal upper bounds for the eigenvalues of the Laplacian and Steklov problem, J. Funct. Anal. 261 (2011), no. 12, 3419–3436, DOI 10.1016/j.jfa.2011.08.003. MR2838029 [He] Antoine Henrot, Extremum problems for eigenvalues of elliptic operators, Frontiers in Mathematics, Birkh¨ auser Verlag, Basel, 2006. MR2251558 [Ko] Nicholas Korevaar, Upper bounds for eigenvalues of conformal metrics, J. Differential Geom. 37 (1993), no. 1, 73–93. MR1198600 [CES]

40

[Kok] [La] [Lo] [LY]

[Mu] [Pe1]

[Pe2] [Pe3]

[PW]

[Ro]

[Sa]

[San]

[Sc]

[ST]

[Uh] [Y] [YY]

[Zu]

BRUNO COLBOIS

Gerasim Kokarev, Variational aspects of Laplace eigenvalues on Riemannian surfaces, Adv. Math. 258 (2014), 191–239, DOI 10.1016/j.aim.2014.03.006. MR3190427 Olivier Labl´ ee, Spectral theory in Riemannian geometry, EMS Textbooks in Mathematics, European Mathematical Society (EMS), Z¨ urich, 2015. MR3309800 Joachim Lohkamp, Discontinuity of geometric expansions, Comment. Math. Helv. 71 (1996), no. 2, 213–228, DOI 10.1007/BF02566417. MR1396673 Peter Li and Shing Tung Yau, Estimates of eigenvalues of a compact Riemannian manifold, Geometry of the Laplace operator (Proc. Sympos. Pure Math., Univ. Hawaii, Honolulu, Hawaii, 1979), Proc. Sympos. Pure Math., XXXVI, Amer. Math. Soc., Providence, R.I., 1980, pp. 205–239. MR573435 Hideo Mutˆ o, The first eigenvalue of the Laplacian on even-dimensional spheres, Tˆ ohoku Math. J. (2) 32 (1980), no. 3, 427–432, DOI 10.2748/tmj/1178229601. MR590038 Romain Petrides, Maximization of the second conformal eigenvalue of spheres, Proc. Amer. Math. Soc. 142 (2014), no. 7, 2385–2394, DOI 10.1090/S0002-9939-2014-12095-8. MR3195761 Romain Petrides, On a rigidity result for the first conformal eigenvalue of the Laplacian, J. Spectr. Theory 5 (2015), no. 1, 227–234, DOI 10.4171/JST/95. MR3340180 Romain Petrides, Existence and regularity of maximal metrics for the first Laplace eigenvalue on surfaces, Geom. Funct. Anal. 24 (2014), no. 4, 1336–1376, DOI 10.1007/s00039014-0292-5. MR3248488 L. E. Payne and H. F. Weinberger, Lower bounds for vibration frequencies of elastically supported membranes and plates, J. Soc. Indust. Appl. Math. 5 (1957), 171–182. MR0092431 Steven Rosenberg, The Laplacian on a Riemannian manifold, London Mathematical Society Student Texts, vol. 31, Cambridge University Press, Cambridge, 1997. An introduction to analysis on manifolds. MR1462892 Takashi Sakai, Riemannian geometry, Translations of Mathematical Monographs, vol. 149, American Mathematical Society, Providence, RI, 1996. Translated from the 1992 Japanese original by the author. MR1390760 G. Santhanam, A sharp upper bound for the first eigenvalue of the Laplacian of compact hypersurfaces in rank-1 symmetric spaces, Proc. Indian Acad. Sci. Math. Sci. 117 (2007), no. 3, 307–315, DOI 10.1007/s12044-007-0026-9. MR2352050 Richard Schoen, Existence and geometric structure of metrics on surfaces which extremize eigenvalues, Bull. Braz. Math. Soc. (N.S.) 44 (2013), no. 4, 777–807, DOI 10.1007/s00574-013-0034-6. MR3167132 Brian Davies and Yuri Safarov (eds.), Spectral theory and geometry, London Mathematical Society Lecture Note Series, vol. 273, Cambridge University Press, Cambridge, 1999. Papers from the ICMS Instructional Conference held in Edinburgh, March 30–April 9, 1998. MR1736863 K. Uhlenbeck, Generic properties of eigenfunctions, Amer. J. Math. 98 (1976), no. 4, 1059–1078, DOI 10.2307/2374041. MR0464332 Shing Tung Yau, Problem section, Seminar on Differential Geometry, Ann. of Math. Stud., vol. 102, Princeton Univ. Press, Princeton, N.J., 1982, pp. 669–706. MR645762 Paul C. Yang and Shing Tung Yau, Eigenvalues of the Laplacian of compact Riemann surfaces and minimal submanifolds, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 7 (1980), no. 1, 55–63. MR577325 Shunhui Zhu, The comparison geometry of Ricci curvature, Comparison geometry (Berkeley, CA, 1993), Math. Sci. Res. Inst. Publ., vol. 30, Cambridge Univ. Press, Cambridge, 1997, pp. 221–262. MR1452876

ˆtel, Institut de Math´ Universit´ e de Neucha ematiques, Rue Emile-Argand 11, CHˆtel, Suisse 2000, Neucha E-mail address: [email protected]

Contemporary Mathematics Volume 700, 2017 http://dx.doi.org/10.1090/conm/700/14182

An elementary introduction to quantum graphs Gregory Berkolaiko Abstract. We describe some basic tools in the spectral theory of Schr¨ odinger operator on metric graphs (also known as “quantum graphs”) by studying in detail some basic examples. The exposition is kept as elementary and accessible as possible. In the later sections we apply these tools to prove some results on the count of zeros of the eigenfunctions of quantum graphs.

1. Introduction Studying operators of Schr¨odinger type on metric graphs is a growing subfield of mathematical physics which is motivated both by direct applications of the graph models to physical phenomena and by use of graphs as a simpler setting in which to study complex phenomena of quantum mechanics, such as Anderson localization, universality of spectral statistics, nodal statistics, scattering and resonances, to name but a few. The name “quantum graphs” is most likely a shortening of the title of the article “Quantum Chaos on Graphs” by Kottos and Smilansky [36]. The model itself has been studied well before the name appeared, for example in [41, 43, 46, 47, 51]. Several reviews and monographs cover various directions within the quantum graphs research [29, 40, 45]. However, when starting a research project with students, both (post-) graduate and undergraduate, the author felt that a more elementary introduction would be helpful. The present manuscript grew out of the same preparatory lecture repeated, at different points of time, to several students. It is basically a collection of minimal examples of quantum graphs which already exhibit behavior typical to larger graphs. We supply the examples with pointers to the more general facts and theorems. Only in the last sections we explore a research topic (the nodal statistics on graphs) in some depth. For obvious reasons the pointers often lead to the monograph [15]; the notation is kept in line with that book, too. 2. Schr¨ odinger equation on a metric graph Consider a graph Γ = (V, E), where V is the set of vertices and E is the set of edges. Each edge connects a pair of vertices; we allow more than one edge running between any two vertices. We also allow edges connecting vertices to themselves 2010 Mathematics Subject Classification. Primary 34B34, 35B05, 81Q35. c 2017 American Mathematical Society

41

42

GREGORY BERKOLAIKO

(loops). This freedom creates some notational difficulties, so we ask the reader to be flexible and forgiving. Each edge e is assigned a positive length Le and is thus identified with an interval [0, Le ] (the direction is chosen arbitrarily and is irrelevant to the resulting theory). This makes Γ a metric graph. Now a function on a graph is just a collection of functions defined on individual edges. The eigenvalue equation for the Schr¨ odinger operator is d2 f + V (x)f (x) = λf (x), dx2 which is to be satisfied on every edge, in addition to the vertex matching conditions as follows (1)

−

(2)

f (x) is continuous,  df (v) = 0. dx e∼v

(3)

The continuity means that the values at the vertex agree among all functions living on the edges attached (or incident) to the vertex. In the second condition (often called current conservation condition), the sum is over all edges attached to the vertex and the derivative are all taken in the same direction: from the vertex into the edge. A looping edge contributes two terms to the sum, one for each end of the edge. The function V (x) is called the electric potential but we will set it identically to zero in all of the examples below. Vertex conditions (2)-(3) are called Neumann conditions 1 ; they can be generalized significantly, but before we give any more theory, let us consider some examples. 2.1. Example: a trivial graph — an interval. An interval [0, L] is the simplest example of a graph; it has two vertices (the endpoints of the interval) and one edge. The continuity condition is empty at every vertex since there is only one edge. The current conservation condition at the vertex 0 becomes f  (0) = 0,

(4) and at the vertex L becomes (5)

−f  (L) = 0.

The minus sign appeared because we agreed to direct the derivatives into the edge; of course it is redundant in this particular case. Let V (x) ≡ 0 and consider first the positive eigenvalues, λ > 0. The eigenvalue equation becomes (6)

−f  = k2 f,

where for convenience we substituted λ = k2 . This is a second order linear equation with constant coefficients which for k > 0 is readily solved by (7)

f (x) = C1 cos(kx) + C2 sin(kx).

1 Other names present in the literature are “Kirchhoff”, “Neumann-Kirchhoff”, “standard”, “natural” etc.

AN ELEMENTARY INTRODUCTION TO QUANTUM GRAPHS

43

Applying the first vertex condition f  (0) = 0 we get C2 = 0 and f (x) = C1 cos(kx). The second vertex condition becomes (8)

C1 k sin(kL) = 0,

which imposes a condition on k but does nothing to determine C1 (naturally we are not interested in the trivial solution f (x) ≡ 0). We thus get the eigenvalues 2 λ = k2 = (πn/L) , n = 1, 2, . . . with the corresponding eigenfunctions f (x) = cos (πnx/L) defined up to an overall constant multiplier (as befits eigenvectors and eigenfunctions). There is one other eigenvalue in the spectrum that we missed: λ = 0 with the eigenfunction f (x) ≡ 1. While this agrees with the above formulas with n = 0, the premise of equation (7) is no longer correct when λ = 0. Exercise 2.1. Solve the eigenvalue equation with λ < 0 and show that the vertex conditions (4) and (5) are never satisfied simultaneously (ignore the trivial solution f (x) ≡ 0). Exercise 2.2. Integrate by parts the scalar product  L f (x) (−f  (x)) dx, (9) f, −f  = 0

to obtain an expression that is obviously non-negative, and thus show that it is not necessary to solve (6) to conclude that there are no negative eigenvalues. We did not try to look for complex eigenvalues. This is because the Schr¨ odinger operator we defined is self-adjoint (see Thm 1.4.4 of [15]) and therefore has real spectrum. The spectrum in the above example is discrete: all eigenvalues are isolated and of finite multiplicity. This is true for any graph which is compact (has finitely many edges, all of which have finite length), see Thm 3.1.1 of [15]. The proof outlined in Exercise 2.2 works for general graphs with Neumann conditions at all vertices. The multiplicity of the eigenvalue 0 in the spectrum can be shown to equal the number of the connected components of the graph. 2.2. Example: a trivializable graph with a vertex of degree two. Consider now a graph consisting of two consecutive intervals, [0, L1 ] and [L1 , L1 + L2 ]. We do not really have to parametrize the edges starting from 0, so in this example we will employ the “natural” parametrization. Denote the components of eigenfunction living on the two intervals by f1 and f2 correspondingly. Solving the equation on the first edge and enforcing the Neumann condition at 0 results in f1 (x) = C cos(kx). The conditions at the point L1 are (10)

f1 (L1 ) = f2 (L1 ),

(11)

− f1 (L1 ) + f2 (L1 ) = 0.

Now, by uniqueness theorem for second order differential equations, the solution on the second edge is fully determined by its value at L1 and the value of its derivative. Thus the solution is still f2 (x) = C cos(kx) and there is no change in the solution happening at L1 . We could have considered the interval [0, L1 + L2 ] without introducing the additional vertex at L1 . This obviously generalizes to the following rule: having a Neumann vertex of degree 2 is equivalent to having an uninterrupted edge.

44

GREGORY BERKOLAIKO

L2 L3

L1

Figure 1. A star graph with three edges. This rule is useful, for example, for when one wants to program a looping edge but is troubled by the notational difficulties of loops or multiple edges. In this case a looping edge can be implemented as a triangle with two “dummy” vertices of degree two. 2.3. Example: star graph with Neumann endpoints. Consider now a first non-trivial example: a star graph with 3 edges meeting at a central vertex, see Fig. 1. Parametrizing the edges from the endpoints towards the central vertex, we get (12)

− f1 = k2 f1 ,

−f2 = k2 f2 ,

(13)

f1 (0) = 0,

(14)

f1 (L1 ) = f2 (L2 ) = f3 (L3 ),

(15)

− f1 (L1 ) − f2 (L2 ) − f3 (L3 ) = 0,

f2 (0) = 0,

−f3 = k2 f3 ,

f3 (0) = 0,

where in addition to the already familiar equations (12) and (13) (in three copies), we have continuity condition at the central vertex in equation (14) and current conservation at the central vertex in equation (15). Note that in equation (12), the eigenvalue k2 is the same on all three edges. Equations (12)-(13) are solved by (16)

f1 (x) = A1 cos(kx),

f2 (x) = A2 cos(kx),

f3 (x) = A3 cos(kx),

for some constants A1 , A2 and A3 . Now the remaining two equations become, after a minor simplification, (17)

A1 cos(kL1 ) = A2 cos(kL2 ) = A3 cos(kL3 ),

(18)

A1 sin(kL1 ) + A2 sin(kL2 ) + A3 sin(kL3 ) = 0. Dividing equation (18) by (17) cancels the unknown constants, resulting in

(19)

tan(kL1 ) + tan(kL2 ) + tan(kL3 ) = 0.

Squares of the roots k of this equation (which cannot be solved explicitly except when all Ls are equal) are the eigenvalues of the star graph. Exercise 2.3. We ignored the possibility that one or more of the cosines in equation (17) are zero. Show that the more robust (but much longer!) version of equation (19) is (20)

sin(kL1 ) cos(kL2 ) cos(kL3 ) + cos(kL1 ) sin(kL2 ) cos(kL3 ) + cos(kL1 ) cos(kL2 ) sin(kL3 ) = 0.

AN ELEMENTARY INTRODUCTION TO QUANTUM GRAPHS

D

=

45

D

Figure 2. A Dirichlet condition imposed at a vertex of degree d = 3 (here and in subsequent figures, the Dirichlet vertices are denoted by empty circles) is equivalent to splitting the vertex into three and imposing the condition at every new vertex. Moreover, the order of the root k of (20) is equal to the dimension of the corresponding eigenspace. For example, if L1 = L2 = L3 = π/2, the left-hand side of (20) vanishes at k = 1 to the second order. This corresponds to two linearly independent solutions, (21)

(f1 , f2 , f3 ) = (cos(x), − cos(x), 0) and

(cos(x), 0, − cos(x)).

There is actually a lot more that can be (and will be said) about this simple graph, but we first need to extend the set of possible vertex conditions that we consider. 3. Dirichlet condition Another possible vertex condition which is compatible with self-adjointness of the Schr¨odinger operator is the so-called Dirichlet condition, (22)

fe (v) = 0 for all e incident to v.

It is usually used only at vertices of degree 1 for the following reason. A Dirichlet condition imposed at a vertex of degree 2 or more fails to relate in any way the individual functions living on the incident edges. Thus a graph with a Dirichlet condition at a vertex v of degree dv is equivalent to a graph with v substituted with dv vertices of degree one, see Fig. 2. Note that the difference between a Neumann and a Dirichlet condition at a vertex of degree dv is minimal: the current conservation condition is substituted with the condition that one of the function values is equal to zero; the rest is taken care of by continuity. Exercise 3.1. Show that the eigenvalues of the interval [0, L] with Dirichlet conditions at both ends are given by λn = (πn/L)2 , n = 1, 2, . . . with the eigenfunctions f (n) (x) = sin (πnx/L). 3.1. Example: a star graph with Dirichlet conditions at endpoints. Consider a star graph with N edges. We parametrize the edges from the endpoints towards the central vertex, as before. Solving the eigenvalue equation −f  = k2 f and imposing the Dirichlet condition at x = 0 results in fi (x) = Ai sin(kx), where the constant Ai depends on the edge.

46

GREGORY BERKOLAIKO

100 80 60 40

F(k)

20 0 -20 -40 -60 -80 -100 0

0.5

1

1.5

2

2.5

3

3.5

k

Figure 3.√Plot of√the right-hand side of (25) with 3 edges of lengths 1, 2 and 3. It is a function that decreases monotonely between each poles. The poles visible on the plot are 0, √ pair of its √ √ π/ 3, π/ 2, π and 2π/ 3. At the central vertex we have (23)

A1 sin(kL1 ) = . . . = AN sin(kLN ),

(24)

A1 cos(kL1 ) + . . . + AN cos(kLN ) = 0.

If we assume that the lengths Li are incommensurate, we will not be missing any roots by dividing equation (24) by (23), leading to the eigenvalue condition (25)

N 

cot(kLi ) = 0.

i=1

This condition is very similar to equation (19) we derived for the star graph with Neumann conditions at the endpoints. However, it is now easier to see a connection between the star graph and the eigenvalue problem of an interval. The left-hand side of equation (25) has derivative of constant sign (negative) except at the poles k ∈ {nπ/Li }, i = 1, . . . , N , n ∈ Z. Therefore, between each pair of consecutive poles there is a single root of equation (25), see Fig. 3. The poles can be interpreted as the square roots of the eigenvalues of the individual edges of the graph, see Exercise 3.1. Furthermore, the collection of the edges with Dirichlet conditions can be obtained from the original star graph by changing the central vertex condition from Neumann to Dirichlet, see Fig. 4. As we mentioned already,

AN ELEMENTARY INTRODUCTION TO QUANTUM GRAPHS

N

D

=

47

D

Figure 4. A star graph with the central vertex condition changed to Dirichlet splits into a collection of intervals. this can be effected by changing only one equation in the Neumann conditions, which is a rank-one perturbation. To summarize, we found that there is exactly one eigenvalue of a star graph between any two consecutive eigenvalues of its rank 1 perturbation. 4. Interlacing inequalities Naturally, the observation of Section 3.1 applies not only to star graphs but to any graphs with discrete spectrum. Lemma 4.1 (Neumann–Dirichlet interlacing). Let Γ0 be a quantum graph with a vertex v which is endowed with Neumann conditions. Let Γ∞ denote the graph obtained by changing the conditions at v to Dirichlet. Numbering the eigenvalues of both graphs in ascending order starting from 1, we have (26)

λn (Γ0 ) ≤ λn (Γ∞ ) ≤ λn+1 (Γ0 ) ≤ λn+1 (Γ∞ ).

An equality between a Neumann and a Dirichlet eigenvalue is possible only if the eigenspace of the Neumann eigenvalue contains a function vanishing at v, or, equivalently, the eigenspace of the Dirichlet eigenvalue contains a function satisfying the current conservation condition. The proof, which can be found in [14, 15], uses the standard arguments built upon the minimax characterisation of eigenvalues of a self-adjoint operator. It is analogous to the proofs of Cauchy’s Interlacing Theorem or rank-one perturbations for matrices, see, for example [32]. The following exercise contains an application of Lemma 4.1 to the nodal count of eigenfunctions. Exercise 4.2. Show that if the n-th eigenvalue of a star graph with Dirichlet endpoints is simple and the corresponding eigenfunction is non-vanishing at the central vertex, the eigenfunction has precisely n − 1 zeros in the interior of the graph. This statement can be obtained by combining the strict version of inequality (26) with the following observation. If λ satisfies λn (I) ≤ λ ≤ λn+1 (I),√where I is the interval [0, L] with Dirichlet boundary conditions, then f (x) = sin( λx) has n zeros on the interval (0, L). A more general version of this statement holds for tree graphs. This theorem has a rich history, originally appearing in [44] and [49]; the shortest proof along the lines outlined in Exercise 4.2 appeared in [14] and in Section 5.2.2 of [15].

48

GREGORY BERKOLAIKO

v1

v2 Figure 5. The operation of merging two vertices into one. Another useful interlacing inequality arises when we join two Neumann vertices to form a single Neumann vertex, see Fig. 5. Since the change in the vertex conditions can be described as imposing another continuity equality2 , we expect the eigenvalues to increase as a result. Lemma 4.3. Let Γ be a quantum graph (not necessarily connected) with two vertices v1 and v2 with Neumann conditions. Modify the graph by merging the two vertices into one, to obtain the graph Γ . Then (27)

λn (Γ) ≤ λn (Γ ) ≤ λn+1 (Γ).

An equality between an eigenvalue of Γ and an eigenvalue of Γ is only possible if the eigenspace of Γ contains an eigenfunction whose values at v1 and v2 are equal or, equivalently, the eigenspace of Γ contains an eigenfunction which additionally satisfy the current conservation condition with respect to the subset Ev1 of the edges incident to vertex v1 in the graph Γ. This lemma and Lemma 4.1 have many applications to counting zeros of a graph’s eigenfunctions, one of which will be presented in Section 7.3. Another application is to the eigenvalue counting which is the subject of the next section. 4.1. An application to eigenvalue counting: Weyl’s law. Let us define the eigenvalue counting function NΓ (k) as the number of eigenvalues of the graph Γ which are smaller than k2 ,   (28) NΓ (k) = # λ ∈ σ(Γ) : λ ≤ k2 . This number is guaranteed to be finite since the spectrum of a quantum graph is discrete and bounded from below (Sec 3.1.1 and Thm 1.4.19 of [15]). We count the √ eigenvalues in terms of k = λ as this is more convenient and can be easily related back to λ. The counting function NΓ (k) grows linearly in k, with the slope proportional to the “size” of the graph. This type of result is known as the “Weyl’s Law”. 2 This is not entirely correct, as the two current conservations conditions are also relaxed into one. However, in terms of quadratic forms, which impose the current conservation automatically, the change is indeed a one-dimensional reduction of the domain of the form.

AN ELEMENTARY INTRODUCTION TO QUANTUM GRAPHS

49

Lemma 4.4. Let Γ be a graph with Neumann or Dirichlet conditions at every vertex. Then L (29) N (k) = k + O(1), π where L = L1 + . . . + L|E| is the total length of the graph’s edges and the remainder term is bounded above and below by constants independent of k. Proof. Let us first consider an interval of length L with Dirichlet conditions. We know the eigenvalues are (πn/L)2 , n ∈ N, therefore we can express NL (k) using the integer part function,   kL NL (k) = , π and thus bound it, L L k − 1 ≤ NL (k) ≤ k. π π Let us now consider the setting of Lemma 4.1: Γ0 is a quantum graph with a vertex v which is endowed with Neumann conditions and Γ∞ is the graph obtained by changing the conditions at v to Dirichlet. Inequality (26) can be rewritten as

(30)

(31)

NΓ∞ (k) ≤ NΓ0 (k) ≤ NΓ∞ (k) + 1.

Starting with a graph Γ, we can change conditions at every vertex to Dirichlet. Applying the interlacing inequality |V | times (or less, if some vertices are already Dirichlet), we get (32)

NΓD (k) ≤ NΓ (k) ≤ NΓD (k) + |V |,

where by ΓD we denote the graph with every vertex conditions changed to Dirichlet. The graph ΓD is just a collection of disjoint intervals, each with Dirichlet conditions at the endpoints. The eigenvalue spectrum of ΓD is the union (in the sense of multisets) of the spectra of the intervals; the counting function is the sum of the interval counting functions. By adding |E| inequalities of type (30), we get L1 + . . . + L|E| L1 + . . . + L|E| k − |E| ≤ NΓD (k) ≤ k, π π leading to the final estimate

(33)

(34)

L L k − |E| ≤ NΓ (k) ≤ k + |V |. π π 

Remark 4.5. The bounds on the remainder term in the Weyl’s law for a graph obtained in the proof are of order |E|. However, numerically it appears that the counting function follows the Weyl’s term much more closely. Getting the optimal bound remains an open question at the time of writing. 5. Secular determinant We will now describe another procedure for deriving an equation for the eigenvalues of a quantum graph. Before we describe the general case, we shall tackle a simple but useful example.

50

GREGORY BERKOLAIKO

L2

L1

Figure 6. A lasso (or lollipop) graph, consisting of an edge and a loop. 5.1. Example: lasso (lollipop) graph. Consider the graph depicted in Fig. 6, an edge attached to a loop. We will impose Neumann conditions at both the attachment point and the endpoint of the edge. Let the edge be parametrized by [0, L1 ] with 0 corresponding to the attachment point and the loop be parametrized by [0, L2 ]. The solution of the eigenvalue equation −f  = k2 f on the edge can be written as f1 (x) = a1 eikx + a¯1 eik(L1 −x) ,

(35)

valid as long as k = 0 (we take care of this special case separately). The Neumann condition at the endpoint leads to f1 (L1 ) = ika1 eikL1 − ika¯1 = 0,

(36) and therefore

a¯1 = a1 eikL1 .

(37)

The solution on the loop we express similarly as (38)

f2 (x) = a2 eikx + a¯2 eik(L2 −x) ,

At the attachment point, the continuity condition reads (39)

a1 + a¯1 eikL1 = a2 + a¯2 eikL2 = a¯2 + a2 eikL2 ,

while the current conservation is (40)

f1 (0) + f2 (0) − f2 (L2 ) = 0,

which, after simplification, yields (41)

a1 − a¯1 eikL1 + a2 − a¯2 eikL2 + a¯2 − a2 eikL2 = 0.

Rearranging equations (39) and (41) we get the system 1 2 2 (42) a1 = − a¯1 eikL1 + a¯2 eikL2 + a2 eikL2 , 3 3 3 2 2 1 (43) a2 = a¯1 eikL1 + a2 eikL2 − a¯2 eikL2 , 3 3 3 2 1 2 (44) a¯2 = a¯1 eikL1 − a2 eikL2 + a¯2 eikL2 . 3 3 3 This system has an interesting “dynamical” interpretation, see Fig. 7. Take, for example, the coefficient a¯1 and interpret it as the amplitude of the plain wave leaving the endpoint vertex in the direction of the loop. Traversing the edge (of length L1 ), it acquires the phase factor of eikL1 . Hitting the vertex, it scatters in three directions: back into the edge with back-scattering amplitude −1/3 contributing to the right-hand side of equation (42), and forward into the two ends of the loop, with forward scattering amplitude 2/3, contributing to equations (43) and (44).

AN ELEMENTARY INTRODUCTION TO QUANTUM GRAPHS

2 ikL1 1e 3 a¯

a¯1 eikL1

a¯1

51

− 13 a¯1 eikL1

2 ikL1 1e 3 a¯

Figure 7. Scattering of a wave on a vertex. The waves leaves the endpoint with amplitude a¯1 , acquires the phase eikL1 by traversing the edge and scatters in three directions on the vertex.

The amplitudes a1 , a2 and a¯2 can be interpreted similarly and undergo the same process. Equations (37) and (42)-(44) can be interpreted as describing a stationary state of such a dynamical process. They can be written as ⎞ ⎛ ikL ⎞⎛ ⎞ ⎛ ⎞ ⎛ 2 2 e 1 0 0 0 0 − 13 a1 a1 3 3 ikL1 ⎟⎜ 0 ⎟ ⎜a¯1 ⎟ ⎜a¯1 ⎟ ⎜1 0 e 0 0 0 0 ⎟⎜ ⎟⎜ ⎟ = ⎜ ⎟. ⎜ (45) 2 2 ⎝0 − 13 ⎠ ⎝ 0 0 eikL2 0 ⎠ ⎝a2 ⎠ ⎝a2 ⎠ 3 3 2 2 a¯2 a¯2 0 − 13 0 0 0 eikL2 3 3 Defining two matrices ⎛ 2 0 − 13 3 ⎜1 0 0 (46) S = ⎜ 2 2 ⎝0 3 3 2 0 − 13 3

2 3

⎞

0 ⎟ ⎟ − 13 ⎠ 2 3

⎛

and

eikL1 ⎜ 0 D=⎜ ⎝ 0 0

0 eikL1 0 0

0 0 eikL2 0

⎞ 0 0 ⎟ ⎟, 0 ⎠ eikL2

we can interpret (45) as saying that the matrix SD(k) has 1 as its eigenvalue. Moreover, each eigenvector of the eigenvalue 1 gives rise to a solution of the eigenvalue equation for the original differential operator, via equations (35) and (38). In other words, the geometric multiplicity of the eigenvalue 1 of the matrix SD(k) is equal to the geometric multiplicity of the eigenvalue k2 of the differential operator d2 − dx 2 . Furthermore, both matrices S and D(k) are unitary, thus for their product SD(k), the algebraic and geometric multiplicity coincide. The only point where this relationship can break down is at k = 0; this is because solutions (35) and (38) are not valid at this point. To summarize, we have the following criterion: the value k2 = 0 is an eigenvalue of the lasso quantum graph if and only if k is a root of the equation Σ(k) := det (I − SD(k)) = 0.

(47)

The multiplicity of k2 in the spectrum of the graph coincides with the multiplicity of k as a root of Σ(k). The function Σ(k) is called the secular determinant of the graph. To finish this example, we mention that explicit evaluation results in (48)

Σ(k) =

1 (z2 − 1)(3z12 z2 − z12 + z2 − 3), 3

where z1 = eikL1 , z2 = eikL2 .

We will understand the reason for the factorization of Σ(k) in Section 6. Note that the value k = 0 is a double root of Σ(k) whereas λ = 0 is a simple eigenvalue of the graph (with constant as the eigenfunction).

52

GREGORY BERKOLAIKO

5.2. Secular determinant for a general Neumann graph. Let us now consider a general vertex with Neumann conditions and d edges incident to it. Writing the solution on j-th edge as fj (x) = aj eikx + a¯j eik(Lj −x) ,

(49)

we get from the vertex conditions (50)

a1 + a¯1 eikL1 = . . . = ad + ad¯eikLd , d 

(51)

j=1

aj −

d 

a¯j eikLj = 0.

j=1

For any n, 1 ≤ n ≤ d, equations (50) imply (52)

d  j=1

aj +

d 

a¯j eikLj = d an + an¯ eikLn .

j=1

Subtracting from this equation (51) and solving for an results in 2 a¯eikLj , d j=1 j d

(53)

an = −an¯ eikLn +

which is a generalization of both (37) and (42)-(44), with d = 1 and d = 3 correspondingly. Now, it is clear how to generalize the matrices S and D(k) in equation (46). Every edge of the graph gives rise to two directed edges which inherit the length of the edge. The two directed edges corresponding to the same undirected edge are reversals of each other. The reversal of a directed edge j is denoted by ¯j. Consider the 2|E|-dimensional complex space, with dimensions indexed by the directed edges. The matrix D(k) is diagonal with entries (54)

D(k)j,j = eikLj ,

while the matrix S has the entries ⎧ 2  ⎪ ⎨ dv − 1, if j = ¯j, (55) Sj  ,j = d2v , if j  follows j and j  = ¯j, ⎪ ⎩ 0, otherwise. The edge j  follows j if the end-vertex of j is the start vertex of j  ; dv denotes the degree of the end-vertex of j. The matrix S is sometimes called the bond scattering matrix. Exercise 5.1. Prove that the matrix S defined by (55) on a graph is unitary. As before, every eigenvector of SD(k) with k = 0 corresponds to an eigenfunction of the graph via equation (49). We therefore have the following theorem. Theorem 5.2. Consider a graph with Neumann conditions at every vertex. The value λ = k2 = 0 is an eigenvalue of the operator −d2 /dx2 if and only if k is the solution of (56)

Σ(k) := det (I − SD(k)) = 0,

where S and D(k) are defined in equations (54) and (55).

AN ELEMENTARY INTRODUCTION TO QUANTUM GRAPHS

53

Exercise 5.3. Incorporate vertices of degree 1 with Dirichlet conditions by showing that back-scattering from such a vertex has amplitude −1 (in contrast with 1 from a Neumann vertex, see the first case of (55) with d = 1). This matches with the classic reflection principle of the wave equation on an interval. Remark 5.4. Another method used to prove the Weyl’s Law (Lemma 4.4) is to observe [30] that the eigenvalues of the matrix SD(k) lie on the unit circle and move in the counter-clockwise direction as k is increased. There are 2|E| of the eigenvalues and their average angular speed can be calculated to be L/|E|. Thus the frequency of an eigenvalue crossing the positive real axis is the average speed times the number of eigenvalues divided by the length of the circle, giving 2|E| × L/|E|/(2π) = L/π. 5.3. Example: secular determinant of star graphs with three edges. Consider again a star graph with 3 edges and Neumann conditions everywhere. Ordering the directed edges as [1, 2, 3, ¯ 1, ¯ 2, ¯3], where j is directed away from the ¯ central vertex and j is directed towards the central vertex, the bond scattering matrix is ⎞ ⎛ 0 0 0 −1/3 2/3 2/3 ⎜0 0 0 2/3 −1/3 2/3 ⎟ ⎟ ⎜ ⎜0 0 0 2/3 2/3 −1/3⎟ ⎟. (57) S=⎜ ⎜1 0 0 0 0 0 ⎟ ⎟ ⎜ ⎝0 1 0 0 0 0 ⎠ 0 0 1 0 0 0 Evaluating the secular determinant Σ(k) from equation (56), we get (58)

1

1 Σ(k) = −z12 z22 z32 − z12 z22 + z22 z32 + z32 z12 + z12 + z22 + z32 +1, where zj = eikLj . 3 3 Dividing this expression by iz1 z2 z3 = ieik(L1 +L2 +L3 ) and using the Euler’s formula we can transform equation (58) to the form (20) obtained previously (up to a constant). According to Exercise 5.3, supplying the star graph with Dirichlet conditions at the endpoints results in the bond scattering matrix ⎞ ⎛ 0 0 0 −1/3 2/3 2/3 ⎜0 0 0 2/3 −1/3 2/3 ⎟ ⎟ ⎜ ⎜0 0 0 2/3 2/3 −1/3⎟ ⎟, ⎜ (59) S=⎜ 0 0 0 0 ⎟ ⎟ ⎜−1 0 ⎝ 0 −1 0 0 0 0 ⎠ 0 0 −1 0 0 0 and the secular determinant

1 2

1 2 2 (60) Σ(k) = z12 z22 z32 − z1 z2 + z22 z32 + z32 z12 − z1 + z22 + z32 + 1. 3 3 Again, dividing it by z1 z2 z3 = eik(L1 +L2 +L3 ) brings it close to the previously obtained form, equation (25), namely to (61)

e−ik(L1 +L2 +L3 ) Σ(k) = sin(kL1 ) sin(kL2 ) cos(kL3 ) + cos(kL1 ) sin(kL2 ) sin(kL3 ) + sin(kL1 ) cos(kL2 ) sin(kL3 ).

54

GREGORY BERKOLAIKO

5.4. Real secular determinant. Theorem 5.2 gives a handy tool for looking for the eigenvalues of a quantum graph. However, Σ(k), as defined by equation (56) is a complex valued function (that needs to be evaluated on the real line — at least when looking for positive eigenvalues). A complex function is equal to zero when both real and imaginary part are equal to zero. It would be nicer to have one equation instead of two. There are several indications that it should be possible. First, the fact that we do have roots of a complex function on the real line3 is atypical; it suggests that the function has some symmetries. Second, in the two examples that we considered in Section 5.3 we succeeded in making the secular determinant real. It turns out that the same method works in general. Lemma 5.5. Let L=

(62)



Le

e∈E

denote the total length of the graph. The analytic function (63)

e−ikL ζ(k) = √ det (I − SD(k)) , det S

is real on the real line and has the same zeros as the secular determinant Σ(k). Proof. We remark that det(D(k)) = e2ikL ,

(64)

and the matrix D(k) is unitary for real k. Denoting the unitary matrix SD(k) by U , and using the identity det(AB) = det A det B, we can rewrite (65)

ζ(k) = (det U )−1/2 det (I − U ) ,

Using the unitarity of U , we now evaluate ζ(k) = (det U )1/2 det (I − U ∗ ) = (det U )1/2 det (U − I) det U ∗ = ζ(k), where we used the identities (66)

I − U ∗ = U U ∗ − U ∗ = (U − I)U ∗

and

det (U ∗ ) = (det U )

−1

. 

Exercise 5.6. Show that for a graph with Neumann conditions at every vertex except for n vertices of degree 1 where Dirichlet conditions are imposed, the determinant of S is det S = (−1)|E|−|V |+n . A detailed study of the secular determinant, including the interpretation of the coefficients of the polynomials like (58) and (60) in terms of special closed paths on the graph, appears in [6]. 3 Since our operator is self-adjoint, the eigenvalues λ = k 2 must be real and therefore the roots of Σ(k) are restricted to real and imaginary axes. Since the operator is bounded from below and unbounded from above, infinitely many of the eigenvalues must lie on the real axis

AN ELEMENTARY INTRODUCTION TO QUANTUM GRAPHS

55

−2

−3

−4

−5

−6

−7

−8

−9

0

5

10

15 k

20

25

30

Figure 8. The difference between the counting function of the numerically found eigenvalue and the Weyl’s estimate. The plot shows that two pairs of eigenvalues were missed, around k = 15.5 and k = 22.

5.5. Remarks on numerical calculation of graph eigenvalues. Expression (63) which is guaranteed to be real for real k allows for a simple way to compute eigenvalues of a quantum graph: find roots of a real function. The naive method is to evaluate the function ζ(k) on a dense enough set of points to get bounds for the eigenvalues, within which the bisection method can be employed. The bisection method can be substituted by a more sophisticated tool such as Brent-Dekker method. For a small graph, the function ζ(k), which is a combination of trigonometric functions of incommensurate frequency, can be derived explicitly, giving access to its derivative and therefore Newton-like methods. However, the initial evaluation may miss a pair of almost-degenerate eigenvalues. To check for this possibility, it is very effective to plot the difference between the counting function for the computed eigenvalues and the Weyl’s Law. The approximate location of the missed eigenvalues (if any) can be seen very clearly, see Fig. 8 for a typical example. See also [48] for another method. A smarter method for the initial step is to use the interlacing inequalities of Lemma 4.1 to bracket the eigenvalues. Unfortunately, this involves computing eigenvalues of another graph, but it may be much simpler, as in the case of the star graphs.

56

GREGORY BERKOLAIKO

Γ

Γeven symm

asymm Γodd Figure 9. The mandarin graph with 3 edges (left) and its decomposition into the even and odd quotients (right). The Dirichlet vertices are distinguished as empty cicrles. 6. Symmetry and isospectrality 6.1. Example: 3-mandarin graph. The 3-mandarin graph is a graph with two vertices and three edges connecting them, see Fig. 9. If one is uncomfortable with multiple edges running between a pair of vertices, extra Neumann vertices of degree two may be placed on some edges, see Section 2.2. Labelling the edges running down by 1, 2 and 3, and using the ordering [1, 2, 3, ¯ 1, ¯ 2, ¯ 3], the matrix S becomes ⎞ ⎛ 0 0 0 −1/3 2/3 2/3 ⎜ 0 0 0 2/3 −1/3 2/3 ⎟ ⎟ ⎜ ⎜ 0 0 0 2/3 2/3 −1/3⎟ ⎟. (67) S=⎜ ⎜−1/3 2/3 2/3 0 0 0 ⎟ ⎟ ⎜ ⎝ 2/3 −1/3 2/3 0 0 0 ⎠ 2/3 2/3 −1/3 0 0 0 Denoting, as before, zj = eikLj , j = 1, 2, 3, we get D(k) = diag(z1 , z2 , z3 , z1 , z2 , z3 ). The secular determinant simplifies to   1 1 (68) Σ(k) = −z1 z2 z3 − (z1 z2 + z2 z3 + z3 z1 ) + (z1 + z2 + z3 ) + 1 3 3   1 1 × z1 z2 z3 − (z1 z2 + z2 z3 + z3 z1 ) − (z1 + z2 + z3 ) + 1 , 3 3 where the factors coincide with the secular determinants we obtained for the star graphs with three edges and Neumann and Dirichlet conditions at the endpoints, correspondingly, see equations (58) and (60), modulo the change zj2 ↔ zj . The reason for this factorization is the symmetry. The graph, as shown in Fig. 9, is symmetric with respect to the vertical (up-down) reflection. This means that the reflected eigenfunction is still an eigenfunction. Denote by Rf the reflected version of an eigenfunction f . By linearity, fe = f + Rf and fo = f − Rf satisfy the eigenvalue equation with the same λ. They may be identically zero, but not both at the same time, since (fe + fo )/2 = f ≡ 0. And under the action of R, fe is even and fo is odd: (69)

Rfe = fe ,

Rfo = −fo .

AN ELEMENTARY INTRODUCTION TO QUANTUM GRAPHS

57

Using this idea one can show that every eigenspace has a basis of eigenfunctions each of which is either even or odd. Indeed, starting with an arbitrary basis of size m, we produce 2m even/odd combinations from them. These combinations span the eigenspace (since f = (fe + fo )/2), so it remains to choose m linearly independent vectors among them. Consider an odd eigenfunction on the mandarin graph. Let xm be the midpoint of the first edge. This point is fixed under the action of reflection, thus (Rf )(xm ) = f (xm ). On the other hand, Rf = −f , therefore f (xm ) = −f (xm ) = 0. The same applies for the midpoint of every edge. Therefore every odd eigenfunction is the eigenfunction of the half of the mandarin with Dirichlet conditions at the midpoints. The converse is also true: starting from an eigenfunction of the half with Dirichlet boundary, we obtain an eigenfunction of the full graph by planting two copies and multiplying one of them by −1. A similar reasoning for even eigenfunctions of the mandarin graph shows that they are in one-to-one correspondence with the eigenfunctions of the half of the graph with Neumann boundary. The half of a 3-mandarin is a star graph with three edges, see Fig. 9; the edge lengths are half of the mandarin’s. It turns out that this symmetry of the mandarin graph (and the corresponding factorization (68)) leads to interesting anomalies in the number of of zeros of the eigenfunctions. This subject will be visited again in Section 7.4.

6.2. Quotient graphs. To put the observations of the previous section on a more formal footing, the Hilbert space H(Γ) of functions on the mandarin graph (that are sufficiently smooth and satisfy correct vertex conditions) can be decomposed into the direct sum of two orthogonal subspaces Ho and He , which are invariant with respect to operator −d2 /dx2 acting on the graph. Restrictions of the operator to these subspaces can be identified with this operator acting on two smaller graphs, a star with Dirichlet ends and a star with Neumann ends. Such a smaller graph, together with its vertex conditions, is called quotient graph and was introduced by Band, Parzanchevski and Ben-Shach [9, 42] to study isospectrality. To produce a quotient graph one needs a quantum graph, a group of symmetries (not necessarily the largest possible) and a representation of this group. We will not describe the full procedure, which can be learned from the already mentioned papers and the forthcoming article [3]. Instead we will briefly describe its consequences and explain one particular construction that leads to a pair of graphs with identical spectra (i.e. an isospectral pair). Let Γ be a graph with a finite group of symmetries G. To each irreducible representation ρ there corresponds a subspace Hρ of the Hilbert space H(Γ). This is the subspace of functions that transform according to the representation ρ when acted upon by the symmetries from G. In some sources, such functions are called equivariant vectors; the subspace Hρ is called the isotypic component. The subspaces corresponding to different irreducible representations are orthogonal, the space H(Γ) is a direct sum of Hρ over all irreps of the group G. If ρ has dimension d > 1, then every eigenvalue of the Hamiltonian restricted to Hρ has multiplicity which is a multiple of d. Moreover, the secular determinant Σ(k) factorizes into factors that correspond to the irreps ρ of G. Each factor is raised to a power which is the dimension of the corresponding ρ (hence the degeneracy of

58

GREGORY BERKOLAIKO

3

b

a

a 4

b 1

a

b 2

Figure 10. A tetrahedron graph which is invariant under rotation by 2π/3 and horizontal reflection. Its group of symmetry is S3 : an arbitrary permutation of vertices 1, 2 and 3. the corresponding eigenvalue), (70)

ΣΓ (k) =

dim(ρ)   Σρ (k) . irreps ρ

Example 6.1 (from [3]). Consider the tetrahedron graph from Fig. 10. This graph is symmetric under reflection and rotation by 2π/3. The corresponding group of symmetries is S3 , the group of permutations of 3 objects; in this case it can be thought of as permuting the vertices number 1, 2 and 3. The group S3 has three irreducible representations, trivial, alternating and 2d. The first two are one-dimensional, while the latter is 2-dimensional (as suggested by its name). The secular determinant Σ(k) has the corresponding factorization 1 (za − 1)(3za zb2 − zb2 + za − 3)(3za2 zb2 + 2za zb2 − za2 + zb2 − 2za − 3)2 . (71) Σ(k) = 27 Note that it may happen that for some ρ the subspace Hρ is trivial. In this case the corresponding factor Σρ (k) is 1 and there are no eigenvalues corresponding to this representation. 6.3. Example: dihedral graphs. The present example originates from [10] and is the origin of the theory of [9, 42]. Consider the graph on Fig. 11(a). It has the symmetries of the square, thus its full group of symmetries is the dihedral group of degree four D4 (and order eight, hence another notation, D8 , which causes much confusion). We will first consider the subgroup generated by the vertical (up-down) reflection τ1 and the horizontal (left-right) reflection τ2 . One of the irreducible representations of this subgroup is (72)

τ1 → (1),

τ2 → (−1),

where (1) (correspondingly (−1)) stands for the operation of multiplication by 1 (correspondingly −1). To understand the functions that transform according to representation (72), we choose as the fundamental domain (a subgraph that covers the entire graph under the action of the subgroup) the top right quarter of the graph shown in Fig. 11(b), and plant there a function F . Applying the vertical reflection τ1 , we find that in the bottom right quarter of the graph, the function must be equal to F multiplied by 1 and suitably reflected. Applying the horizontal reflection τ2 , we

AN ELEMENTARY INTRODUCTION TO QUANTUM GRAPHS

2a

b

a

2c

b

59

b

2c

b a

τ1

τ2 (a)

(b)

Figure 11. (a) A graph with symmetry group D4 , the dihedral group of degree 4; (b) symmetry axes of the first subgroup and a choice of fundamental domain. a

b

2c

b

a

b

D a

2c

b a N

(a)

(b)

Figure 12. Constructing a quotient graph with respect to the representation (72). find that the left side of the graph must be populated by the copies of the function F multiplied by −1, see Fig. 12(a). We have already discovered in Section 6.1 that at the point where F meets −F , the function must vanish (i.e. have the Dirichlet condition), whereas at the point where F meets F the condition must be Neumann. Thus we obtain the quotient graph in Fig. 12(c). Let us now repeat the same procedure but choose the diagonal reflections d1 and d2 as the generators of the subgroup, together with the representation (73)

d1 → (1),

d2 → (−1).

60

GREGORY BERKOLAIKO

b

2a

b

c

c b

2a

b

N

D c

c

N

D

d1

d2 (a)

(b)

Figure 13. Constructing a quotient graph with respect to the representation (73). 1

2 1

3

3

2

4

5

4

Figure 14. Numbering of edges of the two isospectral graph. Starting with a fundamental domain, reflecting and multiplying it as prescribed by (73), we fill the entire graph as in Fig. 13(a). The corresponding conditions on the fundamental domain are shown in Fig. 13(b). The most interesting feature of the two quotient subgraphs is that they are isospectral, i.e. have exactly the same eigenvalues. This can be shown by a transplantation procedure a la Buser [18, 19], which describes a unitary equivalence of the corresponding operators. Another possibility (admittedly more tedious) is to find the secular determinants of the two graphs. To this end, number the edges of the two graphs as shown in Fig. 14. Starting with the graph with a cycle (which we will now call the dihedral graph, and its partner the dihedral tree), order its edges as [1, 2, 3, 4, ¯1, ¯2, ¯3, ¯4]. Then ⎞ ⎛ 0 0 0 0 −1 0 0 0 ⎜ 2/3 0 0 0 0 −1/3 2/3 0 ⎟ ⎟ ⎜ ⎜ 2/3 0 0 0 0 2/3 −1/3 0 ⎟ ⎟ ⎜ ⎜ 0 2/3 2/3 0 0 0 0 −1/3⎟ ⎟, ⎜ (74) S=⎜ 0 0 0 0 2/3 2/3 0 ⎟ ⎟ ⎜−1/3 ⎜ 0 −1/3 2/3 0 0 0 0 2/3 ⎟ ⎟ ⎜ ⎝ 0 0 0 2/3 ⎠ 2/3 −1/3 0 0 0 0 0 1 0 0 0 0

AN ELEMENTARY INTRODUCTION TO QUANTUM GRAPHS

61

and D(k) = diag(za , zb2 , zc2 , za , za , zb2 , zc2 , za ), where zs = eiks , s = a, b, c. Evaluating the determinant, we obtain (75)

8 4

1 4 4 1 4 za zb + zb4 zc4 + zc4 za4 + za − 1 zb2 zc2 − za + zb4 + zc4 + 1. Σ(k) = −za4 zb4 zc4 + 9 9 9 Repeating the procedure for the dihedral tree, which has 10 × 10 matrices S and D(k), we arrive to the secular determinant which is again given by expression (75). It is easy to check that 0 is eigenvalue of neither graph and therefore the graphs are isospectral. The glimpse of the underlying reason for the isospectrality can be seen in the secular determinant of the original graph (that of Fig. 11(a)). As predicted by (70), it factorizes:



2 Σ(k) = 9za4 zb4 zc4 − za4 zb4 + zb4 zc4 + zc4 za4 − 8 za4 − 1 zb2 zc2 + za4 + zb4 + zc4 − 1

× 3za2 zb2 zc2 + za2 zb2 + zb2 zc2 + zc2 za2 − za2 − zb2 − zc2 − 3

× 3za2 zb2 zc2 + za2 zb2 − zb2 zc2 + zc2 za2 − za2 + zb2 + zc2 + 3

× 3za2 zb2 zc2 − za2 zb2 + zb2 zc2 − zc2 za2 − za2 + zb2 + zc2 − 3

× 3za2 zb2 zc2 − za2 zb2 − zb2 zc2 − zc2 za2 − za2 − zb2 − zc2 + 3 , up to an overall factor. The last four terms correspond to the four one-dimensional representations of the group D4 , while the first one, squared, corresponds to the two-dimensional representation. The term inside the square also coincides with the secular determinant of the two quotient graphs. This suggests that although we constructed them as quotients by the (one-dimensional) representations of two different symmetry subgroups, they are both realizable as quotients by the twodimensional representation of the whole group. This is indeed shown in [9], together with a general criterion for isospectrality involving induction of representations from subgroups to the whole group. 7. Magnetic Schr¨ odinger operator and nodal count Magnetic field is introduced into the Schr¨odinger equation via the magnetic vector potential usually denoted A(x). In our case, A(x) is a one-dimensional vector: it changes sign if the direction of the edge is reversed. The Schr¨odinger eigenvalue equation then takes the form 2  d − iA(x) f (x) + V (x)f (x) = k2 f (x), (76) − dx where the square is interpreted in the sense of operators, i.e. 2     d d − iA(x) f (x) = − iA(x) f  (x) − iA(x)f (x) dx dx (77)    = f (x) − i A(x)f (x) − iA(x)f  (x) − A2 (x)f (x). To understand the “strange” definition of A(x) a little better, consider the equation  2 d (78) − − iA f (x) = k2 f (x), dx

62

GREGORY BERKOLAIKO

where A is a “normal” constant, on the interval [0, L]. Solutions of this equation, e±ikx+iAx , under the change of variables x → L − x become solutions of a slightly different equation, 2  d + iA f (L − x) = k2 f (L − x). (79) − dx But this variable change is just a reparametrization of the interval in terms of the distance from the other end and should not affect the laws of physics. Letting A to be the “one-form” which transforms according to A(L − x) = −A(x) addresses this problem. To understand the effect of the magnetic potential A(x) on the secular determinant derived in Section 5.2, we write the solution of (76) with V ≡ 0 on the j-th edge in the form (80)

fj (x) = aj eikx+i

x 0

A(x)

ik(Lj −x)+i

+ a¯j e

x

Lj

A(x)

.

Taking A(x) to be the constant vector on the edge (as we will see, this results in no loss of generality), we obtain a somewhat more manageable form (81)

fj (x) = aj ei(k+A)x + a¯j ei(k−A)(Lj −x) ,

which also highlights the fact that the wave travelling in the negative direction (the second term) feels the magnetic potential as −A. The vertex conditions also change with the introduction of the magnetic field. It is convenient to define the operator d − iA(x), dx so that the Schr¨odinger operator can be written as −D2 + V and the Neumann vertex conditions become (82)

(83) (84)

D=

f (x) is continuous,  Df (v) = 0, e∼v

which is to be compared to (2)-(3). Applying these conditions to the solution form (81), we get (85) (86)

a1 + a¯j eikL1 +iA¯1 L1 = . . . = ad + ad¯eikLd +iAd¯Ld , d  j=1

aj −

d 

a¯j eikLj +iA¯j Lj = 0,

j=1

which assumes that all edges attached to v are oriented outward and also introduces the notation A¯j = −Aj in agreement with the nature of A. In fact the same answer would be obtained had we started with equation (80) instead, and defined  Lj  0 1 1 (87) Aj = A(x), A¯j = A(x). Lj 0 Lj Lj Either way, the only change in the definition of the secular determinant Σ(k) = det(I − SD(k)) is in the diagonal matrix D(k) which becomes (88)

D(k)b,b = eikLb +i



b

A(x)

.

AN ELEMENTARY INTRODUCTION TO QUANTUM GRAPHS

63

As we just saw, the precise nature of A(x) is not important; the only quantity that enters Σ(k) is the integral of A(x). In fact, even more is true: only the values of the integral of A(x) around the cycles of the graph are important. Definition 7.1. The flux of the magnetic field given by A(x) through an oriented cycle γ of Γ is the integral of A(x) over the cycle,  (89) αγ = A(x). γ

Theorem 7.2. Consider two operators on the same quantum graph Γ differing only in their magnetic potentials A1 (x) and A2 (x). Then they are unitarily equivalent if their fluxes through every cycle on Γ are equal modulo 2π. In fact, it is enough to compute the fluxes through a fundamental set of β = |E| − |V | + 1 cycles. A magnetic perturbation of a Schr¨ odinger operator on a graph Γ is thus fully specified by a set of β numbers between −π and π. The proof of this simple theorem can be found in, for example, [34] or [15], Section 2.6. 7.1. Example: Dihedral graph. To study the influence of the magnetic perturbation on the eigenvalues of the dihedral graph, we put the magnetic flux α through its loop by using D(k) = diag(za , eiα zb2 , zc2 , za , za , e−iα zb2 , zc2 , za ) together with the matrix S as given in (74). This results in the secular determinant

1 4 4 za zb + zb4 zc4 + zc4 za4 Σ(k) = −za4 zb4 zc4 + 9 (90)

8 1 4 za + zb4 + zc4 + 1. + cos(α) za4 − 1 zb2 zc2 − 9 9 We plot the first few eigenvalues λn (α) of the resulting graph as a function of the magnetic flux α, Fig. 15. Since the flux is only important modulo 2π, we plot the eigenvalues over the interval [−π, π]. We observe that the eigenvalues are symmetric (even) functions with respect to α = 0: this can be seen directly by complex conjugating equation (76): if f (x) is an eigenfunction with potential A(x) then f (x) is an eigenfunction with potential −A(x) with the same eigenvalue. Sometimes the eigenvalue has a minimum at α = 0 and sometimes a maximum. These events do not alternate as in the Hill’s equation4 ; there is no strict periodicity there, but we will be able to extract some information about them. 7.2. Magnetic–nodal connection. Let us for a moment come back to the case of no magnetic field, α = 0, and study the eigenfunctions of the dihedral graph, see Fig. 16. In Table 1 we give the results of a numerical calculation for one choice of the graph’s lengths. We list the sequential number of the eigenvalue, starting from 1, its value, the number of zeros of the corresponding eigenfunction, and, somewhat arbitrarily, a description of the behavior of the eigenvalue of the magnetic dihedral graph (see Fig. 15), namely whether it has a maximum or a minimum at α = 0. A careful reader will observe a curious pattern: the magnetic eigenvalue λn (α) appears to have a minimum whenever the number of zeros of the eigenfunction, 4 Hill’s differential equation [39] is a Schr¨ odinger equation on R1 with periodic potential; its Floquet reduction is equivalent to a quantum graph in the shape of a circle with a magnetic flux α through it. It is an important result of Hill’s equation theory that the minima and maxima of λn (α) at α = 0 alternate with n.

64

GREGORY BERKOLAIKO

3

2.5

λ1/2

2

1.5

1

0.5

0

−3

−2

−1

0 α

1

2

3

Figure 15. The square roots of the first few eigenvalues of the dihedral graph as function of the magnetic flux through the cycle. √ The lengths are a = π, b = 1, c = 2. 1.5

1.5

1

1

0.5

0.5

0 −0.5 −5

0 0

5

−0.5 −5

0

5

Figure 16. Location of zeros of two eigenfunctions of the dihedral graph. Eigenfunctions number n = 2 and n = 3 are displayed. The zero at the left endpoint is due to the Dirichlet condition; such zeros are not counted when we report the nodal count of an eigenfunction.

which we will denote φn is less than n and a maximum whenever φn = n. The connection between the two was discovered in [13] for discrete Laplacians, an alternative proof given in [20] and an extension to quantum graphs proved in [17]. Before we formulate this result we need to recall some definitions from multivariate calculus. Definition 7.3. Let F (x1 , . . . , xβ ) be a twice differentiable function of β variables.

AN ELEMENTARY INTRODUCTION TO QUANTUM GRAPHS

n λn (0) # zeros λn (α) at 0

1 0.1708 0 min

2 0.5359 1 min

3 0.9126 3 max

4 1.2294 4 max

5 1.3398 4 min

6 1.6225 5 min

7 1.9877 7 max

8 2.3349 8 max

65

9 2.5680 9 max

Table 1. Eigenvalues of a dihedral graph with the lengths a = π, √ b = 1 and c = 2, the number of zeros of the corresponding eigenfunction, and the behavior of the magnetic eigenvalue around the point of no magnetic field.

• The point x∗ = (x∗1 , . . . , x∗β ) is a critical point of the function F if all first derivatives of F vanish at x∗ , ∂F ∗ (x , . . . , x∗β ) = 0. ∂xj 1 • The Hessian matrix of F at x∗ is the matrix of all second derivatives of F, β  2 ∂ F . Hess(F ) = ∂xj ∂xk j,k=1 The matrix Hess(F ) is symmetric therefore all of its eigenvalues are real. • A critical point is called non-degenerate if the Hessian evaluated at the point is a non-degenerate matrix (has no zero eigenvalues). • The Morse index of a critical point is the number of the negative eigenvalues of its Hessian. The Second Derivative Test says that if the Morse index of a non-degenerate critical point is zero, the point is a local minimum; if it equals the dimension of the space, the point is a maximum. Theorem 7.4 (Berkolaiko–Weyand [17]). Let Γα be a quantum graph with magnetic Schr¨ odinger operator characterized by magnetic fluxes α = (α1 , . . . , αβ ) through a fundamental set of cycles. Let ψ be the eigenfunction of Γ0 that corresponds to a simple eigenvalue λ = λn (Γ0 ) of the graph with zero magnetic field. We assume that ψ is non-zero on vertices of the graph. Then α = (0, . . . , 0) is a non-degenerate critical point of the function λn (α) := λn (Γα ) and its Morse index is equal to the nodal surplus φ − (n − 1), where φ is the number of internal zeros of ψ on Γ. As a corollary we get a simple but useful bound on the number of zeros of n-th eigenfunction. Corollary 7.5. Let Γ be a quantum graph with a Schr¨ odinger operator, λn be its n-th eigenvalue and ψn the corresponding eigenfunction. If λn is simple and ψn does not vanish on vertices, the number of zeros φn of the function ψn is a well-defined quantity which satisfies (91)

0 ≤ φn − (n − 1) ≤ β = |E| − |V | + 1.

Remark 7.6. The result of Corollary 7.5 actually predates Theorem 7.4 by a considerable time. It goes back to the results on trees [44, 49], their extension to β > 0 for nodal domains [12] and to number of zeros [4]. But it also follows easily from Theorem 7.4 due to the Morse index being an integer between 0 and the dimension of the space of parameters.

66

GREGORY BERKOLAIKO

Exercise 7.7. Let Γ be a quantum graph with a Schr¨odinger operator, λ be its simple eigenvalue and ψ the corresponding eigenfunction which does not vanish on vertices. Prove that ψ has an even number of zeros on every cycle of the graph Γ. 7.3. Nodal count of the dihedral graph. It turns out there is an explicit formula for the number of zeros of the n-th eigenfunction of the dihedral graph. This formula was discovered by Aronovitch, Band, Oren and Smilansky [8]. The discovery was remarkable as no explicit formula for the eigenvalues of the dihedral graph is known. The formula was proved in [4] using a fairly involved construction of opening the graph by attaching two phase-synchronized infinite leads. It would be nice to be able to prove this result from Theorem 7.4 directly, but we know of no such proof. Instead, we will give here a relatively simple proof showcasing the power of the interlacing results of Section 4.1, Corollary 7.5 and the simple observation that the number of zeros on any cycle of the graph must be even. This proof has not previously appeared in any other source. Theorem 7.8 (Conjectured in [8], first proved in [4]). Let n-th eigenvalue of a dihedral graph be simple and the corresponding eigenfunction not vanish at the vertices (except at the Dirichlet vertex). Then the number of zeros of the eigenfunction is   b+c n . (92) φn = n − mod2 a+b+c Remark 7.9. It can be shown that the hypothesis of the theorem is satisfied for all eigenvalues and eigenfunction for a generic choice of lengths a, b and c [16]. It is interesting to observe that the sequence of nodal counts {φn } contains the information about the relative length of the central loop in the graph. Before we can prove the theorem, we need two auxiliary lemmas. Lemma 7.10. Let λn denote the eigenvalue of the dihedral graph with parame˜ n denote the ordered numbers from the set ters a, b and c and let λ   π  π n1 n2 (93) σ ˜ := ∪ . 2a 2(b + c) n1 ∈N n2 ∈N Then (94)

˜ n ≤ λn+1 , λn ≤ λ

n = 1, 2, . . .

Proof. Denote by σ the spectrum of the dihedral graph. Consider the following sequence of modifications of the graph Γ consisting of two copies of the dihedral graph, Fig. 17(a). First we impose the Dirichlet condition on the left attachment point of one copy and the right attachment point of the other copy, obtaining the ˆ see Fig. 17(b). Second we separate the right edge of the first dihedral graph graph Γ, and the left edge of the second dihedral graph, imposing Neumann conditions on the newly formed vertices, see Fig. 17(c). The first modification is covered by two applications of Lemma 4.1, while the second by two applications of Lemma 4.3 (in reverse). Observe that the spectrum of the graph Γ is σ ∪ σ (in the sense of multisets) and the spectrum of the final graph Γ is {0} ∪ σ ˜ ∪σ ˜ . We will denote the eigenvalues

AN ELEMENTARY INTRODUCTION TO QUANTUM GRAPHS

67

Figure 17. Modifications of the graph Γ, consisting of two copies of the dihedral graph, leading to interlacing inequalities (95) and (96). The Dirichlet vertices are distinguished as empty cicrles. of the three stages by sn , sˆn and s˜n correspondingly. Because of the degeneracies in the spectrum and the interlacing Lemmas, we have the following inequalities, 0 < s2 = s1 ≤ sˆ1 ≤ sˆ2 ≤ s4 = s3 ≤ sˆ3 ≤ sˆ4 ≤ . . .

(95) and (96)

s˜1 = 0 < sˆ1 ≤ s˜3 = s˜2 ≤ sˆ2 ≤ sˆ3 ≤ s˜5 = s˜4 ≤ sˆ4 ≤ . . .

Combining the two, we obtain s˜1 = 0 < s2 = s1 ≤ s˜3 = s˜2 ≤ s4 = s3 ≤ s˜5 = s˜4 ≤ . . . ˜ 1 , s 4 = λ2 , Now the claim of the Lemma follows by observing that s1 = λ2 , s˜2 = λ ˜  s˜4 = λ2 etc.    n1  with positive α and β. Lemma 7.11. Let S1 = α n1 ∈N and S2 = nβ2 n2 ∈N

If (97)

 # {S1 ∪ S2 ≤ λ} = n − 1

then

# {S1 ≤ λ} =

 α n . α+β

Remark 7.12. This Lemma was offered as an exercise to the attendees of 2015 S´eminaire de Math´ematiques Sup´erieures “Geometric and Computational Spectral Theory” at CRM, University of Montreal. At that time, the author did not know a good proof. Several students, including Luc Petiard, Arseny Rayko, Lise Turner and Saskia Voß submitted proofs. The proof below is based on Arseny’s proof with elements borrowed from other attendees’ versions. Proof. Let (98)

# {S1 ≤ λ} = λα =: N1 ,

# {S2 ≤ λ} = λβ =: N2 ,

then N1 + N2 = n − 1 and also N1 ≤ λα < N1 + 1, N2 ≤ λβ < N2 + 1. Multiplying the first inequality by β and the second by α and going through the middle term, we get βN1 < α(N2 + 1),

αN2 < β(N1 + 1).

68

GREGORY BERKOLAIKO

Adding αN1 to the first and α(N1 + 1) to the second inequalities we get (α + β)N1 < α(N1 + N2 + 1) < (α + β)(N1 + 1). Dividing by α + β and using N1 + N2 + 1 = n we get α n < N1 + 1, N1 < α+β 

which implies the desired result.

Proof of Theorem 7.8. The eigenvalue λn of the dihedral graph has n − 1 eigenvalues from σ ˜ below !it, so by" Lemma 7.11, applied with α = 2a/π and π a , which β = 2(b + c)/π it has N1 = a+b+c n eigenvalues from the sequence 2a are precisely the eigenvalues of the side edges of the dihedral graph but with the Dirichlet conditions at the attachment points (i.e. the eigenvalues of the signle edge ˆ components of the graph Γ). By Sturm’s theorem, the eigenfunction of the dihedral graph corresponding to λn has N1 zeros on the side edges. Therefore φn − N1 zeros lie on the cycle and this quantity must be even. But we know from Corollary 7.5 that φn is either n − 1 or n. To make φn − N1 even we must choose   b+c n . φn = n − mod2 (n − N1 ) = n − mod2 a+b+c  7.4. Nodal count of a mandarin graph. When the bounds of equation (91) were discovered, the following question arose: for an arbitrary graph, do all allowed numbers in the range 0, 1, . . . , β appear as the nodal surplus φn − (n − 1) of some eigenfunction? It turns out the answer is no. Theorem 7.13 (Band–Berkolaiko–Weyand [5]). Let Γ be a mandarin graph of 2 vertices connected by d edges. If the eigenvalue number n > 1, is simple and the corresponding eigenfunction does not vanish on vertices, then the nodal surplus σn satisfies (99)

1 ≤ σn ≤ β − 1.

In particular, the nodal surplus of the 3-mandarin graph is equal to 1 for all eigenfunctions except the first, whose nodal surplus is always 0. It is interesting to combine this observation with the result of Theorem 7.4. The Morse index is never equal to 0 (for n > 1) or to the dimension β, therefore the point α = 0 is never a minimum or a maximum. But the space of all possible magnetic fluxes is a β-dimensional torus, which is compact. Therefore the extrema must be achieved somewhere! There are other “standard” points where the eigenvalue λn (α) always have a critical point. It is relatively straightforward to extend Theorem 7.4 to points (b1 , . . . , bβ ), where each bi is 0 or π. But it is also straightforward to extend Theorem 7.13 (see [5] for both extensions) which will show that those critical point are also never extrema. The missing extrema turn out to be achieved at singular points of λn (α), see Fig. 18 for an example. Such conical singularities are sometimes called the Dirac points and their appearance in 3-mandarin graphs is intimately related to their

AN ELEMENTARY INTRODUCTION TO QUANTUM GRAPHS

69

Figure 18. The first four eigenvalues of the mandarin graph as functions of two magnetic fluxes α1 and α2 . The surfaces can be seen to be touching at conical singularities, the so-called Dirac points. appearance in the dispersion relation of graphene [38]. This connection, however, lies outside the scope of this article. 8. Concluding remarks There are many interesting topics within the area of spectral theory of metric graphs that we did not cover. A very partial list (heavily biased towards personal preferences of the author) is as follows: • Generic properties of eigenfunctions and eigenvalues. Many results, such as Theorem 7.4 require the eigenvalue to be simple and the eigenfunction to be non-vanishing on vertices. These properties are generic with respect to small perturbations of the edge lengths (that need to break all symmetries of the graph). Results in this direction can be found in [27], [21](for eigenvalues) and [16] (for both eigenvalues and eigenfunctions). • Ergodic flow on the secular manifold. Barra and Gaspard [11] introduced an interpretation of the secular determinant equation as an ergodic flow piercing a compact manifold (more precisely, an algebraic variety) given by solutions of an equation like (75) on a torus. This interpretation leads to many surprising and very general results, including those of [2] and [21]. • Spectral theory of infinite periodic graphs yields a fruitful connection to the theory of compact graphs with magnetic field. The background is covered in Chapter 4 of [15]; a sample of results can be found in [23, 31], [38], and [2, 5].

70

GREGORY BERKOLAIKO

• Graphs make a very interesting setting to study resonances. Work in this direction has attracted many researchers from across the field [37], [50], [22], [35], [28], [24–26]. • There is an ongoing effort to find bounds on the graph eigenvalues (especially the lowest non-zero ones) in terms of the geometric properties of the graph, see [1, 7, 33] for the latest results and references.

References [1] S. Ariturk. Eigenvalue estimates on quantum graphs. preprint arXiv:1609.07471, 2016. [2] R. Band and G. Berkolaiko. Universality of the momentum band density of periodic networks. Phys. Rev. Lett., 111:130404, Sep 2013. [3] R. Band, G. Berkolaiko, C. Joyner, and W. Liu. Symmetry of quantum graphs and factorization of the spectral determinant. in preparation, 2016. [4] R. Band, G. Berkolaiko, and U. Smilansky. Dynamics of nodal points and the nodal count on a family of quantum graphs. Annales Henri Poincare, 13(1):145–184, 2012. [5] R. Band, G. Berkolaiko, and T. Weyand. Anomalous nodal count and singularities in the dispersion relation of honeycomb graphs. J. Math. Phys., 56(12):122111, 2015. [6] R. Band, J. M. Harrison, and C. H. Joyner. Finite pseudo orbit expansions for spectral quantities of quantum graphs. J. Phys. A, 45(32):325204, 19, 2012. [7] R. Band and G. L´evy. Quantum graphs which optimize the spectral gap. preprint arXiv:1608.00520, 2016. [8] R. Band, I. Oren, and U. Smilansky. Nodal domains on graphs—how to count them and why? In Analysis on graphs and its applications, volume 77 of Proc. Sympos. Pure Math., pages 5–27. Amer. Math. Soc., Providence, RI, 2008. [9] R. Band, O. Parzanchevski, and G. Ben-Shach. The isospectral fruits of representation theory: quantum graphs and drums. J. Phys. A, 42(17):175202, 42, 2009. [10] R. Band, T. Shapira, and U. Smilansky. Nodal domains on isospectral quantum graphs: the resolution of isospectrality? J. Phys. A, 39(45):13999–14014, 2006. [11] F. Barra and P. Gaspard. On the level spacing distribution in quantum graphs. J. Statist. Phys., 101(1–2):283–319, 2000. [12] G. Berkolaiko. A lower bound for nodal count on discrete and metric graphs. Comm. Math. Phys., 278(3):803–819, 2008. [13] G. Berkolaiko. Nodal count of graph eigenfunctions via magnetic perturbation. Anal. PDE, 6:1213–1233, 2013. preprint arXiv:1110.5373. [14] G. Berkolaiko and P. Kuchment. Dependence of the spectrum of a quantum graph on vertex conditions and edge lengths. In Spectral Geometry, volume 84 of Proceedings of Symposia in Pure Mathematics. American Math. Soc., 2012. preprint arXiv:1008.0369. [15] G. Berkolaiko and P. Kuchment. Introduction to Quantum Graphs, volume 186 of Mathematical Surveys and Monographs. AMS, 2013. [16] G. Berkolaiko and W. Liu. Simplicity of eigenvalues and non-vanishing of eigenfunctions of a quantum graph. J. Math. Anal. Appl., 445(1):803–818, 2017. preprint arXiv:1601.06225. [17] G. Berkolaiko and T. Weyand. Stability of eigenvalues of quantum graphs with respect to magnetic perturbation and the nodal count of the eigenfunctions. Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 372(2007):20120522, 17, 2014. [18] P. Buser. Isospectral Riemann surfaces. Ann. Inst. Fourier (Grenoble), 36(2):167–192, 1986. [19] P. Buser, J. Conway, P. Doyle, and K.-D. Semmler. Some planar isospectral domains. Internat. Math. Res. Notices, (9):391–400, 1994. [20] Y. Colin de Verdi` ere. Magnetic interpretation of the nodal defect on graphs. Anal. PDE, 6:1235–1242, 2013. preprint arXiv:1201.1110. [21] Y. Colin de Verdi` ere. Semi-classical measures on quantum graphs and the Gauß map of the determinant manifold. Annales Henri Poincar´ e, 16(2):347–364, 2015. also arXiv:1311.5449. [22] E.B. Davies and A. Pushnitski. Non-Weyl resonance asymptotics for quantum graphs. Analysis & PDE, 4:729–756, 2011. [23] P. Exner, P. Kuchment, and B. Winn. On the location of spectral edges in Z-periodic media. J. Phys. A, 43(47):474022, 8, 2010.

AN ELEMENTARY INTRODUCTION TO QUANTUM GRAPHS

71

[24] P. Exner and J. Lipovsk´ y. Equivalence of resolvent and scattering resonances on quantum graphs. In Adventures in mathematical physics, volume 447 of Contemp. Math., pages 73–81. Amer. Math. Soc., Providence, RI, 2007. [25] P. Exner and J. Lipovsk´ y. Resonances from perturbations of quantum graphs with rationally related edges. J. Phys. A, 43(10):105301, 21, 2010. [26] P. Exner and J. Lipovsk´ y. Non-Weyl resonance asymptotics for quantum graphs in a magnetic field. Phys. Lett. A, 375(4):805–807, 2011. [27] L. Friedlander. Genericity of simple eigenvalues for a metric graph. Israel J. Math., 146:149– 156, 2005. [28] S. Gnutzmann, H. Schanz, and U. Smilansky. Topological resonances in scattering on networks (graphs). Phys. Rev. Lett., 110:094101, Feb 2013. [29] S. Gnutzmann and U. Smilansky. Quantum graphs: Applications to quantum chaos and universal spectral statistics. Adv. Phys., 55(5–6):527–625, 2006. [30] Daniel Grieser. Monotone unitary families. Proc. Amer. Math. Soc., 141(3):997–1005, 2013. preprint arXiv:0711.2869. [31] J. M. Harrison, P. Kuchment, A. Sobolev, and B. Winn. On occurrence of spectral edges for periodic operators inside the Brillouin zone. J. Phys. A, 40(27):7597–7618, 2007. [32] R. A. Horn and C. R. Johnson. Matrix analysis. Cambridge University Press, Cambridge, second edition, 2013. [33] J. B. Kennedy, P. Kurasov, G. Malenov´ a, and D. Mugnolo. On the spectral gap of a quantum graph. Ann. Henri Poincar´ e, 17(9):2439–2473, 2016. [34] V. Kostrykin and R. Schrader. Quantum wires with magnetic fluxes. Comm. Math. Phys., 237(1-2):161–179, 2003. Dedicated to Rudolf Haag. [35] T. Kottos and H. Schanz. Statistical properties of resonance widths for open quantum graphs. Waves Random Media, 14(1):S91–S105, 2004. Special section on quantum graphs. [36] T. Kottos and U. Smilansky. Quantum chaos on graphs. Phys. Rev. Lett., 79(24):4794–4797, 1997. [37] T. Kottos and U. Smilansky. Chaotic scattering on graphs. Phys. Rev. Lett., 85(5):968–971, 2000. [38] P. Kuchment and O. Post. On the spectra of carbon nano-structures. Comm. Math. Phys., 275(3):805–826, 2007. [39] Wilhelm Magnus and Stanley Winkler. Hill’s equation. Interscience Tracts in Pure and Applied Mathematics, No. 20. Interscience Publishers John Wiley & Sons New York-LondonSydney, 1966. [40] D. Mugnolo. Semigroup methods for evolution equations on networks. Understanding Complex Systems. Springer, Cham, 2014. [41] S. Nicaise. Some results on spectral theory over networks, applied to nerve impulse transmission. In Orthogonal polynomials and applications (Bar-le-Duc, 1984), volume 1171 of Lecture Notes in Math., pages 532–541. Springer, Berlin, 1985. [42] O. Parzanchevski and R. Band. Linear representations and isospectrality with boundary conditions. J. Geom. Anal., 20(2):439–471, 2010. [43] L. Pauling. The diamagnetic anisotropy of aromatic molecules. J. Chem. Phys., 4(10):673– 677, 1936. [44] Yu. V. Pokorny˘ı, V. L. Pryadiev, and A. Al-Obe˘ıd. On the oscillation of the spectrum of a boundary value problem on a graph. Mat. Zametki, 60(3):468–470, 1996. [45] O. Post. Spectral Analysis on Graph-like Spaces, volume 2039 of Lecture Notes in Mathematics. Springer Verlag, Berlin, 2012. [46] J.-P. Roth. Spectre du laplacien sur un graphe. C. R. Acad. Sci. Paris S´ er. I Math., 296(19):793–795, 1983. [47] K. Ruedenberg and C. W. Scherr. Free-electron network model for conjugated systems. i. J. Chem. Phys., 21(9):1565–1581, 1953. [48] H. Schanz. A relation between the bond scattering matrix and the spectral counting function for quantum graphs. In Quantum graphs and their applications, volume 415 of Contemp. Math., pages 269–282. Amer. Math. Soc., Providence, RI, 2006. [49] P. Schapotschnikow. Eigenvalue and nodal properties on quantum graph trees. Waves Random Complex Media, 16(3):167–178, 2006. [50] C. Texier and G. Montambaux. Scattering theory on graphs. J. Phys. A, 34(47):10307–10326, 2001.

72

GREGORY BERKOLAIKO

[51] J. von Below. A characteristic equation associated to an eigenvalue problem on c2 -networks. Linear Algebra Appl., 71:309–325, 1985. Department of Mathematics, Texas A&M University, College Station, Texas 778433368

Contemporary Mathematics Volume 700, 2017 http://dx.doi.org/10.1090/conm/700/14183

A free boundary approach to the Faber-Krahn inequality Dorin Bucur and Pedro Freitas Abstract. The purpose of this survey article is to present a complete, comprehensive, proof of the Faber-Krahn inequality for the Dirichlet Laplacian from the perspective of free boundary problems. The proof is of a purely variational nature, proceeding along the following steps: proof of the existence of a domain which minimizes the first eigenvalue among all domains of prescribed volume, proof of (partial) regularity of the optimal domain and usage of a reflection argument in order to prove radiality. As a consequence, no rearrangement arguments are used and, although not the simplest of proofs of this statement, it has the advantage of its adaptability to study the symmetry properties of higher eigenvalues and also to other isoperimetric inequalities (Faber-Krahn or Saint-Venant) involving Robin boundary conditions.

1. Introduction It was conjectured by Rayleigh in 1877 that among all fixed membranes with a given area, the ball would minimize the first eigenvalue [26]. This assertion was proved by Faber and Krahn in the nineteen twenties using a rearrangement technique, and since then several proofs have appeared in the literature. The purpose of this survey article is to give a comprehensive proof of the Rayleigh–Faber–Krahn inequality in the context of free boundary problems. It may be argued that, in spite of the fact that this is a basic result in the theory, no more proofs of this statement are needed, particularly if they are not simpler than other existing proofs. However, it is our belief that because the proof we propose here is somehow of a different nature, has applications to other problems, and follows a natural sequence of intuitive (but mathematically not easy) steps, it deserves some attention. These steps are the following: existence of an optimal shape, proof of its (partial) regularity, and use of its optimality in order to conclude that the optimizer must be radially symmetric. One has in mind the incomplete proof of the isoperimetic inequality by Steiner in 1836 (see [28] and the survey article [4]). Steiner proved (in two–dimensional space) that if a smooth domain is not the ball, then there must exist another domain with the same area but lower perimeter. Of course, the missing step is precisely the proof of the existence of a sufficiently smooth set which minimizes the perimeter among all domains of fixed area! Moreover, existence alone is not enough, This article surveys the lectures given by the first author at Universit´ e de Montr´ eal in June 2015 in the framework of the “Geometric and Computational Spectral Geometry” summer school. P.F. was partially supported by FCT (Portugal) through project PTDC/MAT-CAL/4334/2014. c 2017 American Mathematical Society

73

74

DORIN BUCUR AND PEDRO FREITAS

since in order to use his argument, Steiner implicitly needed some smoothness. In fact, Steiner provided no fewer than five proofs and some of these have since been completed by other mathematicians from Carath´eodory (1909) to De Giorgi (1957), for instance – see [4] and [15], respectively. While Carath´eodory’s approach relied on proving convergence of the sequence of domains used in the proof, it is De Giorgi’s approach which is more relevant to us here, as it consisted in proving that Steiner’s argument could be carried out in the class where existence holds, precisely the sets which have a finite perimeter defined using functions of bounded variation. We also refer to the open problem on the isoperimetric inequality for the buckling load of a clamped plate, where the same couple of questions remain unsolved. Willms and Weinberger (see [19, Theorem 11.3.7]) noticed that if a smooth, simply connected set would minimise the buckling load among all domains with fixed area (in two dimensions of the space), then necessarily the minimizer is the ball. In the proof we give of the Faber–Krahn inequality, we use only variational arguments developed in the context of free boundary problems (see for instance [1,6, 29]). Our purpose is to present the basic tools which allow to complete the sequence : existence-regularity-radiality and give the lecturer the fundamental ideas hidden behind this scheme. This approach has the advantage to be adaptable to other isoperimetric inequalities where rearrangement or mass transport techniques fail. We have in mind isoperimetric inequalities involving Robin boundary conditions, as for example the minimization of the first Robin eigenvalue or the maximization of the Robin torsional rigidity among domains of fixed volume [9, 10]. In the Robin case, the techniques to prove existence-regularity-radiality steps are definitely more involved and require finer analysis arguments on special functions of bounded variations, developed in the framework of free discontinuity problems. In particular, it may even happen that the optimizer is no longer the ball, as was recently shown by David Krejˇciˇr´ık and the second author [16]. Let Ω ⊆ RN be an open (or quasi-open) set of finite measure, but otherwise with no assumptions either on smoothness or on boundedness. Thanks to the compact embedding H01 (Ω) → L2 (Ω), the spectrum of Dirichlet-Laplacian on Ω is discrete and consists on a sequence of eigenvalues which can be ordered (counting multiplicities) as 0 < λ1 (Ω) ≤ λ2 (Ω) ≤ · · · ≤ λk (Ω) ≤ . . . → +∞. If Ω is open, H01 (Ω) is the classical Sobolev space consisting on the completion of C0∞ (Ω) for the L2 -norm of the gradients. The case where Ω is a quasi-open set will be considered in more detail in the next section. The Faber–Krahn inequality asserts that λ1 (Ω) ≥ λ1 (B), where B is the ball having the same measure as Ω. Equality holds if and only if Ω is a ball (up to a negligible set of points, which may be expressed in terms of capacity). Roughly speaking, the proof will be split into the following four steps: Step 1. Prove that there exists a domain Ω∗ which minimizes the first Dirichlet eigenvalue among all domains of fixed volume, i.e. ∃Ω∗ ⊆ RN , |Ω∗ | = m, ∀Ω ⊆ RN , |Ω| = m λ1 (Ω∗ ) ≤ λ1 (Ω).

A FREE BOUNDARY APPROACH TO THE FABER-KRAHN INEQUALITY

75

At this first step, the existence result is carried in the family of quasi-open sets. Proving existence among open sets requires special attention. Step 2. Prove that the set Ω∗ is open and connected. This result can be seen, by itself, as a first regularity result. No smoothness of the boundary is required by the next steps. Step 3. Use a reflection argument in order to deduce that Ω∗ is radially symmetric, hence it is one annulus. Step 4. Prove that among all annuli of prescribed volume, the ball gives the lowest first eigenvalue. It is clear that the proof of the Faber–Krahn inequality we propose in this note is not simple. The proof of Steps 1 and 2 requires some techniques developed in the context of free boundary problems, that we intend to present in a simplified way. Step 3 is done using an idea that may be traced to Steiner’s original manuscript [4, 28] and was named by P´ olya the method of alternative symmetrisation [25, Section 16.13]. Following an idea of Payne and Weinberger, P´olya used this method to prove that the ball is the unique domain for which equality holds in the FaberKrahn inequality. Alternative symmetrisation has also been used in the context of the minimization of integral functionals in H 1 -Sobolev spaces [22, 23]. We show in detail how this can be adapted to shape optimization in a functional context. Step 4 consists in a one dimensional analysis argument for which a precise computation can be carried out. We note that if in Step 2 one is able to prove the smoothness of the boundary of ∂Ω∗ , by the Hadamard argument relying on the vanishing of the shape derivative, one can extract an overdetermined boundary condition. Precisely, one would get that |∇u∗ | is constant on ∂Ω∗ and come to a Serrin problem which could be solved by moving plane techniques (see the pioneering paper [27]). Nevertheless, proving the smoothness of ∂Ω∗ requires definitely more involved regularity techniques, as references [1, 12, 29] show. Proving only the openness of Ω∗ is quite elementary, as shown in the sequel (see [12, 18, 29] for a complete analysis of the regularity question). In the last section of the paper we discuss briefly the problem of minimizing the k-th eigenvalue of the Dirichlet Laplacian among all quasi-open sets of prescribed measure. We present some new results where, in particular, we focus on the symmetry of a minimizer of the k-th eigenvalue and show how our arguments may also be used there. We point out that although one might expect minimizers of this type of spectral problems to always have some symmetry, say at least for the reflection with respect to one hyperplane, recent numerical evidence on this problem has raised the issue of whether or not this is actually true [2]. More precisely, the planar domain found (numerically) in that paper which minimises the thirteenth Dirichlet eigenvalue of the Laplace operator does not have any such symmetry and, furthermore, if a restriction is imposed enforcing that the optimiser does have some symmetry, namely invariance under reflection with respect to an axis, the resulting value of the optimal eigenvalue is worse than the unrestricted case. Although a proof of such a statement, if true, is likely to be quite elusive, there has been some independent numerical confirmation of this observation given in [3, 5]. In any case, these results do beg the question as to what is the minimum symmetry which we

76

DORIN BUCUR AND PEDRO FREITAS

can guarantee these optimal domains will have. In this sense, the results presented here are a first step in this direction. 2. Setting the variational framework Solving Step 1 requires to set up a very large framework for the existence question. As one cannot impose a priori constraints on the competing sets, in order to achieve existence the largest class of admissible shapes should be considered. This is a classical principle in shape optimization. The most natural framework where the Dirichlet-Laplacian operator is well defined is the family of quasi-open sets. More precisely, Ω ⊆ RN is called quasiopen if for all ε > 0 there exists an open set Uε such that the set Ω ∪ Uε is open, where   |∇u|2 + |u|2 dx : u ∈ H 1 (RN ), u ≥ 1 a.e. on Uε < ε. cap(Uε ) := inf RN

Above, cap(Uε ) denotes the capacity of the set Uε . Roughly speaking, quasi-open sets are precisely the level sets {˜ u > 0} of the “most continuous” representatives of Sobolev functions u ∈ H 1 (RN ), i.e. the ones given by  u(y)dy Br (x) . (1) u ˜(x) = lim r→0 |Br (x)| The limit above exists for all points except a set of capacity zero. If Ω is an open set, the Sobolev space H01 (Ω) is defined as the completion of the space of C ∞ -functions with compact support in Ω, with respect to the norm uL2 (Ω) + ∇uL2 (Ω,RN ) . If Ω is a quasi-open set, then the Sobolev space H01 (Ω) associated to the quasi-open set Ω is defined as a subspace of H 1 (RN ), by # H01 (Ω) = H01 (Ω ∪ Uε ). ε>0

The definition above does not depend on the choice of the sets Uε . Morover, if Ω is open, both definitions above lead to the same space. If Ω is a quasi-open set of finite measure, the spectrum of the Dirichlet-Laplacian on Ω is defined in the same way as for open sets, being the inverse of the spectrum of the compact, positive, self-adjoint resolvent operator RΩ : L2 (Ω) → L2 (Ω), RΩ f = u, where u ∈ H01 (Ω) satisfies   (2) ∀ϕ ∈ H01 (Ω) ∇u∇ϕdx = f ϕdx. Ω

In particular

Ω

 λ1 (Ω) :=

min

u∈H01 (Ω)

Ω

|∇u|2 dx . |u|2 dx

Ω

The minimizing function solves the equation  −Δu = λ1 (Ω)u in Ω u= 0 on ∂Ω in the weak sense (2). Clearly, if Ω is open we find the classical definition of the first eigenvalue.

A FREE BOUNDARY APPROACH TO THE FABER-KRAHN INEQUALITY

77

The following inequality, which holds for every quasi-open set of finite measure (see for instance [14, Example 2.1.8]), will be useful for the local study of the optimal sets N

u∞ ≤ CN λ1 (Ω) 4 uL2 .

(3)

The problem we intend to solve is the following: given m > 0, prove that the unique solution of min{λ1 (Ω) : Ω ⊆ RN quasi-open, |Ω| = m}

(4)

is the ball. Uniqueness is understood up to a set of zero capacity, which is precisely the size of a set for which one cannot distinguish the “precise” values of a Sobolev function. Since for every t > 0, one has λ1 (tΩ) = t12 λ1 (Ω), there exists C > 0 such that problem (4) is equivalent to min{λ1 (Ω) + C|Ω| : Ω ⊆ RN quasi-open}.

(5)

Indeed, let us denote α(m) = inf{λ1 (Ω) : Ω ⊆ RN quasi-open, |Ω| = m}. From the Sobolev inequality, we know that α(m) is strictly positive. Indeed, we have for every u ∈ H01 (Ω)    NN−1 2N |∇u2 |dx ≥ CN |u| N −1 , RN

RN

and so from Cauchy-Schwarz on the left hand side and H¨ older on the right hand side 1 2uL2 (RN ) ∇uL2 (RN ) ≥ CN u2L2 (RN ) 1 , |Ω| N which gives that λ1 (Ω) ≥

2 CN 2

|Ω| N

.

Moreover, (6)

α(tN m) = t−2 α(m).

On the other hand, for every C, the minimizer Ω in (5) satisifies 2λ1 (Ω) = CN |Ω|, as a consequence of the fact that the function t → λ1 (tΩ) + C|tΩ| attains its minimum at t = 1. Consequently, 2α(m) = CN m, and so problems (4) and (5) are equivalent, as soon as (7)

m=

 2α(1)  NN+2 CN

.

The equivalence of (4) and (5) proved above plays a crucial role in the proof of the existence of an open minimizer, in Step 2 below. The fact that the constraint is penalized gives more freedom to perform local perturbations.

78

DORIN BUCUR AND PEDRO FREITAS

3. Proof of the Faber-Krahn inequality Proposition 3.1 (Step 1). Problem (4) has a solution, i.e. there exists a quasiopen set Ω∗ such that |Ω∗ | = m and for every quasi-open set Ω ⊆ RN , |Ω| = m we have λ1 (Ω∗ ) ≤ λ1 (Ω). Proof. The idea is very simple and is based on the concentration-compactness principle of P.L. Lions [21]. Assume that (Ωn )n is a minimizing sequence and let us denote (un )n a sequence of L2 -normalized, non negative, associated first eigenfunctions. Then the sequence (un )n ⊆ H 1 (RN ) is bounded. Since H 1 (RN ) is not compactly embedded in L2 (RN ), for a subsequence (still denoted using the same index) one of the three possibilities below occurs: i) compactness: ∃yn ∈ RN such that un (· + yn ) −→ u strongly in L2 (RN ) and weakly in H 1 (RN ). ii) dichotomy: there exists α ∈ (0, 1) and two sequences {u1n }, {u2n } ∈ H 1 (RN ), supp u1n ∪ supp u2n ⊆ supp un , such that  RN

and (8)

un − u1n − u2n L2 (RN ) → 0 ,  1 2 |un | dx → α |u2n |2 dx → 1 − α , RN 1 dist(supp un , supp u2n )

→ +∞ ,

 lim inf n→∞

RN

|∇un |2 − |∇u1n |2 − |∇u2n |2 dx ≥ 0 .

iii) vanishing: for every 0 < R < ∞  lim sup n→+∞ y∈RN

u2n dx = 0 .

B(y,R)

Situations ii) and iii) cannot occur to a minimizing sequence. Indeed, situation ii) leads to searching the minimizer in a class of domains of measure strictly lower than m. First, we notice that the measures of the sets {u1n > 0}, {u2n > 0} cannot vanish as n → +∞. One of the two sequences ({u1n > 0})n , ({u2n > 0})n has also to be minimizing for problem (4), in view of the algebraic inequality a b  a+b ≥ min , , (9) c+d c d for positive numbers a, b, c, d. This allows to select (for a suitable subsequence, still denoted with the same index) either ({u1n > 0})n or ({u2n > 0})n as minimizing sequence of measure not larger than m − ε, for some ε > 0, in contradiction to the strict monotonicity (6). Situation iii) can be excluded by an argument due to Lieb [20] which asserts that if iii) occurs then λ1 (Ωn ) → +∞, in contradiction with the choice of a minimizing sequence (see also [11]). For the sake of the clearness, we shall provide a short argument to prove this fact. Without restricting the generality, we can assume that λ1 (Ωn ) ≤ M1 , for some M1 > 0 independent on n. As un ≥ 0, we get −Δun ≤ λ1 (Ωn )un in D (RN ),

A FREE BOUNDARY APPROACH TO THE FABER-KRAHN INEQUALITY

79

and so (from (3)) N

−Δun ≤ CN λ1 (Ωn ) 4 +1 ≤ M2 . This implies that for every x0 ∈ RN , the function un + M2

|x − x0 |2 2N

is subharmonic in RN . By direct computation we get that for every ball Bx (r)  udx M2 r 2 Bx (r) un (x) ≤ + . |Bx (r)| 2N Choosing first r such that every x ∈ RN

M2 r 2 2N

=

1 √ , 3 m

 udx Bx (r)

|Bx (r)|



and then n large enough such that for

1 ≤ √ , 3 m

4 , in contradiction with the L2 -normalization of un . 9 Ωn Only situation i) can occur and this leads to the existence of an optimal domain which is a quasi-open set. Indeed, we consider the set Ω := {u > 0}. Then, if we choose the representative defined by (1), the set Ω is quasi-open and has a measure less than or equal to m. This latter assertion is a consequence of the strong L2 convergence of un which has as a consequence that (at least for a subsequence) 1Ω (x) ≤ lim inf n→+∞ 1Ωn (x) a.e. x ∈ RN . Moreover, we have   |∇u|2 dx |∇un |2 dx Ω Ω ≤ lim inf  = lim inf λ1 (Ωn ). λ1 (Ω) ≤  n→∞ n→∞ |u|2 dx |un |2 dx we get that

u2n dx ≤

Ω

Ω ∗

If necessary, taking a suitable dilation Ω = tΩ, for some t ≥ 1 such that |Ω∗ | = m and using the rescaling properties of λ1 , we conclude that Ω∗ is a minimizing domain.  Proposition 3.2 (Step 2). The optimal set Ω∗ is open and connected. Proof. Assuming one knows that Ω∗ is open, the proof of the connectedness is immediate. Indeed, assume Ω∗ = Ω1 ∪ Ω2 with Ω1 , Ω2 open, disjoint and nonempty. Then λ1 (Ω∗ ) is either equal to λ1 (Ω1 ) or to λ1 (Ω2 ), hence one could find a set, say Ω1 , with the first Dirichlet eigenvalue equal to λ1 (Ω∗ ) but with measure strictly less than m. This is in contradiction to the strict monotonicity (6). The proof of the openness has a technical issue and is obtained using a local perturbation argument developed by Alt and Caffarelli in the context of free boundary problems [1, Lemma 3.2]. The idea is to prove that u∗ is at least continuous, so that Ω∗ = {u∗ > 0} is an open set. We refer the reader to [29, Theorem 3.2] and [12, Proposition 1.1] for a complete description of the method below.

80

DORIN BUCUR AND PEDRO FREITAS

Let R > 0 and x0 ∈ RN . We introduce the harmonic extension of u∗ in BR (x0 ), by u ˜(x) = u∗ (x) in Ω \ BR (x0 ), Δ˜ u(x) = 0 in BR (x0 ). In view of (5), we have

 |∇˜ u|2 dx λ1 (Ω ) + C|Ω | ≤  2 + C|{˜ u > 0}|, |˜ u| dx ∗

or

 Ω∗

∗

 ∗ 2

|∇u | dx ≤

Ω∗ \BR (x0 )

1+



|∇u|2 dx +

BR (x0 )



|∇˜ u|2 dx

BR (x0 ) (|˜ u|2 − |u∗ |2 )dx

+ C|BR (x0 ) \ Ω∗ |.

 From the L bound of the eigenfunction, we get | BR (x0 ) (|˜ u|2 − |u∗ |2 )dx| ≤ M3 RN so that after an easy computation and for R smaller than a suitable constant independent of the point, we get   |∇u∗ |2 dx ≤ |∇˜ u|2 dx + M4 RN . ∞

BR (x0 )

BR (x0 )

Since u ˜ is harmonic and equal to u∗ on ∂BR , we get    |∇u∗ |2 dx − |∇˜ u|2 dx = BR (x0 )

BR (x0 )

|∇(u∗ − u ˜)|2 dx.

BR (x0 )

Consequently, for every 0 < r < R we have    |∇u∗ |2 dx ≤ |∇(u∗ − u ˜)|2 dx + Br (x0 )

Br (x0 )



|∇(u∗ − u ˜)|2 dx +

≤ BR (x0 )

 r N  R

|∇˜ u|2 dx ≤

Br (x0 )

|∇˜ u|2 dx.

BR (x0 )

The last inequality is due to the fact that |∇˜ u|2 is subharmonic in BR (x0 ), as a consequence of the harmonicity of u ˜ in BR (x0 ). Finally, ∀x0 ∈ RN , ∀0 < r < R ≤ R0 we have   r N  ∗ 2 N |∇u | dx ≤ M4 R + |∇u∗ |2 dx. (10) R Br (x0 ) BR (x0 ) This inequality implies in a classical way that u∗ is H¨older continuous. Indeed, one can prove in a first step that  |∇u∗ |2 dx ≤ M5 r N −1 . (11) Br (x0 )

The proof of the passage from (10) to (11) is classical, see for instance [17, Lemma 2.1, Chapter 3]. The fact that u∗ is H¨older continuous of order 12 in RN is a consequence of the Dirichlet growth theorem (see [17, Theorem 1.1, Chapter3]). For a detailed proof of all steps, we refer to [29] and the references therein.  Analyzing the optimality of Ω∗ by comparison with the sets Ω∗ \ B R (x0 ), one can deduce that Ω∗ satisfies an inner density property and therefore has to be bounded. Comparing with the set {u∗ > ε}, with vanishing ε, one can find an upper bound of the generalized perimeter. Further analysis, leads to the smoothness of

A FREE BOUNDARY APPROACH TO THE FABER-KRAHN INEQUALITY

81

the boundary. The precise smoothness depends on the dimension, but we do not insist on this point since openness alone is enough for our purposes. Proposition 3.3 (Step 3). The optimal set Ω∗ has radial symmetry. Proof. As mentioned in the Introduction, the idea to prove radial symmetry for minimizers of integral functionals in H 1 (RN ) has been used before (see, for instance, [22, 23]), while its usage in a geometrical setting appears in Steiner’s arguments [4, 28]. Roughly speaking, if Ω∗ is a minimizer then one can cut it by a hyperplane H in two pieces of equal measure. Up to a translation of Ω∗ , we can assume that H is given by the equation x1 = 0. Then both the left and right parts together with their respective reflections are admissible (they have the correct measure) and are also minimizers. Indeed, let u be a non-zero first eigenfunction on Ω∗ . We put the indices l, r to the corresponding quantities on the left, right parts of Ω∗ , respectively. We denote Ωl = Ω ∩ {x ∈ RN : x1 ≤ 0}.

Figure 1. A region and one of its symmetrised counterparts with respect to a hyperplane Then,

 Ω∗ l

λ1 (Ω∗ ) =

 |∇ul |2 dx + Ω∗ |∇ur |2 dx   r . u2 dx + Ω∗ u2r dx Ω∗ l r

l

Using the algebraic inequality (9), we get (assuming for instance the left part is minimal)  |∇ul |2 dx Ω∗ ∗ λ1 (Ω ) ≥ l . u2 dx Ω∗ l l

Defining the reflection transformation R : R introduce the reflected domain

N

→ RN R(x) = (−x1 , x2 , . . . , xN ), we

Ωl := Ω∗l ∪ RΩ∗l , together with the reflected test function u(x) = ul (x) if x1 ≤ 0 and u(x) = ul (Rx) if x1 ≥ 0. Then |Ωl | = m, ul ∈ H01 (Ωl ) (this is immediate using the density of C0∞ -function in H 1 ) and we get   |∇ul |2 dx |∇u|2 dx Ω∗ ∗ l Ωl  λ1 (Ω ) ≥  = ≥ λ(Ωl ). 2 dx 2 dx u |u| ∗ l l Ω Ω l

82

DORIN BUCUR AND PEDRO FREITAS

Relying on the minimality of Ω∗ and on the inequalities above, we conclude that Ωl is also a minimizer and that u is an eigenfunction on Ωl . Finally, this means that ul has two analytic extensions: one in the open set Ωr which is ur and another one in RΩl which is u. Using the maximum principle, there cannot be a point of the complement of Ωr where ur is vanishing which is interior for RΩl , and vice-versa. Finally, this implies that Ωr = RΩl and so Ω∗ is symmetric with respect to H. We continue the procedure with the hyperplanes {xi = 0}, for i = 2, . . . , N . At this point, up to a translation, we know that the optimal domain Ω∗ is symmetric with respect to all the hyperplanes {xi = 0}, for i = 1, . . . , N . Now, we can continue the procedure with an arbitrary hyperplane passing through the origin (without any translation), since such a hyperplane divides Ω∗ in two pieces of equal  measure. We conclude that Ω∗ has to be radially symmetric. Let us point out that the openness of the optimal domain Ω∗ was necessary in the proof above when using the analyticity of the eigenfunction. If the optimal set was only quasi-open, the analyticity would have failed, and so our reflection argument could not have been used. Let us also say that in the proof above we implicitly obtained that the minimizer is radially symmetric, not only the domain Ω∗ . This observation goes in the direction of [22, 23] where radial symmetry is proved for the minimizers of integral functionals, and can be successfully used in other situations. Proposition 3.4 (Step 4). The optimal set Ω∗ is the ball. Proof. Since the optimal set is connected, using the proposition above, we know it is an annulus. Precisely, for some t ≥ 0 this annulus can be written Ω∗ = A(0, t, r(t)) := {x ∈ RN : t < |x| < r(t)}, where ωN −1 (r N (t) − tN ) = m. In order to prove that the solution of the FaberKrahn inequality is the ball, it is enough to study the mapping t → λ1 (A(0, t, r(t))) and to prove it is increasing. Using the the shape derivative formula for λ1 (and denoting u an L2 -normalized eigenfunction on A(0, t, r(t)), we get    ∂u 2  ∂u 2 d tN −1 N −1 λ1 (A(0, t, r(t)))) = dH − N −1 dHN −1 . dt r (t) ∂B(0,r(t)) ∂n ∂B(0,t) ∂n d λ1 (A(0, t, r(t)))) > 0 is equivalent to Since u is radially symmetric, proving that dt  2  2 proving that |u (t)| > |u (r(t))| (in this notation u depends only on the radius). The equation satisfied by the radial function u is

−u (s) −

N −1  u (s) = λ1 u(s), on (t, r(t)), s u(t) = u(r(t)) = 0.

Denoting v(s) = |u (s)|2 , we get that 2(N − 1)  |u (s)|2 − 2λ1 u (s)u(s), s and summing between t and r(t) we get  2(N − 1) r(t)  |u (s)|2 ds < 0. v(r(t)) − v(t) = − s t v  (s) = 2u (s)u (s) = −

A FREE BOUNDARY APPROACH TO THE FABER-KRAHN INEQUALITY

83

The last inequality is obvious since u is not constant, hence the mapping t → λ1 (A(0, t, r(t))) is strictly increasing on (0, +∞). Consequently the ball, corresponding to t = 0, is the global minimizer.  4. Further remarks: higher order eigenvalues For k ≥ 2 one can also consider the isoperimetric problem (12)

min{λk (Ω) : Ω ⊆ RN quasi-open, |Ω| = m}.

When k = 2 the minimizer consists of two equal and disjoint balls of measure m/2, this being a direct consequence of the inequality for the first eigenvalue and which was already considered by Krahn. For k ≥ 3, and apart from numerical results (see, for instance, [2]) only few facts are known: • A solution to problem (12) exists (let us call it Ω∗k ), it is a bounded set and has finite perimeter (see [6] and [24]). • There exists an eigefunction u∗k of the optimal set Ω∗k , corresponding to the k-th eigenvalue λk (Ω∗k ) which is a Lipschitz function (see [8]). Note that this information does not imply that the optimal set Ω∗k is open, because Ω∗k contains both the open set {u∗k = 0} and the nodal set {u∗k = 0}, which has a structure, not yet completely understood. Relying on the reflection argument, one can get some more information on the symmetry of Ω∗k , depending on the dimension of the space. Roughly speaking, the larger the space dimension is, the more symmetric must the minimizer be. Assume that u1 , . . . , uk are L2 -normalized eigenfunctions corresponding to λ1 (Ω∗k ), . . . , λk (Ω∗k ) such that   ∀i, j = 1, . . . , k, i = j, ui uj dx = 0, ∇ui ∇uj = 0. Problem (12) can be re-written as

 |∇u|2 dx , min1 max Ω min |Ω|=m Sk ⊂H0 (Ω) u∈Sk |u|2 dx Ω

where Sk denotes any subspace of dimension k. Assume H is a hyperplane splitting Ω∗k into Ωl and Ωr such that |Ωl | = |Ωr |,  ui uj dx = 0, ∀i, j = 1, . . . , k, i = j, Ωl



∇ui ∇uj dx = 0,

∀i, j = 1, . . . , k, i = j, Ωl

∀i = 1, . . . , k − 1,

 |∇ui |2 dx Ωl = λi (Ω∗k ). 2 dx |u | i Ωl

Notice that the number of constraints equals k2 . Assuming that   |∇uk |2 dx |∇uk |2 dx Ωl Ωr ≤ , |uk |2 dx |uk |2 dx Ωl Ωr

84

DORIN BUCUR AND PEDRO FREITAS

and reflecting Ωl together with the functions u1 |Ωl , . . . , uk |Ωl , we get  2 l |∇uk | dx l l ∗ λk (Ω ∪ RΩ ) ≤ max{λk−1 (Ωk ), Ω } ≤ λk (Ω∗k ). 2 dx |u | k l Ω Consequently, the set Ωl ∪ RΩl is also a minimizer and is symmetric with respect to H. If Ω∗k were open, then we could conclude that 1Ω∗k = 1Ωl ∪RΩl . Following the same arguments as [23, Theorem 2], we would get that the set Ω∗k would be symmetric with respect to an affine subspace of dimension k2 − 1. As Ω∗k is not known to be open, we can only assert the existence of a minimizer which has N − (k2 − 1) hyperplanes of symmetry. In either case, we see that it is only for k equal to one that full symmetry may be obtained in this way. Remark 4.1. Similar existence questions can be raised for more general functions of eigenvalues. Let F : Rk → R be a lower semicontinuous function, non decreasing in each variable. Then, the following problem   (13) min F (λ1 (Ω), . . . , λk (Ω)) : Ω ⊆ RN quasi-open, |Ω| = m . has a solution. If moreover F is strictly increasing in at least one variable, then any optimal set has to be bounded with finite perimeter. We refer the reader to [6, 7, 13, 24]. A key argument is contained in the following surgery result, for which we refer the reader to [7]. Surgery result for the spectrum: for every K > 0, there exists D, C > 0 depending ˜ ⊂ RN only on K and the dimension N , such that for every open (quasi-open) set Ω there exists an open (quasi-open, respectively) set Ω satisfying ˜ • the measure of Ω equals the measure of Ω, 1 ˜ C|Ω| NN−1 }, • diam (Ω) ≤ D|Ω| N and P er(Ω) ≤ min{P er(Ω), ˜ ≤ K|Ω| ˜ − N2 , then λi (Ω) ≤ λi (Ω) ˜ for all • if for some k ∈ N it holds λk (Ω) i = 1, . . . , k. The perimeter of a non-smooth set has to be understood in a weak sense (see [7] for the details). Acknowledgements The authors would like to thank the anonymous referee for a careful reading of the manuscript, and Mark Ashbaugh for pointing out P´ olya’s work [25] to us. References [1] H. W. Alt and L. A. Caffarelli, Existence and regularity for a minimum problem with free boundary, J. Reine Angew. Math. 325 (1981), 105–144. MR618549 [2] Pedro R. S. Antunes and Pedro Freitas, Numerical optimization of low eigenvalues of the Dirichlet and Neumann Laplacians, J. Optim. Theory Appl. 154 (2012), no. 1, 235–257, DOI 10.1007/s10957-011-9983-3. MR2931377 [3] Berger, A. Optimisation du spectre du Laplacien avec conditions de Dirichlet et Neumann atel, May 2015. dans R2 et R3 , PhD thesis, University of Neuchˆ [4] Viktor Bl˚ asj¨ o, The isoperimetric problem, Amer. Math. Monthly 112 (2005), no. 6, 526–566, DOI 10.2307/30037526. MR2142606 [5] Bogosel, B. Optimisation de formes et probl` emes spectraux, PhD thesis, University of Grenoble, December 2015. [6] Dorin Bucur, Minimization of the k-th eigenvalue of the Dirichlet Laplacian, Arch. Ration. Mech. Anal. 206 (2012), no. 3, 1073–1083, DOI 10.1007/s00205-012-0561-0. MR2989451

A FREE BOUNDARY APPROACH TO THE FABER-KRAHN INEQUALITY

85

[7] Dorin Bucur and Dario Mazzoleni, A surgery result for the spectrum of the Dirichlet Laplacian, SIAM J. Math. Anal. 47 (2015), no. 6, 4451–4466, DOI 10.1137/140992448. MR3427044 [8] Dorin Bucur, Dario Mazzoleni, Aldo Pratelli, and Bozhidar Velichkov, Lipschitz regularity of the eigenfunctions on optimal domains, Arch. Ration. Mech. Anal. 216 (2015), no. 1, 117–151, DOI 10.1007/s00205-014-0801-6. MR3305655 [9] Dorin Bucur and Alessandro Giacomini, Faber-Krahn inequalities for the Robin-Laplacian: a free discontinuity approach, Arch. Ration. Mech. Anal. 218 (2015), no. 2, 757–824, DOI 10.1007/s00205-015-0872-z. MR3375539 [10] Dorin Bucur and Alessandro Giacomini, The Saint-Venant inequality for the Laplace operator with Robin boundary conditions, Milan J. Math. 83 (2015), no. 2, 327–343, DOI 10.1007/s00032-015-0243-0. MR3412286 [11] Dorin Bucur and Nicolas Varchon, Global minimizing domains for the first eigenvalue of an elliptic operator with non-constant coefficients, Electron. J. Differential Equations (2000), No. 36, 10. MR1764712 [12] Tanguy Brian¸con and Jimmy Lamboley, Regularity of the optimal shape for the first eigenvalue of the Laplacian with volume and inclusion constraints, Ann. Inst. H. Poincar´ e Anal. Non Lin´eaire 26 (2009), no. 4, 1149–1163, DOI 10.1016/j.anihpc.2008.07.003. MR2542718 [13] Giuseppe Buttazzo and Gianni Dal Maso, An existence result for a class of shape optimization problems, Arch. Rational Mech. Anal. 122 (1993), no. 2, 183–195, DOI 10.1007/BF00378167. MR1217590 [14] E. B. Davies, Heat kernels and spectral theory, Cambridge Tracts in Mathematics, vol. 92, Cambridge University Press, Cambridge, 1990. MR1103113 [15] Ennio De Giorgi, Sulla propriet` a isoperimetrica dell’ipersfera, nella classe degli insiemi aventi frontiera orientata di misura finita (Italian), Atti Accad. Naz. Lincei. Mem. Cl. Sci. Fis. Mat. Nat. Sez. I (8) 5 (1958), 33–44. MR0098331 [16] Pedro Freitas and David Krejˇ ciˇr´ık, The first Robin eigenvalue with negative boundary parameter, Adv. Math. 280 (2015), 322–339, DOI 10.1016/j.aim.2015.04.023. MR3350222 [17] Mariano Giaquinta, Multiple integrals in the calculus of variations and nonlinear elliptic systems, Annals of Mathematics Studies, vol. 105, Princeton University Press, Princeton, NJ, 1983. MR717034 [18] Mohammed Hayouni, Sur la minimisation de la premi` ere valeur propre du laplacien (French, with English and French summaries), C. R. Acad. Sci. Paris S´ er. I Math. 330 (2000), no. 7, 551–556, DOI 10.1016/S0764-4442(00)00229-9. MR1760437 [19] Antoine Henrot, Extremum problems for eigenvalues of elliptic operators, Frontiers in Mathematics, Birkh¨ auser Verlag, Basel, 2006. MR2251558 [20] Elliott H. Lieb, On the lowest eigenvalue of the Laplacian for the intersection of two domains, Invent. Math. 74 (1983), no. 3, 441–448, DOI 10.1007/BF01394245. MR724014 [21] P.-L. Lions, The concentration-compactness principle in the calculus of variations. The limit case. I, Rev. Mat. Iberoamericana 1 (1985), no. 1, 145–201, DOI 10.4171/RMI/6. MR834360 [22] Orlando Lopes, Radial symmetry of minimizers for some translation and rotation invariant functionals, J. Differential Equations 124 (1996), no. 2, 378–388, DOI 10.1006/jdeq.1996.0015. MR1370147 [23] Mihai Mari¸s, On the symmetry of minimizers, Arch. Ration. Mech. Anal. 192 (2009), no. 2, 311–330, DOI 10.1007/s00205-008-0136-2. MR2486598 [24] Dario Mazzoleni and Aldo Pratelli, Existence of minimizers for spectral problems, J. Math. Pures Appl. (9) 100 (2013), no. 3, 433–453, DOI 10.1016/j.matpur.2013.01.008. MR3095209 [25] George P´ olya, Circle, sphere, symmetrization, and some classical physical problems., Modern mathematics for the engineer: Second series, McGraw-Hill, New York, 1961, pp. 420–441. MR0129169 [26] John William Strutt Rayleigh Baron, The Theory of Sound, Dover Publications, New York, N. Y., 1945. 2d ed. MR0016009 [27] James Serrin, A symmetry problem in potential theory, Arch. Rational Mech. Anal. 43 (1971), 304–318, DOI 10.1007/BF00250468. MR0333220 [28] J. Steiner, Einfache Beweise der isoperimetrischen Haupts¨ atze (German), J. Reine Angew. Math. 18 (1838), 281–296, DOI 10.1515/crll.1838.18.281. MR1578194 [29] Alfred Wagner, Optimal shape problems for eigenvalues, Comm. Partial Differential Equations 30 (2005), no. 7-9, 1039–1063, DOI 10.1081/PDE-200064443. MR2180294

86

DORIN BUCUR AND PEDRO FREITAS

´matiques (LAMA), Universit´ Laboratoire de Mathe e de Savoie, Campus Scientifique, 73376 Le-Bourget-Du-Lac, France E-mail address: [email protected] ´tica, Instituto Superior T´ Departamento de Matema ecnico, Universidade de Lisboa, ´tica, Av. Rovisco Pais, 1049-001 Lisboa, Portugal — and — Grupo de F´ısica Matema Faculdade de Ciˆ encias, Universidade de Lisboa, Campo Grande, Edif´ıcio C6, 1749-016 Lisboa, Portugal E-mail address: [email protected]

Contemporary Mathematics Volume 700, 2017 http://dx.doi.org/10.1090/conm/700/14184

Some nodal properties of the quantum harmonic oscillator and other Schr¨ odinger operators in R2 Pierre B´erard and Bernard Helffer Abstract. For the spherical Laplacian on the sphere and for the Dirichlet Laplacian in the square, Antonie Stern claimed in her PhD thesis (1924) the existence of an infinite sequence of eigenvalues whose corresponding eigenspaces contain an eigenfunction with exactly two nodal domains. These results were given complete proofs respectively by Hans Lewy in 1977, and the authors in 2014 (see also Gauthier-Shalom–Przybytkowski, 2006). In this paper, we obtain similar results for the two dimensional isotropic quantum harmonic oscillator. In the opposite direction, we construct an infinite sequence of regular eigenfunctions with as many nodal domains as allowed by Courant’s theorem, up to a factor 14 . A classical question for a 2-dimensional bounded domain is to estimate the length of the nodal set of a Dirichlet eigenfunction in terms of the square root of the energy. In the last section, we consider some Schr¨ odinger operators −Δ + V in R2 and we provide bounds for the length of the nodal set of an eigenfunction with energy λ in the classically permitted region {V (x) < λ}.

1. Introduction and main results Given a finite interval ]a, b[ and a continuous function q : [a, b] → R, consider the one-dimensional self-adjoint eigenvalue problem (1.1)

−y  + qy = λy in ]a, b[, y(a) = y(b) = 0.

Arrange the eigenvalues in increasing order, λ1 (q) < λ2 (q) < · · · . A classical theorem of C. Sturm [22] states that an eigenfunction u of (1.1) associated with λk (q) has exactly (k − 1) zeros in ]a, b[ or, equivalently, that the zeros of u divide ]a, b[ into k sub-intervals. In higher dimensions, one can consider the eigenvalue problem for the LaplaceBeltrami operator −Δg on a compact connected Riemannian manifold (M, g), with Dirichlet condition in case M has a boundary ∂M , (1.2)

−Δu = λu in M, u|∂M = 0.

Arrange the eigenvalues in non-decreasing order, with multiplicities, λ1 (M, g) < λ2 (M, g) ≤ λ3 (M, g) ≤ . . . 2010 Mathematics Subject Classification. Primary 35B05, 35Q40, 35P99, 58J50, 81Q05. Key words and phrases. Quantum harmonic oscillator, Schr¨ odinger operator, nodal lines, nodal domains, Courant nodal theorem. c 2017 American Mathematical Society

87

88

´ PIERRE BERARD AND BERNARD HELFFER

Denote by M0 the interior of M , M0 := M \ ∂M . Given an eigenfunction u of −Δg , denote by N (u) := {x ∈ M0 | u(x) = 0}

(1.3) the nodal set of u, and by (1.4)

μ(u) := # { connected components of M0 \ N (u)}

the number of nodal domains of u i.e., the number of connected components of the complement of N (u). Courant’s theorem [12] states that if −Δg u = λk (M, g)u, then μ(u) ≤ k. In this paper, we investigate three natural questions about Courant’s theorem in the framework of the 2D isotropic quantum harmonic oscillator. Question 1. In view of Sturm’s theorem, it is natural to ask whether Courant’s upper bound is sharp, and to look for lower bounds for the number of nodal domains, depending on the geometry of (M, g) and the eigenvalue. Note that for orthogonality reasons, for any k ≥ 2 and any eigenfunction associated with λk (M, g), we have μ(u) ≥ 2. We shall say that λk (M, g) is Courant-sharp if there exists an eigenfunction u, such that −Δg u = λk (M, g)u and μ(u) = k. Clearly, λ1 (M, g) and λ2 (M, g) are always Courant-sharp eigenvalues. Note that if λ3 (M, g) = λ2 (M, g), then λ3 (M, g) is not Courant-sharp. The first results concerning Question 1 were stated by Antonie Stern in her 1924 PhD thesis [36] written under the supervision of R. Courant. Theorem 1.1 (A. Stern, [36]). (1) For the square [0, π] × [0, π] with Dirichlet boundary condition, there is a sequence of eigenfunctions {ur , r ≥ 1} such that −Δur = (1 + 4r 2 )ur , and μ(ur ) = 2. (2) For the sphere S2 , there exists a sequence of eigenfunctions u ,  ≥ 1 such that −ΔS2 u = ( + 1)u , and μ(u ) = 2 or 3, depending on whether  is odd or even. Stern’s arguments are not fully satisfactory. In 1977, H. Lewy [25] gave a complete independent proof for the case of the sphere, without any reference to [36] (see also [4]). More recently, the authors [2] gave a complete proof for the case of the square with Dirichlet boundary conditions (see also Gauthier-Shalom–Przybytkowski [17]). The original motivation of this paper was to investigate the possibility to extend Stern’s results to the case of the two-dimensional isotropic quantum harmonic os$ := −Δ + |x|2 acting on L2 (R2 , R) (we will say “harmonic oscillator” for cillator H short). After the publication of the first version of this paper [3], T. HoffmannOstenhof informed us of the unpublished master degree thesis of J. Leydold [26].

SOME NODAL PROPERTIES OF THE QUANTUM HARMONIC OSCILLATOR

89

$ is given by An orthogonal basis of eigenfunctions of the harmonic oscillator H x2 + y 2 ), 2 for (m, n) ∈ N2 , where Hn denotes the Hermite polynomial of degree n. For Hermite polynomials, we use the definitions and notation of Szeg¨o [37, §5.5].

(1.5)

φm,n (x, y) = Hm (x)Hn (y) exp(−

The eigenfunction φm,n corresponds to the eigenvalue 2(m + n + 1), (1.6)

$ m,n = 2(m + n + 1) φm,n . Hφ

ˆ $ associated with the eigenvalue λ(n) It follows that the eigenspace En of H = 2(n+1) has dimension (n+1), and is generated by the eigenfunctions φn,0 , φn−1,1 , . . . , φ0,n . We summarize Leydold’s main results in the following theorem. Theorem 1.2 (J. Leydold, [26]). (1) For n ≥ 2, and for any nonzero u ∈ En , n2 + 2. 2 (2) The lower bound on the number of nodal domains is given by ⎧ ⎨ 1 if n = 0, 3 if n ≡ 0 (mod 4), n ≥ 4, (1.8) min {μ(u) | u ∈ En , u = 0} = ⎩ 2 if n ≡ 0 (mod 4).

(1.7)

μ(u) ≤ μL n :=

Remark. When n ≥ 3, the estimate (1.7) is better than Courant’s bound which is n2 n μC n := 2 + 2 + 1. The idea of the proof is to apply Courant’s method separately to odd and to even eigenfunctions (with respect to the map x → −x). A consequence of (1.7) is that the only Courant-sharp eigenvalues of the harmonic oscillator are the first, second and fourth eigenvalues. The same ideas work for the sphere as well [26, 27]. For the analysis of Courant-sharp eigenvalues of the square with Dirichlet boundary conditions, see [2, 30]. Leydold’s proof that there exist eigenfunctions of the harmonic oscillator satisfying (1.8) is quite involved. In this paper, we give a simple proof that, for any odd integer n, there exists a one-parameter family of eigenfunctions with exactly two nodal domains in En . More precisely, for θ ∈ [0, π[, we consider the following curve in En , (1.9)

Φθn := cos θ φn,0 + sin θ φ0,n , Φθn (x, y) = (cos θ Hn (x) + sin θ Hn (y)) exp(−

x2 + y 2 ). 2

We prove the following theorems (Sections 4 and 5). Theorem 1.3. Assume that n is odd. Then, there exists an open interval I π4 containing π4 , and an open interval I 3π , containing 3π 4 , such that for 4 π 3π θ ∈ I π4 ∪ I 3π }, \{ , 4 4 4 the nodal set N (Φθn ) is a connected simple regular curve, and the eigenfunction Φθn has two nodal domains in R2 .

90

´ PIERRE BERARD AND BERNARD HELFFER

Theorem 1.4. Assume that n is odd. Then, there exists θc > 0 such that, for 0 < θ < θc , the nodal set N (Φθn ) is a connected simple regular curve, and the eigenfunction Φθn has two nodal domains in R2 . Remark. The value θc and the intervals can be computed numerically. The proofs of the theorems actually show that ]0, θc [ ∩I π4 = ∅. Question 2. How good/bad is Courant’s upper bound on the number of nodal domains? Consider the eigenvalue problem (1.2). Given k ≥ 1, define μ(k) to be the maximum value of μ(u) when u is an eigenfunction associated with the eigenvalue λk (M, g). Then, (1.10)

lim sup k→∞

μ(k) ≤ γ(n) k

where γ(n) is a universal constant which only depends on the dimension n of M . Furthermore, for n ≥ 2, γ(n) < 1. The idea of the proof, introduced by Pleijel in 1956, is to use a Faber-Krahn type isoperimetric inequality and Weyl’s asymptotic law. Note that the constant γ(n) is not sharp. For more details and references, see [14, 30]. As a corollary of the above result, the eigenvalue problem (1.2) has only finitely many Courant-sharp eigenvalues. The above result gives a quantitative improvement of Courant’s theorem in the case of the Dirichlet Laplacian in a bounded open set of R2 . When trying to implement the strategy of Pleijel for the harmonic oscillator, we get into trouble because of the absence of a reasonable Faber-Krahn inequality. P. Charron [9, 10] has obtained the following theorem. $ then Theorem 1.5. If (λn , un ) is an infinite sequence of eigenpairs of H, (1.11)

lim sup

4 μ(un ) ≤ γ(2) = 2 , n j0,1

where j0,1 is the first positive zero of the Bessel function of order 0 . This is in some sense surprising that the statement is exactly the same as in the case of the Dirichlet realization of the Laplacian in a bounded open set in R2 . The proof does actually not use the isotropy of the harmonic potential, can be extended to the n-dimensional case, but strongly uses the explicit knowledge of the eigenfunctions. We refer to [11] for further results in this direction. A related question concerning the estimate (1.10) is whether the order of magnitude is correct. In the case of the 2-sphere, using spherical coordinates, one can find decomposed spherical harmonics u of degree , with associated eigenvalue ( + 1), 2 such that μ(u ) ∼ 2 when  is large, whereas Courant’s upper bound is equivalent to 2 . These spherical harmonics have critical zeros and their nodal sets have selfintersections. In [16, Section 2], the authors construct spherical harmonics v , of degree , without critical zeros i.e., whose nodal sets are disjoint closed regular 2 curves, such that μ(v ) ∼ 4 . These spherical harmonics v have as many nodal domains as allowed by Courant’s theorem, up to a factor 14 . Since the v ’s are regular, this property is stable under small perturbations in the same eigenspace.

SOME NODAL PROPERTIES OF THE QUANTUM HARMONIC OSCILLATOR

91

In Section 6, in a direction opposite to Theorems 1.3 and 1.4, we construct eigen$ with “many” nodal domains. functions of the harmonic oscillator H $ in L2 (R2 ), there exists a sequence Theorem 1.6. For the harmonic oscillator H ˆ ˆ $ k = λ(k)u of eigenfunctions {uk , k ≥ 1} such that Hu k , with λ(4k) = 2(4k + 1), uk as no critical zeros (i.e. has a regular nodal set), and μ(uk ) ∼

(4k)2 . 8

Remarks. 1. The above estimate is, up to a factor 14 , asymptotically the same as the upper bounds for the number of nodal domains given by Courant and Leydold. 2. A related question is to analyze the zero set when θ is a random variable. We refer to [18] for results in this direction. The above questions are related to the question of spectral minimal partitions [19]. In the case of the harmonic oscillator similar questions appear in the analysis of the properties of ultracold atoms (see for example [33]). Question 3. Consider the eigenvalue problem (1.2), and assume for simplicity that M is a bounded domain in R2 . Fix any small number r, and a point x ∈ M such that B(x, r) ⊂ M . Let u be a Dirichlet eigenfunction associated with the eigenvalue πj 2

λ and assume that λ ≥ r0,1 2 . Then N (u) ∩ B(x, r) = ∅. This fact follows from the monotonicity of the Dirichlet eigenvalues, and indicates that the length of the nodal set should tend to infinity as the eigenvalue tends to infinity. The first results in this direction are due to Br¨ uning and Gromes [6, 7] who show √ that the length of the nodal set N (u) is bounded from below by a constant times λ. For further results in this direction (Yau’s conjecture), we refer to [15, 28, 29, 34, 35]. In Section 7, we investigate this question for the harmonic oscillator. Theorem 1.7. Let δ ∈]0, 1[ be given. Then, there exists a positive constant Cδ such that for λ large enough, and for any nonzero eigenfunction of the isotropic 2D quantum harmonic oscillator, $ := −Δ + |x|2 , Hu $ = λu, H √  3 the length of N (u) ∩ B δλ is bounded from below by Cδ λ 2 .

(1.12)

As a matter of fact, we prove a lower bound for more general Schr¨ odinger operators in R2 (Propositions 7.2 and 7.8), shedding some light on the exponent 32 in the above estimate. In Section 7.3, we investigate upper and lower bounds for the length of the nodal sets, using the method of Long Jin [29]. 2. A reminder on Hermite polynomials We use the definition, normalization, and notation of Szeg¨o’s book [37]. With these choices, the Hermite polynomial Hn has the following properties, [37, § 5.5 and Theorem 6.32]. (1) Hn satisfies the differential equation y  (t) − 2t y  (t) + 2n y(t) = 0.

´ PIERRE BERARD AND BERNARD HELFFER

92

(2) Hn is a polynomial of degree n which is even (resp. odd) for n even (resp. odd). (3) Hn (t) = 2t Hn−1 (t) − 2(n − 1) Hn−2 (t), n ≥ 2, H0 (t) = 1, H1 (t) = 2t. (4) Hn has n simple zeros tn,1 < tn,2 < · · · < tn,n . (5)  (t). Hn (t) = 2t Hn−1 (t) − Hn−1 (6) Hn (t) = 2nHn−1 (t).

(2.1)

(7) The coefficient of tn in Hn is 2n . (8)  +∞ 2 1 e−t |Hn (t)|2 dt = π 2 2n n!. −∞

(9) The first zero tn,1 of Hn satisfies (2.2)

tn,1 = (2n + 1) 2 − 6− 2 (2n + 1)− 6 (i1 + n ), 1

1

1

where i1 is the first positive real zero of the Airy function, and limn→+∞ n = 0. The following result (Theorem 7.6.1 in Szeg¨o’s book [37]) will also be useful. Lemma 2.1. The successive relative maxima of t → |Hn (t)| form an increasing sequence for t ≥ 0. Proof. It is enough to observe that the function Θn (t) := 2nHn (t)2 + Hn (t)2 satisfies Θn (t) = 4t (Hn (t))2 .  3. Stern-like constructions for the harmonic oscillator in the case n-odd 3.1. The case of the square. Consider the square [0, π]2 , with Dirichlet boundary conditions, and the following families of eigenfunctions associated with ˆ 2r) := 1 + 4r 2 , where r is a positive integer, and θ ∈ [0, π/4], the eigenvalues λ(1, (x, y) → cos θ sin x sin(2ry) + sin θ sin(2rx) sin y. According to [36], for any given r ≥ 1, the typical evolution of the nodal sets when θ varies is similar to the case r = 4 shown in Figure 1 [2, Figure 6.9]. Generally speaking, the nodal sets deform continuously, except for finitely many values of θ, for which self-intersections of the nodal set appear or disappear or, equivalently, for which critical zeros of the eigenfunction appear/disappear. We would like to get similar results for the isotropic quantum harmonic oscillator.

SOME NODAL PROPERTIES OF THE QUANTUM HARMONIC OSCILLATOR

93

ˆ 8) of the square. Figure 1. Nodal sets for the Dirichlet eigenvalue λ(1, 3.2. Symmetries. Recall the notation, Φθn (x, y) := cos θ φn,0 (x, y) + sin θ φ0,n (x, y).

(3.1) More simply,

 2  x + y2 Φθn (x, y) = exp − (cos θ Hn (x) + sin θ Hn (y)) . 2

= −Φθn , it suffices to vary the parameter θ in the interval [0, π[. Since Φθ+π n Assuming n is odd, we have the following symmetries. ⎧ θ π−θ ⎪ ⎪ Φn (−x, y) = Φn (x, y), ⎨ Φθn (x, −y) = −Φπ−θ (x, y), n (3.2) ⎪ π −θ ⎪ ⎩ Φθ (y, x) = Φn2 (x, y). n

When n is odd, it therefore suffices to vary the parameter θ in the interval [0, π4 ]. The case θ = 0 is particular, so that we shall mainly consider θ ∈]0, π4 ].

94

´ PIERRE BERARD AND BERNARD HELFFER

3.3. Critical zeros. A critical zero of Φθn is a point (x, y) ∈ R2 such that both Φθn and its differential dΦθn vanish at (x, y). The critical zeros of Φθn satisfy the following equations. ⎧ ⎪ ⎨ cos θ Hn (x) + sin θ Hn (y) = 0, = 0, cos θ Hn (x) (3.3) ⎪ ⎩  = 0. sin θ Hn (y) Equivalently, using the properties of the Hermite polynomials, a point (x, y) is a critical zero of Φθn if and only if ⎧ ⎪ ⎨ cos θ Hn (x) + sin θ Hn (y) = 0, = 0, cos θ Hn−1 (x) (3.4) ⎪ ⎩ = 0. sin θ Hn−1 (y) The only possible critical zeros of the eigenfunction Φθn are the points (tn−1,i , tn−1,j ) for 1 ≤ i, j ≤ (n−1), where the coordinates are the zeros of the Hermite polynomial Hn−1 . The point (tn−1,i , tn−1,j ) is a critical zero of Φθn if and only if θ = θ(i, j) , where θ(i, j) ∈]0, π[ is uniquely determined by the equation, (3.5)

cos (θ(i, j)) Hn (tn−1,i ) + sin (θ(i, j)) Hn (tn−1,j ) = 0.

The values θ(i, j) will be called critical values of the parameter θ, the other values regular values. Here we have used the fact that Hn and Hn have no common zeros. We have proved the following lemma. Lemma 3.1. For θ ∈ [0, π[, the eigenfunction Φθn has no critical zero, unless θ is one of the critical values θ(i, j) defined by equation (3.5). In particular Φθn has no critical zero, except for finitely many values of the parameter θ ∈ [0, π[. Given a pair (i0 , j0 ) ∈ {1, . . . , n − 1}, let θ0 = θ(i0 , j0 ), be defined by (3.5) for the pair (tn−1,i0 , tn−1,j0 ) . Then, the function Φθn0 has finitely many critical zeros, namely the points (tn−1,i , tn−1,j ) which satisfy (3.6)

cos θ0 Hn (tn−1,i ) + sin θ0 Hn (tn−1,j ) = 0,

among them the point (tn−1,i0 , tn−1,j0 ) . Remarks. From the general properties of nodal lines [2, Properties 5.2], we derive the following facts. (1) When θ ∈ {θ(i, j) | 1 ≤ i, j ≤ n − 1}, the nodal set N (Φθn ) of the eigenfunction Φθn , is a smooth 1-dimensional submanifold of R2 (a collection of pairwise distinct connected simple regular curves). (2) When θ ∈ {θ(i, j) | 1 ≤ i, j ≤ n − 1}, the nodal set N (Φθn ) has finitely many singularities which are double crossings1 . Indeed, the Hessian of the function Φθn at a critical zero (tn−1,i , tn−1,j ) is given by   t2n−1,i + t2n−1,j cos θ Hn (tn−1,i ) 0 θ ) Hess(tn−1,i ,tn−1,j ) Φn = exp (− , 0 sin θ Hn (tn−1,j ) 2 and the assertion follows from the fact that Hn−1 has simple zeros. 1 This result is actually general for any eigenfunction of the harmonic oscillator, as stated in [26], on the basis of Euler’s formula and Courant’s theorem.

SOME NODAL PROPERTIES OF THE QUANTUM HARMONIC OSCILLATOR

95

3.4. General properties of the nodal set N (Φθn ). Denote by L the finite lattice (3.7)

L := {(tn,i , tn,j ) | 1 ≤ i, j ≤ n} ⊂ R2 ,

consisting of points whose coordinates are the zeros of the Hermite polynomial Hn . The horizontal and vertical lines {y = tn,i } and {x = tn,j }, 1 ≤ i, j ≤ n, form a checkerboard like pattern in R2 which can be colored according to the sign of the function Hn (x) Hn (y) (grey where the function is positive, white where it is negative). We will refer to the following properties as the checkerboard argument, compare with [2, 36]. For symmetry reasons, we can assume that θ ∈]0, π4 ]. (i) We have the following inclusions for the nodal sets N (Φθn ) , (3.8)

  L ⊂ N (Φθn ) ⊂ L ∪ (x, y) ∈ R2 | Hn (x) Hn (y) < 0 .

(ii) The nodal set N (Φθn ) does not meet the vertical lines {x = tn,i }, or the horizontal lines {y = tn,i } away from the set L. (iii) The lattice point (tn,i , tn,j ) is not a critical zero of Φθn (because Hn and Hn have no common zero). As a matter of fact, near a lattice point, the nodal set N (Φθn ) is a single arc through the lattice point, with a tangent which is neither horizontal, nor vertical. Figure 2 shows the evolution of the nodal set of Φθn , for n = 7, when θ varies in the interval ]0, π4 ]. The pictures in the first column correspond to regular values of θ whereas the pictures in the second column correspond to critical values of θ. The form of the nodal set is stable in the open interval between two consecutive critical values of the parameter θ. In the figures, the thick curves represent the nodal sets N (Φθ7 ), the thin lines correspond to the zeros of H7 , and the grey lines to the zeros of H7 , i.e. to the zeros of H6 . We now describe the nodal set N (Φθn ) outside a large enough square which contains the lattice L. For this purpose, we give the following two barrier lemmas which describe the intersections of the nodal set with horizontal and vertical lines. Lemma 3.2. Assume that θ ∈]0, π4 ]. For n odd, define tn−1,0 to be the unique point in ] − ∞, tn,1 [ such that Hn (tn−1,0 ) = −Hn (tn−1,1 ). Then, (1) ∀t ≤ tn,1 , the function y → Φθn (t, y) has exactly one zero in the interval [tn,n , +∞[ ; (2) ∀t < tn−1,0 , the function y → Φθn (t, y) has exactly one zero in the interval ] − ∞, +∞[. Using the symmetry with respect to the vertical line {x = 0}, one has similar statements for t ≥ tn,n and for t > −tn−1,0 .

96

´ PIERRE BERARD AND BERNARD HELFFER

Figure 2. Evolution of the nodal set N (Φθn ), for n = 7 and θ ∈]0, π4 ].

2

2

θ  Proof. Let v(y) := exp( t +y 2 ) Φn (t, y). In ]tn,n , +∞[ , v (y) is positive, and v(tn,n ) ≤ 0 . The first assertion follows. The local extrema of v occur at the points tn−1,j , for 1 ≤ j ≤ (n − 1) . The second assertion follows from the definition of

SOME NODAL PROPERTIES OF THE QUANTUM HARMONIC OSCILLATOR

97

tn−1,0 , and from the inequalities,

 1  cos θ Hn (t) + sin θ Hn (tn−1,j ) ≤ √ Hn (t) + |Hn |(tn−1,j ) 2  1  < − √ Hn (tn−1,1 ) − |Hn |(tn−1,j ) ≤ 0, 2 for t < tn−1,0 , where we have used Lemma 2.1.



Lemma 3.3. Let θ ∈]0, Define ∈]tn,n , ∞[ to be the unique point such θ that tan θ Hn (tn−1,n ) = Hn (tn−1,1 ). Then, (1) ∀t ≥ tn,n , the function x → Φθn (x, t) has exactly one zero in the interval ] − ∞, tn,1 ] ; (2) ∀t > tθn−1,n , the function x → Φθn (x, t) has exactly one zero in the interval ] − ∞, ∞[. 2 1 < tθn−1,n . (3) For θ2 > θ1 , we have tθn−1,n Using the symmetry with respect to the horizontal line {y = 0}, one has similar statements for t ≤ tn,1 and for t < −tθn−1,n . π 4 ].

tθn−1,n

2

2

θ Proof. Let h(x) := exp( x +t 2 ) Φn (x, t). In the interval ] − ∞, tn,1 ] , the deriv ative h (x) is positive, h(tn,1 ) > 0 , and limx→−∞ h(x) = −∞ , since n is odd. The first assertion follows. The local extrema of h are achieved at the points tn−1,j . Using Lemma 2.1, for t ≥ tθn−1,n , we have the inequalities,

Hn (tn−1,j ) + tan θ Hn (t) ≥ tan θ Hn (tθn−1,n ) − |Hn (tn−1,j )| = Hn (tn−1,1 ) − |Hn (tn−1,j )| ≥ 0.



As a consequence of the above lemmas, we have the following description of the nodal set far enough from (0, 0). Proposition 3.4. Let θ ∈]0, π4 ]. In the set R2 \] − tθn−1,n , tθn−1,n [×]tn−1,0 , |tn−1,0 |[, the nodal set N (Φθn ) consists of two regular arcs. The first arc is a graph y(x) over the interval ] − ∞, tn,1 ], starting from the point (tn,1 , tn,n ) and escaping to infinity with, √ y(x) n = − cot θ. lim x→−∞ x The second arc is the image of the first one under the symmetry with respect to (0, 0) in R2 . 3.5. Local nodal patterns. As in the case of the Dirichlet eigenvalues for the square, we study the possible local nodal patterns taking into account the fact that the nodal set contains the lattice points L, can only visit the connected components of the set {Hn (x) Hn (y) < 0} (colored white), and consists of a simple arc at the lattice points. The following figure summarized the possible nodal patterns in the interior of the square [2, Figure 6.4], Except for nodal arcs which escape to infinity, the local nodal patterns for the quantum harmonic oscillator are similar (note that in the present case, the connected components of the set {Hn (x) Hn (y) < 0} are rectangles, no longer equal squares). The checkerboard argument and the location of the possible critical zeros determine the possible local patterns: (A), (B) or (C). Case (C) occurs near a critical zero.

´ PIERRE BERARD AND BERNARD HELFFER

98

Figure 3. Local nodal patterns for Dirichlet eigenfunctions of the square.

Following the same ideas as in the case of the square, in order to decide between cases (A) and (B), we use the barrier lemmas, Lemma 3.2 or 3.3, the vertical lines {x = tn−1,j }, or the horizontal lines {y = tn−1,j }.

4. Proof of Theorem 1.3 Note that √ √ π 3π φn,0 (x, y) − φ0,n (x, y) = − 2 Φn4 (x, y) = − 2Φn4 (x, −y). Hence, up to symmetry, it is the same to work with θ = π4 and the anti-diagonal, or to work with θ = 3π 4 and the diagonal. For notational convenience, we work with 3π . 4 3π

4.1. The nodal set of Φn4 . The purpose of this section is to prove the following result which is the starting point for the proof of Theorem 1.3. Proposition 4.1. Let {tn−1,i , 1 ≤ i ≤ n − 1} denote the zeroes of Hn−1 . For n odd, the nodal set of φn,0 − φ0,n consists of the diagonal x = y, and of n−1 2 disjoint simple closed curves crossing the diagonal at the (n − 1) points (tn−1,i , tn−1,i ), and the anti-diagonal at the (n − 1) points (tn,i , −tn,i ). To prove Proposition 4.1, we first observe that it is enough to analyze the zero set of (x, y) → Ψn (x, y) := Hn (x) − Hn (y). 4.1.1. Critical zeros. The only possible critical zeros of Ψn are determined by Hn (x) = 0, Hn (y) = 0. Hence, they consist of the (n − 1)2 points (tn−1,i , tn−1,j ) , for 1 ≤ i, j ≤ (n − 1) , where tn−1,i is the i-th zero of the polynomial Hn−1 . The zero set of Ψn contains the diagonal {x = y} . Since n is odd, there are only n points belonging to the zero set on the anti-diagonal {x + y = 0}. On the diagonal,

SOME NODAL PROPERTIES OF THE QUANTUM HARMONIC OSCILLATOR

99

there are (n − 1) critical points. We claim that there are no critical zeros outside the diagonal. Indeed, let (tn−1,i , tn−1,j ) be a critical zero. Then, Hn (tn−1,i ) = Hn (tn−1,j ). Using Lemma 2.1 and the parity properties of Hermite polynomials, we see that |Hn (tn−1,i )| = |Hn (tn−1,j )| occurs if and only if tn−1,i = ±tn−1,j . Since n is odd, we can conclude that Hn (tn−1,i ) = Hn (tn−1,j ) occurs if and only if tn−1,i = tn−1,j . 3π

4.1.2. Existence of disjoint simple closed curves in the nodal set of Φn4 . The second part in the proof of the proposition follows closely the proof in the case of the Dirichlet Laplacian for the square (see Section 5 in [2]). Essentially, the Chebyshev polynomials are replaced by the Hermite polynomials. Note however that the checkerboard is no more with equal squares, and that the square [0, π]2 has to be replaced in the argument by the rectangle [tn−1,0 , −tn−1,0 ] × [−tθn−1,n , tθn−1,n ], for some θ ∈]0, 3π 4 [ , see Lemmas 3.2 and 3.3. The checkerboard argument holds, see (3.8) and the properties at the beginning of Section 3.4. The separation lemmas of our previous paper [2] must be substituted by Lemmas 3.2 and 3.3, and similar statements with the lines {x = tn−1,j } and {y = tn−1,j }, for 1 ≤ j ≤ (n − 1) . One needs to control what is going on at infinity. As a matter of fact, outside a specific rectangle centered at the origin, the zero set is the diagonal {x = y}, see Proposition 3.4. Hence in this way (like for the square), we obtain that the nodal set of Ψn consists of the diagonal and n−1 disjoint simple closed curves turning around the origin. 2 The set L is contained in the union of these closed curves. 3π

4.1.3. No other closed curve in the nodal set of Φn4 . It remains to show that there are no other closed curves which do not cross the diagonal. The “energy” considerations of our previous papers [2, 4] work here as well. Here is a simple alternative argument. We look at the line y = α x for some α = 1. The intersection of the zero set with this line corresponds to the zeroes of the polynomial x → Hn (x) − Hn (α x) which has at most n zeroes. But in our previous construction, we get at least n zeroes. So the presence of extra curves would lead to a contradiction for some α. This argument solves the problem at infinity as well.  4.2. Perturbation argument. Figure 4 shows the desingularization of the 3π nodal set N (Φn4 ), from below and from above. The picture is the same as in the case of the square (see Figure 1), all the critical points disappear at the same time and in the same manner, i.e. all the double crossings open up horizontally or vertically depending whether θ is less than or bigger than 3π 4 . As in the case of the square, in order to show that the nodal set can be desingularized 3π under small perturbation, we look at the signs of the eigenfunction Φn4 near the critical zeros. We use the cases (I) and (II) which appear in Figure 5 below (see also [2, Figure 6.7]).

100

´ PIERRE BERARD AND BERNARD HELFFER

Figure 4. The nodal set of N (Φθn ) near

3π 4

(here n = 7).

Figure 5. Signs near a critical zero. The sign configuration for φn,0 (x, y)−φ0,n (x, y) near the critical zero (tn−1,i , tn−1,i ) is given by Figure 5  case (I), if i is even, case (II), if i is odd. Looking at the intersection of the nodal set with the vertical line {y = tn−1,i }, we have that (−1)i (Hn (t) − Hn (tn−1,i )) ≥ 0, for t ∈]tn,i , tn,i+1 [. For positive small, we write (−1)i (Hn (t) − (1 + )Hn (tn−1,i )) = (−1)i (Hn (t) − Hn (tn−1,i )) + (−1)i+1 Hn (tn−1,i ), so that (−1)i (Hn (t) − (1 + )Hn (tn−1,i )) ≥ 0, for t ∈]tn,i , tn,i+1 [. A similar statement can be written for horizontal line {x = tn−1,i } and − , with > 0 , small enough. These inequalities describe how the crossings all open up at the same time, and in the same manner, vertically (case I) or horizontally (case II), see Figure 6, as in the case of the square [2, Figure 6.8]. We can then conclude as in the case of the square, using the local nodal patterns, Section 3.5.

SOME NODAL PROPERTIES OF THE QUANTUM HARMONIC OSCILLATOR

101

Figure 6. Desingularization at a critical zero. Remark. Because the local nodal patterns can only change when θ passes through one of the values θ(i, j) defined in (3.5), the above arguments work for θ ∈ J \ { 3π 4 }, for any interval J containing 3π and no other critical value θ(i, j). 4 5. Proof of Theorem 1.4 Proposition 5.1. The conclusion of Theorem 1.4 holds with (5.1)

θc := inf {θ(i, j) | 1 ≤ i, j ≤ n − 1} ,

where the critical values θ(i, j) are defined by (3.5). Proof. The proof consists in the following steps. For simplicity, we call N the nodal set N (Φθn ). • Step 1. By Proposition 3.4, the structure of the nodal set N is known outside a large coordinate rectangle centered at (0, 0) whose sides are defined by the ad hoc numbers in Lemmas 3.2 and 3.3. Notice that the sides of the rectangle serve as barriers for the arguments using the local nodal patterns as in our paper for the square. • Step 2. For 1 ≤ j ≤ n − 1, the line {x = tn−1,j } intersects the set N at exactly one point (tn−1,j , yj ), with yj > tn,n when j is odd, resp. with yj < tn,1 when j is even. The proof is given below, and is similar to the proofs of Lemmas 3.2 or 3.3. • Step 3. Any connected component of N has at least one point in common with the set L. This follows from the argument with y = αx or from the energy argument (see Subsection 4.1.3). • Step 4. Follow the nodal set from the point (tn,1 , tn,n ) to the point (tn,n , tn,1 ), using the analysis of the local nodal patterns as in the case of the square. Proof of Step 2. For 1 ≤ j ≤ (n − 1), define the function vj by vj (y) := cos θ Hn (tn−1,j ) + sin θ Hn (y). The local extrema of vj are achieved at the points tn−1,i , for 1 ≤ i ≤ (n − 1), and we have vj (tn−1,i ) = cos θ Hn (tn−1,j ) + sin θ Hn (tn−1,i ),

102

´ PIERRE BERARD AND BERNARD HELFFER

which can be rewritten, using (3.5), as vj (tn−1,i ) =

Hn (tn−1,j ) sin (θ(j, i) − θ) . sin θ(j, i)

The first term in the right-hand side has the sign of (−1)j+1 and the second term is positive provided that 0 < θ < θc . Under this last assumption, we have (5.2)

(−1)j+1 vj (tn−1,i ) > 0, ∀i, 1 ≤ i ≤ (n − 1). 

The assertion follows.

6. Eigenfunctions with “many” nodal domains, proof of Theorem 1.6 This section is devoted to the proof of Theorem 1.6 i.e., to the constructions of $ with regular nodal sets (no self-intersections) and “many” nodal eigenfunctions of H domains. We work in polar coordinates. An orthogonal basis of E is given by the functions Ω±

,n , (6.1)

Ω±

,n (r, ϕ) = exp(−

r 2 −2n ( −2n) 2 )r Ln (r ) exp (±i( − 2n)ϕ) , 2

% & (α) with 0 ≤ n ≤ 2 , see [26, Section 2.1]. In this formula Ln is the generalized Laguerre polynomial of degree n and parameter α, see [37, Chapter 5]. Recall that (0) the Laguerre polynomial Ln is the polynomial Ln . Assumption 6.1. From now on, we assume that  = 4k , with k even. 2

Since  is even, we have a rotation invariant eigenfunction exp(− r2 ) L2k (r 2 ) which has (2k + 1) nodal domains. We also look at the eigenfunctions ω ,n , r2 (r 2 ) sin (( − 2n)ϕ) , ω ,n (r, ϕ) = exp(− ) r −2n L( −2n) n 2 % & with 0 ≤ n < 2 . The number of nodal domains of these eigenfunctions is μ(ω ,n ) = 2(n + 1)( − 2n), because the Laguerre polynomial of degree n has n simple positive roots. When  = 4k, the largest of these numbers is (6.2)

(6.3)

μ := 4k(k + 1),

and this is achieved for n = k . When k tends to infinity, we have μ ∼

2 upper bound μL

∼ 2 .

2 4 ,

the same order of magnitude as Leydold’s

We want now to construct eigenfunctions uk ∈ E ,  = 4k with regular nodal sets, and “many” nodal domains (or equivalently, “many” nodal connected components), 2 more precisely with μ(uk ) ∼ 8 . The construction consists of the following steps. (1) Choose A ∈ E such that μ(A) = μ . (2) Choose B ∈ E such that for a small enough, the perturbed eigenfunction Fa := A + a B has no critical zero except the origin, and a nodal set with many components. Fix such an a.

SOME NODAL PROPERTIES OF THE QUANTUM HARMONIC OSCILLATOR

103

(3) Choose C ∈ E such that for b small enough (and a fixed), Ga,b := A + a B + b C has no critical zero. From now on, we fix some , 0 < < 1 . We assume that a is positive (to be chosen small enough later on), and that b is non zero (to be chosen small enough, either positive or negative later on). In the remaining part of this section, we skip the exponential factor in the eigenfunctions since it is irrelevant to study the nodal sets. Under Assumption 6.1, define (2k)

(6.4)

A(r, ϕ) := r 2k Lk

(r 2 ) sin(2kϕ),

B(r, ϕ) := r 4k sin(4kϕ − π),

C(r, ϕ) := L2k (r 2 ).

We consider the deformations Fa = A + a B and Ga,b = A + a B + b C. Both functions are invariant under the rotation of angle πk , so that we can restrict to ϕ ∈ [0, πk ] . For later purposes, we introduce the angles ϕj = jπ k , for 0 ≤ j ≤ 4k − 1, and (m+)π ψm = , for 0 ≤ m ≤ 8k − 1. We denote by ti , 1 ≤ i ≤ k the zeros of 4k (2k) Lk , listed in increasing order. They are simple and positive, so that the numbers √ (2k) the ri = ti are well defined. For notational convenience, we denote by L˙ k (2k) derivative of the polynomial Lk . This polynomial has (k − 1) simple zeros, which  we denote by ti , 1 ≤ i ≤ k − 1, with ti < ti < ti+1 . We define ri := ti . 6.1. Critical zeros. Clearly, the origin is a critical zero of the eigenfunction Fa = A + a B , while Ga,b = A + a B + b C does not vanish at the origin. Away from the origin, the critical zeros of Fa are given by the system = 0, Fa (r, ϕ) ∂r Fa (r, ϕ) = 0, ∂ϕ Fa (r, ϕ) = 0.

(6.5)

The first and second conditions imply that a critical zero (r, ϕ) satisfies   (2k) (2k) (6.6) sin(2kϕ) sin(4kϕ − π) kLk (r 2 ) − r 2 L˙ k (r 2 ) = 0, where L˙ is the derivative of the polynomial L . The first and third conditions imply that a critical zero (r, ϕ) satisfies (6.7)

2 sin(2kϕ) cos(4kϕ − π) − cos(2kϕ) sin(4kϕ − π) = 0.

It is easy to deduce from (6.5) that when (r, ϕ) is a critical zero, sin(2kϕ) sin(4kϕ − π) = 0 . It follows that, away from the origin, a critical zero (r, ϕ) of Fa satisfies the system (2k)

(6.8)

kLk

(2k)

(r 2 ) − r 2 L˙ k

(r 2 ) = 0,

2 sin(2kϕ) cos(4kϕ − π) − cos(2kϕ) sin(4kϕ − π) = 0.

The first equation has precisely (k−1) positive simple zeros rc,i , one in each interval ]ri , ri [ , for 1 ≤ i ≤ (k − 1). An easy analysis of the second shows that it has 4k simple zeros ϕc,j , one in each interval ]ϕj , ψ2j+1 [ , for 0 ≤ j ≤ 4k − 1 .

´ PIERRE BERARD AND BERNARD HELFFER

104

Property 6.2. The only possible critical zeros of the function Fa , away from the origin, are the points (rc,i , ϕc,j ) , for 1 ≤ i ≤ k − 1 and 0 ≤ j ≤ 4k − 1 , with corresponding finitely many values of a given by (6.5). In particular, there exists some a0 > 0 such that for 0 < a < a0 , the eigenfunction Fa has no critical zero away from the origin. The function Ga,b does not vanish at the origin (provided that b = 0). Its critical zeros are given by the system G(r, ϕ) = 0, ∂r G(r, ϕ) = 0, ∂ϕ G(r, ϕ) = 0.

(6.9)

We look at the situation for r large. Write (2k)

Lk

(6.10)

(t) =

L2k (t)

=

(−1)k k k! t 1 2k (2k)! t

+ Pk (t), + Qk (t),

where Pk and Qk are polynomials with degree (k − 1) and (2k − 1) respectively. The first and second equations in (6.9) are equivalent to the first and second equations of the system (6.11)

0 = 0 =

(−1)k k! (−1)k k!

sin(2kϕ) + a sin(4kϕ − π) +

b (2k)!

+ O( r12 ),

cos(2kϕ) + 2a cos(4kϕ − π) + O( r12 ),

where the O( r12 ) are uniform in ϕ and a, b (provided they are initially bounded). Property 6.3. There exist positive numbers a1 ≤ a0 , b1 , R1 , such that for 0 < a < a1 , 0 < |b| < b1 , and r > R1 , the function Ga,b (r, ϕ) has no critical zero. It follows that for fixed 0 < a < a1 , and b small enough (depending on a), the function Ga,b has no critical zero in R2 . Proof. Let α :=

(−1)k k!

sin(2kϕ) + a sin(4kϕ − π), β :=

b (2k)!

and

(−1)k cos(2kϕ) + 2a cos(4kϕ − π). k! 1 Compute (α + β)2 + γ 2 . For 0 < a < 2k! , one has γ :=

4|b| 1 4a − . − (2k!)2 k! k!(2k)! The first assertion follows. The second assertion follows from the first one and from Property 6.2.  (α + β)2 + γ 2 ≤

6.2. The checkerboard. Since a in positive, the nodal set of Fa satisfies (6.12)

L ⊂ N (Fa ) ⊂ L ∪ {AB < 0} ,

where L is the finite set N (A) ∩ N (B), more precisely, (6.13)

L = {(ri , ψm ) | 1 ≤ i ≤ k, 0 ≤ m ≤ 8k − 1} .

Let pi,m denote the point with polar coordinates (ri , ψm ). It is easy to check that the points pi,m are regular points of the nodal set N (Fa ). More precisely the nodal set N (Fa ) at these points is a regular arc transversal to the lines {ϕ = ψm } and

SOME NODAL PROPERTIES OF THE QUANTUM HARMONIC OSCILLATOR

105

Figure 7.  = 4k, k even. {r = ri }. Note also that the nodal set N (Fa ) can only cross the nodal sets N (A) or N (B) at the points in L . The connected components of the set {AB = 0} form a “polar checkerboard” whose white boxes are the connected components in which AB < 0 . The global aspect of the checkerboard depends on the parity of k . Recall that our assumption is that  = 4k , with k even. Figure 7 displays a partial view of the checkerboard, using the invariance under the rotation of angle πk . The thin lines labelled “R” correspond to the angles ψm , with m = 0, 1, 2, 3 . The thick lines to the angles ϕj , with j = 0, 1, 2 . The thick arcs of circle correspond to the values ri , with i = 1, 2, 3 and then i = k − 1, k . The light grey part represents the zone ri with i = 4, . . . , k − 2. The intersection points of the thin lines “R” with the thick arcs are the point in L , in the sector 0 ≤ ϕ ≤ ϕ2 . The outer arc of circle (in grey) represents the horizon. 6.3. Behavior at infinity. We now look at the behavior at infinity of the functions Fa and Ga,b . We restrict our attention to the sector {0 ≤ ϕ ≤ ϕ2 }. Recall that k is even. For r > rk , the nodal set N (Fa ) can only visit the white sectors S0 := {ϕ0 < ϕ < ψ0 }, S1 := {ψ1 < ϕ < ϕ1 }, and S2 := {ψ2 < ϕ < ψ3 }, issuing respectively from the points pk,0 , pk,1 or pk,2 , pk,3 . As above, we can write (6.14)

Fa (r, ϕ) = r 4k g(ϕ) + sin(2kϕ)Pk (r 2 ),

with

1 sin(2kϕ) + a sin(4kϕ − π), k! where we have used the fact that k is even. g(ϕ) =

106

´ PIERRE BERARD AND BERNARD HELFFER

π π • Analysis in S0 . We have 0 < ϕ < 4k . Note that g(0) g( 4k ) < 0 . On the  other-hand, g (ϕ) satisfies   1 π  g (ϕ) ≥ 2k cos( ) − 2a . k! 2

It follows that provided that 0 < a < π zero θ0 in the interval ]0, 4k [.

1 2 k!

cos( π 2 ), the function g has exactly one

It follows that for r big enough, the equation Fa (r, ϕ) = 0 has exactly one zero ϕ(r) π [, and this zero tends to θ0 when r tends to infinity. Looking in the interval ]0, 4k at (6.14) again, we see that ϕ(r) = θ0 + O( r12 ). It follows that the nodal set in the sector S0 is a line issuing from pk,0 and tending to infinity with the asymptote ϕ = θ0 . • Analysis in S1 . The analysis is similar to the analysis in S0 . • Analysis in S2 . In this case, we have that (2+)π < ϕ < (3+)π . It follows that 4k 4k π π 1 π − sin(2kϕ) ≥ min{sin( 2 ), cos( 2 )} > 0. If 0 < a < k! min{sin( 2 ), cos( π 2 )}, then Fa (r, ϕ) tends to negative infinity when r tends to infinity, uniformly in ϕ ∈]ψ2 , ψ3 [ . It follows that the nodal set of N (Fa ) is bounded in the sector S2 . 6.4. The nodal set N (Fa ) and N (Ga,b ). Proposition 6.4. For  = 4k, k even, and a positive small enough, the nodal set of Fa consists of three sets of “ovals” (1) a cluster of 2k closed (singular) curves, with a common singular point at the origin, (2) 2k curves going to infinity, tangentially to lines ϕ = ϑa,j (in the case of the sphere, they would correspond to a cluster of closed curves at the south pole), (3) 2k(k−1) disjoint simple closed curves (which correspond to the white cases at finite distance of Stern’s checkerboard for A and B). Proof. Since B vanishes at higher order than A at the origin, the behavior of the nodal set of Fa is well determined at the origin. More precisely, the nodal set of Fa at the origin consists of 4k semi-arcs, issuing from the origin tangentially to the lines ϕ = jπ/2k, for 0 ≤ j ≤ 4k − 1 . At infinity, the behavior of the nodal set of Fa is determined for a small enough in Subsection 6.3. An analysis a` la Stern, then shows that for a small enough there is a cluster of ovals in the intermediate  region {r2 < r < rk−1 }, when k ≥ 4. Fixing a small enough so that the preceding proposition holds, in order to obtain a regular nodal set, it suffices to perturb Fa into Ga,b , with b small enough, choosing its sign so that the nodal set Fa is desingularized at the origin, creating 2k ovals. Figure 8 displays the cases  = 8 (i.e. k = 2). Finally, we have constructed an eigenfunction Ga,b with 2k(k + 1) nodal component 2 so that μ(Ga,b ) ∼ 2k2 = 8 .

SOME NODAL PROPERTIES OF THE QUANTUM HARMONIC OSCILLATOR

107

Figure 8. Ovals for  = 8 . 7. On bounds for the length of the nodal set In Subsection 7.1, we obtain Theorem 1.7 as a corollary of a more general result, Proposition 7.2, which sheds some light on the exponent 32 . The proof is typically 2-dimensional, a` la Br¨ uning-Gromes [6, 7]. We consider more general potentials in Subsection 7.3. In Subsection 7.3, we extend the methods of Long Jin [29] to some Schr¨odinger operators. We obtain both lower and upper bounds on the length of the nodal sets in the classically permitted region, Proposition 7.10. 7.1. Lower bounds, proof ` a la Br¨ uning-Gromes. Consider the eigenvalue problem on L2 (R2 ) HV := −Δ + V (x), HV u = λ u,

(7.1)

for some suitable non-negative potential V such that the operator has discrete spectrum (see [24, Chapter 8]). More precisely, we assume: Assumption 7.1. The potential V is positive, continuous and tends to infinity at infinity. Introduce the sets

  BV (λ) := x ∈ R2 | V (x) < λ ,

(7.2) and, for r > 0 , (7.3)

(−r)

BV

  (λ) := x ∈ R2 | B(x, r) ⊂ BV (λ) ,

where B(x, r) is the open ball with center x and radius r. Proposition 7.2. Fix δ ∈]0, 1[ and ρ ∈]0, 1]. Under Assumption 7.1, for λ large enough, and for any nonzero eigenfunction u of HV , HV u = λu, the length of N (u) ∩ BV (δλ) is not less than   2(1 − δ) √ (−2ρ) (7.4) λ A B (δλ) . V 9π 2 j0,1

´ PIERRE BERARD AND BERNARD HELFFER

108

Proof of Proposition 7.2 Lemma 7.3. Choose some radius 0 < ρ ≤ 1, and let j0,1 ρδ := √ . 1−δ

(7.5)

 2 (−ρ) ρδ ) intersects the Then, for λ > ρρδ , and for any x ∈ BV (δλ), the ball B(x, √ λ nodal set N (u) of the function u. Proof of Lemma 7.3. Let r := √ρδλ . If the ball B(x, r) did not intersect N (u), then it would be contained in a nodal domain D of the eigenfunction u. Denoting by σ1 (Ω) the least Dirichlet eigenvalue of the operator HV in the domain Ω, by monotonicity, we could write λ = σ1 (D) ≤ σ1 (B(x, r)) .  2 (−ρ) Since x ∈ BV (δλ) and λ > ρρδ , the ball B(x, r) is contained in BV (δλ), and we can bound V from above by δλ in this ball. It follows that σ1 (B(x, r)) < This leads to a contradiction with the definition of ρδ .

2 j0,1 r2

+ δλ. 

Consider the set F of finite subsets {x1 , . . . , xn } of R2 with the following properties, ' (−ρ) xi ∈ N (u) ∩ BV (δλ), 1 ≤ i ≤ n, (7.6) B(xi , √ρδλ ), 1 ≤ i ≤ n, pairwise disjoint. For λ large enough, the set F is not empty, and can be ordered by inclusion. It admits a maximal element {x1 , . . . , xN } , where N depends on δ, ρ, λ and u. (−2ρ)

√ δ ), 1 ≤ i ≤ N , cover the set B Lemma 7.4. The balls B(xi , 3ρ V λ

(δλ) .

Proof of Lemma 7.4. Assume the claim in not true, i.e. that there exists some (−2ρ) √ δ for all i ∈ {1, . . . , N }. Since y ∈ (δλ) such that |y − xi | > 3ρ y ∈ BV λ (−ρ)

BV

ρδ (δλ) , by Lemma 7.3, there exists some x ∈ N (u) ∩ B(y, √ ), and we have λ (−ρ)

√δ . x ∈ BV (δλ). Furthermore, for all i ∈ {1, . . . , N }, we have |x − xi | ≥ 2ρ λ The set {x, x1 , x2 , . . . , xN } would belong to F, contradicting the maximality of {x1 , x2 , . . . , xN }. 

Lemma 7.4 gives a lower bound on the number N ,   λ (−2ρ) (7.7) N≥ A B (δλ) , V 9π 2 ρ2δ where A(Ω) denotes the area of the set Ω . Lemma 7.5. For any α < j0,1 , the ball B(x, √αλ ) does not contain any closed connected component of the nodal set N (u).

SOME NODAL PROPERTIES OF THE QUANTUM HARMONIC OSCILLATOR

109

Proof of Lemma 7.5. Indeed, any closed connected component of N (u) contained in B(x, √αλ ) would bound some nodal domain D of u, contained in B(x, √αλ ), and we would have   2 j0,1 α λ = σ1 (D) ≥ σ1 B(x, √ ≥ 2 λ, α λ contradicting the assumption on α .  (−ρ)

Take the maximal set {x1 , . . . , xN } ⊂ N (u) ∩ BV (δλ) constructed above. The balls B(xi , √ρδλ ) are pairwise disjoint, and so are the balls B(xi , √αλ ) for any 0 < α < j0,1 . There are at least two nodal arcs issuing from a point xi , and they must exit B(xi , √αλ ), otherwise we could find a closed connected component of N (u) inside 2α . this ball, contradicting Lemma 7.5. The length of N (u) ∩ B(xi , √αλ ) is at least √ λ 2α Finally, the length of N (u) ∩ BV (δλ) is at least N √λ which is bigger than   2α √ (−2ρ) λA BV (δλ) . 2 2 9π ρδ 

Since this is true for any α < j0,1 .

Proof of Theorem 1.7. We apply the preceding proposition with V (x) = 1 1 (−r) |x|2k and ρ = 1. Then, BV (λ) = B(λ 2k ) and BV (λ) = B(λ 2k − r). In this 1 1 case, the length of the nodal set is bounded by some constant times λ 2 + k ≈ 1  λ 2 A (BV (δλ)). When k = 1, we obtain Proposition 1.7. Remark. The above proof sheds some light on the exponent

3 2

in Proposition 1.7.

7.2. More general potentials. We reinterpret Proposition 7.2 for more general potentials V (x), under natural assumptions which appear in the determination of the Weyl’s asymptotics of HV (see [32], [23]). After renormalization, we assume: Assumption 7.6. V is of class C 1 , V ≥ 1, and there exist some positive constants ρ0 and C1 such that for all x ∈ R2 , |∇V (x)| ≤ C1 V (x)1−ρ0 .

(7.8)

Note that under this assumption there exist positive constants r0 and C0 such that (7.9)

x, y satisfy |x − y| ≤ r0 ⇒ V (x) ≤ C0 V (y).

The proof is easy. We first write V (x) ≤ V (y) + |x − y| sup |∇V (z)|. z∈[x,y]

Applying (7.8) (here we only use ρ0 ≥ 0), we get V (x) ≤ V (y) + C1 |x − y| sup V (z). z∈[x,y]

We now take x ∈ B(y, r) for some r > 0 and get sup x∈B(y,r)

V (x) ≤ V (y) + C1 r

sup x∈B(y,r)

V (x),

´ PIERRE BERARD AND BERNARD HELFFER

110

which we can rewrite, if C1 r < 1, in the form V (y) ≤

sup

V (x) ≤ V (y)(1 − C1 r)−1 .

x∈B(y,r)

This is more precise than (7.9) because we get C0 (r0 ) = (1 − C1 r0 )−1 , which tends to 1 as r0 → 0 . We assume Assumption 7.7. For any δ ∈]0, 1[, there exists some positive constants Aδ and λδ such that (7.10)

1 < A(BV (λ))/A(BV (δλ)) ≤ Aδ , ∀λ ≥ λδ .

Proposition 7.8. Fix δ ∈ (0, 1), and assume that V satisfies the previous assumptions. Then, there exists a positive constant Cδ (depending only on the constants appearing in the assumptions on V ) and λδ such that for any eigenpair (u, λ) of 1 HV with λ ≥ λδ , the length of N (u) ∩ BV (δλ) is larger than Cδ λ 2 A(BV (λ)). Proof. Using (7.10), it is enough to prove the existence of r1 such that, for 0 < r < r1 , there exists C2 (r) and M (r) s.t. BV (μ − C2 μ1−ρ0 ) ⊂ BV−r (μ), ∀μ > M (r). But, if x ∈ BV (μ − C2 μ1−ρ0 ), and y ∈ B(x, r), we have V (y) ≤ V (x) + C1 C0 (r)1−ρ0 rV (x)1−ρ0 ≤ μ − C2 μ1−ρ0 + C1 C0 (r)1−ρ0 rμ1−ρ0 . 1

Taking C2 (r) = C1 C0 (r)1−ρ0 r and M (r) ≥ (C2 (r) + 1) ρ0 gives the result.



Remarks. (1) The method of proof of Proposition 7.2, which is reminiscent of the proof by Br¨ uning [7] (see also [6]) is typically 2-dimensional. (2) The same method could be applied to a Schr¨ odinger operator on a complete noncompact Riemannian surface, provided one has some control on the geometry, the first eigenvalue of small balls, etc.. (3) If we assume that there exist positive constants m0 ≤ m1 and C3 such that for any x ∈ R2 , 1 (7.11) < x >m0 ≤ V (x) ≤ C3 < x >m1 , C3  where < x >:= 1 + |x|2 , then A(BV (λ)) has a controlled growth at ∞. (4) If m0 = m1 in (7.11), then (7.10) is satisfied. The control of Aδ as δ → +1 can be obtained under additional assumptions. 7.3. Upper and lower bounds on the length of the nodal set: the semi-classical approach of Long Jin. In [29], Long Jin analyzes the same question in the semi-classical context for a Schr¨ odinger operator HW,h := −h2 Δg + W (x), where Δg is the Laplace-Beltrami operator on the compact connected analytic Riemannian surface (M, g), with W analytic. In this context, he shows that if (uh , λh ) is an h-family of eigenpairs of HW,h such that λh → E, then the length of the zero set of uh inside the classical region W −1 (] − ∞, E]) is of order h−1 .

SOME NODAL PROPERTIES OF THE QUANTUM HARMONIC OSCILLATOR

111

Although not explicitly done in [29], the same result is also true in the case of M = R2 under the condition that lim inf W (x) > E ≥ inf W , keeping the assumption that W is analytic. Let us show how we can reduce the case M = R2 to the compact situation. Proposition 7.9. Let us assume that W is continuous and that there exists E1 such that W −1 (] − ∞, E1 ]) is compact. Then the bottom of the essential spectrum of HW,h is bigger than E1 . Furthermore, if (λh , uh ) is a family (h ∈]0, h0 ]) of eigenpairs of HW,h such that limh→0 λh = E0 with E0 < E1 and ||uh || = 1, then given K a compact neighborhood of W −1 (] − ∞, E0 ]), there exists K > 0 such that ||uh ||L2 (K) = 1 + O (exp(− K /h)) , as h → 0 . This proposition is a consequence of Agmon estimates (see Helffer-Sj¨ ostrand [21] or Helffer-Robert [20] for a weaker result with a remainder in OK (h∞ )) measuring the decay of the eigenfunctions in the classically forbidden region. This can also be found in a weaker form in the recent book of M. Zworski [38] (Chapter 7), which also contains a presentation of semi-classical Carleman estimates. Observing that in the proof of Long Jin the compact manifold M can be replaced by any compact neighborhood of W −1 (] − ∞, E0 ]), we obtain: Proposition 7.10. Let us assume in addition that W is analytic in some compact neighborhood of W −1 (] − ∞, E0 ]), then the length of the zero set of uh inside the classical region W −1 (] − ∞, E0 ]) is of order h−1 . More precisely, there exist C > 0 and h0 > 0 such that for all h ∈]0, h0 ] we have (7.12)



1 −1 h ≤ length N (uh ) ∩ W −1 (] − ∞, E0 ]) ≤ C h−1 . C

Remark 7.11. As observed in [29] (Remark 1.3), the results of [18] suggest that the behavior of the nodal sets in the classically forbidden region could be very different from the one in the classically allowed region. We can by scaling recover Proposition 1.7, and more generally treat the eigenpairs of −Δx + |x|2k . Indeed, assume that (−Δx + |x|2k )u(x) = λu(x). Write x = ρ y. k+1 Then, (−ρ−2 Δy +ρ2k |y|2k −λ)u(ρy) = 0. If we choose ρ2k = λ, h = ρ−k−1 = λ− 2k 1 1 and let vh (y) = h 2(k+1) y) u(h k+1 y), then, (−h2 Δy + |y|2k − 1)vh (y) = 0. Applying (7.12) to the family vh and rescaling back to the variable x, we find that (7.13)



k+2 1 k+2 λ 2k ≤ length N (u) ∩ {x ∈ R2 | |x|2k < λ} ≤ C λ 2k . C

With this extension of Long Jin’s statement, when V = |x|2k , we also obtain an upper bound of the length of N (u) in BV (λ). Note that when k → +∞ , the problem tends to the Dirichlet√problem in a ball of size 1. We then recover that the length of N (u) is of order λ. The above method can also give results in the non-homogeneous case, at least when (7.11) is satisfied with m0 = m1 . We can indeed prove the following generalization.

´ PIERRE BERARD AND BERNARD HELFFER

112

Proposition 7.12. Let us assume that there exist m ≥ 1, 0 > 0 and C > 0 such that V is holomorphic in D := {z = (z1 , z2 ) ∈ C2 , |z| ≤ 0 < z >} and satisfies |V (z)| ≤ C < z >m , ∀z ∈ D.

(7.14)

Suppose in addition that we have the ellipticity condition 1 < x >m ≤ V (x), ∀x ∈ R2 . (7.15) C Then, for any > 0, the length N (u) ∩ (BV (λ) \ BV ( λ)) for an eigenpair (u, λ) of 1 2 HV , is of the order of λ 2 + m as λ → +∞. Moreover, one can take = 0 when V is a polynomial. Proof. The lower bound was already obtained by a more general direct approach in Proposition 7.8. One can indeed verify using Cauchy estimates that (7.14) and (7.15) imply (7.8) and (7.11), with ρ0 = 1/2m. Under the previous assumptions, we consider Wλ (y) = λ−1 V (λ m y), vλ (y) = λ 4m u(λ m y). 1

1

1

We observe that with (7.16)

h = λ− 2 − m , 1

1

odinger operator −h2 Δy + the pair (vλ , 1) is an eigenpair for the semi-classical Schr¨ Wλ (y): (−h2 Δ + Wλ )vλ = vλ . It remains to see if we can extend the result of Long Jin to this situation. We essentially follow his proof, whose basic idea goes back to Donnelly-Feffermann [15]. The difference being that Wλ depends on h through (7.16). The inspection of the proof2 shows that there are three points to control. Analyticity What we need is to have for any y0 in R2 \ {0} a complex neighborhood V of y0 , h0 > 0 and C such that, for any h ∈]0, h0 ], vλ admits an holomorphic extension in V with   C (7.17) sup |vλ | ≤ C exp ||vλ ||L∞ (R2 ) . h V This can be done by using the FBI transform, controlling the uniformity when W is replaced by Wλ . But this is exactly what is given by Assumption (7.14). Notice that this is not true in general for y0 = 0. We cannot in general find a λ-independent neighborhhod of 0 in C2 where Wλ is defined and bounded. Note here that ||vλ ||L∞ (R2 ) is by standard semiclassical analysis O(h−N ) for some N. When V is in addition a polynomial: m  V (x) = Pj (x) j=0 2 We

refer here to the proof of (2.20) in [29].

SOME NODAL PROPERTIES OF THE QUANTUM HARMONIC OSCILLATOR

113

where Pj is an homogeneous polynomial of degree j, we get Wλ (y) = Pm (y) +

m 

λ− m Pm− (y), 

=1

and we can verify the uniform analyticity property for any y0 . Uniform confining As we have mentioned before, Long Jin’s paper was established in the case of a compact manifold (in this case and for Laplacians, it is worth to mention the papers of Sogge-Zelditch [34,35]) but it can be extended to the case of R2 under the condition that the potential is confining, the length being computed in a compact containing the classically permitted region. This is the case with Wλ . Note that if Wλ (y) ≤ C1 , then we get λ−1 V (λ m y) ≤ C1 , 1

which implies by the ellipticity condition

1 1 −1 m |λ |m |y|m C λ

≤ C1 , that is

|y| ≤ (C  C1 ) . 1 m

Uniform doubling property Here instead of following Long Jin’s proof, it is easier to refer to the results of Bakri-Casteras [1], which give an explicit control in term of the C 1 norm of Wλ . As before, we have to use our confining assumption in order to establish our result in any bounded domain Ω in R2 containing uniformly the classically permitted area Wλ−1 (]−∞, +1]). This last assumption permits indeed to control the L2 -norm of vλ from below in Ω. We actually need the two following estimates (we assume (7.16)): Given Ω like above, for any R > 0, there exists CR such that, for any (x, R) such that B(x, R) ⊂ Ω ,   CR (7.18) ||vλ ||L2 (B(x,R)) ≥ exp − . h Given Ω like above, there exists C such that, for any (x, r) such that B(x, 2r) ⊂ Ω ,   C ||vλ ||L2 (B(x,r)) . (7.19) ||vλ ||L2 (B(x,2r)) ≤ exp h Here we have applied Theorem 3.2 and Proposition 3.3 in [1] with electric potential h−2 (Wλ − 1). These two statements involve the square root of the C 1 norm of the electric potential in Ω, which is O(h−1 ) in our case. End of the proof Hence, considering an annulus A( 0 , R0 ) we get following Long Jin that the length of the nodal set of vλ in this annulus is indeed of order O(h−1 ) and after rescaling we get the proposition for the eigenpair (u, λ). In the polynomial case, we get the  same result but in the ball B(0, R0 ). Remarks. (1) Long Jin’s results hold in dimension n, not only in dimension 2. The above extensions work in any dimension as well, replacing the length by the (n − 1)-Hausdorff measure.

114

´ PIERRE BERARD AND BERNARD HELFFER

(2) As observed in [29], the results in [18] suggest that the behavior of nodal sets in the classically forbidden region could be very different from the one in the classically allowed region. (3) Under the assumptions of Proposition 7.12, one gets from Theorem 1.1 in [1] that the order of a critical point of the zero set of an eigenfunction of HV associated with λ in the classically permitted region is at most of order 1 1 λ 2 + m . Let us emphasize that here no assumption of analyticity for V is used. On the other hand, note that using Courant’s theorem and Euler’s and Weyl’s formulas, one can prove that the number of critical points in 2 the classically allowed region is at most of order λ1+ m . When m = 2, we can verify from the results in Section 6 that this upper bound cannot be improved in general. (4) For nodal sets in forbidden regions, see [8]. Acknowledgements The authors would like to thank P. Charron for providing an earlier copy of [9], T. Hoffmann-Ostenhof for pointing out [26], J. Leydold for providing a copy of his master degree thesis [26], Long Jin for enlightening discussions concerning [29] and Section 7, I. Polterovich for pointing out [9], as well as D. Jakobson and M. PerssonSundqvist for useful comments. During the preparation of this paper, B. H. was Simons foundation visiting fellow at the Isaac Newton Institute in Cambridge. Finally, the authors would like to thank the referee for his remarks which helped improve the paper. References [1] Laurent Bakri and Jean-Baptiste Casteras, Quantitative uniqueness for Schr¨ odinger operator with regular potentials, Math. Methods Appl. Sci. 37 (2014), no. 13, 1992–2008, DOI 10.1002/mma.2951. MR3245115 [2] Pierre B´ erard and Bernard Helffer, Dirichlet eigenfunctions of the square membrane: Courant’s property, and A. Stern’s and ˚ A. Pleijel’s analyses, Analysis and geometry, Springer Proc. Math. Stat., vol. 127, Springer, Cham, 2015, pp. 69–114, DOI 10.1007/978-3-319-174433 6. MR3445517 [3] P. B´ erard and B. Helffer. On the number of nodal domains for the 2D isotropic quantum harmonic oscillator, an extension of results of A. Stern. arXiv:1409.2333v1 (September 8, 2014). [4] P. B´ erard and B. Helffer, A. Stern’s analysis of the nodal sets of some families of spherical harmonics revisited, Monatsh. Math. 180 (2016), no. 3, 435–468, DOI 10.1007/s00605-0150788-6. MR3513215 [5] P. B´ erard and B. Helffer. Edited extracts from Antonie Stern’s thesis. S´ eminaire de Th´eorie spectrale et g´eom´ etrie, Institut Fourier, Grenoble, 32 (2014-2015), 39–72. ¨ [6] Jochen Br¨ uning and Dieter Gromes, Uber die L¨ ange der Knotenlinien schwingender Membranen (German), Math. Z. 124 (1972), 79–82, DOI 10.1007/BF01142586. MR0287202 ¨ [7] Jochen Br¨ uning, Uber Knoten von Eigenfunktionen des Laplace-Beltrami-Operators (German), Math. Z. 158 (1978), no. 1, 15–21, DOI 10.1007/BF01214561. MR0478247 [8] Yaiza Canzani and John A. Toth, Nodal sets of Schr¨ odinger eigenfunctions in forbidden regions, Ann. Henri Poincar´e 17 (2016), no. 11, 3063–3087, DOI 10.1007/s00023-016-0488-3. MR3556516 [9] P. Charron. On Pleijel’s theorem for the isotropic harmonic oscillator. M´emoire de Maˆıtrise, Universit´ e de Montr´ eal (June 2015). [10] P. Charron. A Pleijel type theorem for the quantum harmonic oscillator. To appear in Journal of Spectral Theory. arXiv:1512.07880.

SOME NODAL PROPERTIES OF THE QUANTUM HARMONIC OSCILLATOR

115

[11] P. Charron, B. Helffer and T. Hoffmann-Ostenhof. Pleijel’s theorem for Schr¨ odinger operators with radial potentials. To appear in Annales math´ ematiques du Qu´ebec. arXiv:1604.08372. [12] R. Courant. Ein allgemeiner Satz zur Theorie der Eigenfunktionen selbstadjungierter Differentialausdr¨ ucke. Nachr. Ges. G¨ ottingen (1923), 81-84. [13] R. Courant and D. Hilbert. Methods of Mathematical Physics. Volume 1. First English edition. Translated and revised from the German original, Winheim 2004. [14] Harold Donnelly, Counting nodal domains in Riemannian manifolds, Ann. Global Anal. Geom. 46 (2014), no. 1, 57–61, DOI 10.1007/s10455-013-9408-7. MR3205801 [15] Harold Donnelly and Charles Fefferman, Nodal sets of eigenfunctions on Riemannian manifolds, Invent. Math. 93 (1988), no. 1, 161–183, DOI 10.1007/BF01393691. MR943927 [16] Alexandre Eremenko, Dmitry Jakobson, and Nikolai Nadirashvili, On nodal sets and nodal domains on S 2 and R2 (English, with English and French summaries), Ann. Inst. Fourier (Grenoble) 57 (2007), no. 7, 2345–2360. Festival Yves Colin de Verdi`ere. MR2394544 [17] G. Gauthier-Shalom and K. Przybytkowski. Description of a nodal set on T2 . McGill University Research Report 2006 (unpublished). [18] Boris Hanin, Steve Zelditch, and Peng Zhou, Nodal sets of random eigenfunctions for the isotropic harmonic oscillator, Int. Math. Res. Not. IMRN 13 (2015), 4813–4839, DOI 10.1093/imrn/rnu071. MR3439093 [19] B. Helffer, T. Hoffmann-Ostenhof, and S. Terracini, Nodal domains and spectral minimal partitions, Ann. Inst. H. Poincar´e Anal. Non Lin´eaire 26 (2009), no. 1, 101–138, DOI 10.1016/j.anihpc.2007.07.004. MR2483815 [20] B. Helffer and D. Robert, Calcul fonctionnel par la transformation de Mellin et op´ erateurs admissibles (French, with English summary), J. Funct. Anal. 53 (1983), no. 3, 246–268, DOI 10.1016/0022-1236(83)90034-4. MR724029 [21] B. Helffer and J. Sj¨ ostrand, Multiple wells in the semiclassical limit. I, Comm. Partial Differential Equations 9 (1984), no. 4, 337–408, DOI 10.1080/03605308408820335. MR740094 [22] Don Hinton, Sturm’s 1836 oscillation results evolution of the theory, Sturm-Liouville theory, Birkh¨ auser, Basel, 2005, pp. 1–27, DOI 10.1007/3-7643-7359-8 1. MR2145075 [23] Lars H¨ ormander, On the asymptotic distribution of the eigenvalues of pseudodifferential operators in Rn , Ark. Mat. 17 (1979), no. 2, 297–313, DOI 10.1007/BF02385475. MR608322 [24] R.S. Laugesen. Spectral Theory of Partial Differential Equations. University of Illinois, Urbana-Champain 2014. https://wiki.cites.illinois.edu/wiki/display/MATH595STP/ Math+595+STP [25] Hans Lewy, On the minimum number of domains in which the nodal lines of spherical harmonics divide the sphere, Comm. Partial Differential Equations 2 (1977), no. 12, 1233–1244, DOI 10.1080/03605307708820059. MR0477199 [26] J. Leydold. Knotenlinien und Knotengebiete von Eigenfunktionen. Diplomarbeit 1989, University of Vienna, Austria. Unpublished, available at http://othes.univie.ac.at/34443/ [27] Josef Leydold, On the number of nodal domains of spherical harmonics, Topology 35 (1996), no. 2, 301–321, DOI 10.1016/0040-9383(95)00028-3. MR1380499 [28] A. Logunov and E. Malinnikova. Nodal sets of Laplace eigenfunctions: estimates of the Hausdorff measure in dimension two and three. arXiv:1605.02595. [29] Long Jin, Semiclassical Cauchy estimates and applications, Trans. Amer. Math. Soc. 369 (2017), no. 2, 975–995, DOI 10.1090/tran/6715. MR3572261 [30] ˚ Ake Pleijel, Remarks on Courant’s nodal line theorem, Comm. Pure Appl. Math. 9 (1956), 543–550, DOI 10.1002/cpa.3160090324. MR0080861 [31] Iosif Polterovich, Pleijel’s nodal domain theorem for free membranes, Proc. Amer. Math. Soc. 137 (2009), no. 3, 1021–1024, DOI 10.1090/S0002-9939-08-09596-8. MR2457442 [32] D. Robert, Propri´ et´ es spectrales d’op´ erateurs pseudo-diff´ erentiels (French), Comm. Partial Differential Equations 3 (1978), no. 9, 755–826, DOI 10.1080/03605307808820077. MR504628 [33] Jimena Royo-Letelier, Segregation and symmetry breaking of strongly coupled two-component Bose-Einstein condensates in a harmonic trap, Calc. Var. Partial Differential Equations 49 (2014), no. 1-2, 103–124, DOI 10.1007/s00526-012-0571-7. MR3148108 [34] Christopher D. Sogge and Steve Zelditch, Lower bounds on the Hausdorff measure of nodal sets, Math. Res. Lett. 18 (2011), no. 1, 25–37, DOI 10.4310/MRL.2011.v18.n1.a3. MR2770580

116

´ PIERRE BERARD AND BERNARD HELFFER

[35] Christopher D. Sogge and Steve Zelditch, Lower bounds on the Hausdorff measure of nodal sets II, Math. Res. Lett. 19 (2012), no. 6, 1361–1364, DOI 10.4310/MRL.2012.v19.n6.a14. MR3091613 [36] A. Stern. Bemerkungen u ¨ ber asymptotisches Verhalten von Eigenwerten und Eigenfunktionen. Inaugural-Dissertation zur Erlangung der Doktorw¨ urde der Hohen MathematischNaturwissenschaftlichen Fakult¨ at der Georg August-Universit¨ at zu G¨ ottingen (30 Juli 1924). Druck der Dieterichschen Universit¨ ats-Buchdruckerei (W. Fr. Kaestner). G¨ ottingen, 1925. Partial extracts in the above reference [5]. [37] G´ abor Szeg˝ o, Orthogonal polynomials, 4th ed., American Mathematical Society, Providence, R.I., 1975. American Mathematical Society, Colloquium Publications, Vol. XXIII. MR0372517 [38] Maciej Zworski, Semiclassical analysis, Graduate Studies in Mathematics, vol. 138, American Mathematical Society, Providence, RI, 2012. MR2952218 Institut Fourier, Universit´ e Grenoble Alpes and CNRS, B.P.74, F38402 Saint Martin d’H` eres Cedex, France E-mail address: [email protected] Laboratoire Jean Leray, Universit´ e de Nantes and CNRS, F44322 Nantes Cedex France — and — Laboratoire de Math´ ematiques, Univ. Paris-Sud 11 E-mail address: [email protected]

Contemporary Mathematics Volume 700, 2017 http://dx.doi.org/10.1090/conm/700/14185

Numerical solution of linear eigenvalue problems Jessica Bosch and Chen Greif Abstract. We review numerical methods for computing eigenvalues of matrices. We start by considering the computation of the dominant eigenpair of a general dense matrix using the power method, and then generalize to orthogonal iterations and the QR iteration with shifts. We also consider divide-andconquer algorithms for tridiagonal matrices. The second part of this survey involves the computation of eigenvalues of large and sparse matrices. The Lanczos and Arnoldi methods are developed and described within the context of Krylov subspace eigensolvers. We also briefly present the idea of the Jacobi–Davidson method.

1. Introduction Eigenvalue problems form one of the central problems in Numerical Linear Algebra. They arise in many areas of sciences and engineering. In this survey, we study linear eigenvalue problems. The standard algebraic eigenvalue problem has the form Ax = λx. We consider real eigenvalue problems, i.e., A ∈ Rn×n . In some places we will use the notion of complex matrices, as they are crucial in mathematical as well as computational aspects of eigenvalue solvers. There is a wide range of publications dealing with numerical methods for solving eigenvalue problems, e.g., [22, 32, 39, 47, 54, 55, 62, 66]. Typically, eigensolvers are classified into methods for symmetric (or Hermitian) and nonsymmetric (or non-Hermitian) matrices, or methods for small, dense matrices and large, sparse matrices. This survey reviews popular methods for computing eigenvalues of a given matrix. It follows a minicourse presented by the second author at the 2015 Summer School on “Geometric and Computational Spectral Theory” at the Universit´e de Montr´eal, and can be viewed as a set of comprehensive lecture notes. When it comes to numerical computation of eigenvalues, it is reasonable to classify eigensolvers by the size and the nonzero pattern of the matrix. As opposed to the solution of linear systems, where it is possible to obtain a solution within a finite number of steps, most eigenvalue computations (except trivial cases such as a diagonal or a triangular matrix) require an iterative process. For matrices that are not particularly large and do not have a specific nonzero structure, eigensolvers are 2010 Mathematics Subject Classification. Primary 65F15. c 2017 American Mathematical Society

117

118

JESSICA BOSCH AND CHEN GREIF

often based on matrix decompositions. One may be interested in a small number of the eigenvalues and/or eigenvectors, or all of them, and there are methods that are available for accomplishing the stated goal. On the other hand, when the matrix is large and sparse, it is rare to seek the entire spectrum; in most cases we are interested in just a few eigenvalues and eigenvectors, and typical methods are based on matrix-vector products rather than matrix decompositions. Interestingly, despite the fact that all processes of eigenvalue computations are iterative, methods that are based on matrix decompositions are often referred to as direct, whereas methods that are based on matrix-vector products are termed iterative. This slight abuse of terminology is nonetheless widely understood and typically does not cause any confusion. It is a bit ambitious to talk in general terms about a recipe for solution of eigenvalue problems, but it is legitimate to identify a few main components. A typical eigensolver starts with applying similarity transformations and transforming the matrix into one that has an appealing nonzero structure: for example tridiagonal if the original matrix was symmetric. Once this is accomplished, an iterative process is pursued, whereby repeated orthogonal similarity transformations are applied to get us closer and closer to a diagonal or triangular form. For large and sparse matrices, an additional component, generally speaking, in state of the art methods, is the transformation of the problem to a small and dense one on a projected subspace. This survey devotes a significant amount of space to elaborating on the above principles. It is organized as follows. In Section 2, we briefly review basic concepts of Numerical Linear Algebra that are related to eigenvalue problems. We start with presenting methods for computing a few or all eigenvalues for small to moderatesized matrices in Section 3. This is followed by a review of eigenvalue solvers for large and sparse matrices in Section 4. Conclusions complete the paper. 2. Background in Numerical Linear Algebra 2.1. Theoretical basics. We begin our survey with a review of basic concepts in Numerical Linear Algebra. We introduce some notation used throughout the survey. Let A ∈ Cm×n . The kernel or nullspace of A is given as ker(A) = {x ∈ Cn : Ax = 0} . Another important subspace, which is often related to the kernel, is the range of A, which is given as ran(A) = {Ax : x ∈ Cn } . The rank of A is the maximal number of linearly independent columns (or rows), i.e., rank(A) = dim (ran(A)) . It holds n = rank(A) + dim (ker(A)). A is called rank-deficient if rank(A) < min{m, n}. In what follows, we consider real matrices A ∈ Rn×n if not stated otherwise. Definition 2.1 (Invertibility). A is called invertible or nonsingular if there exists a matrix B ∈ Rn×n such that AB = BA = I.

NUMERICAL SOLUTION OF LINEAR EIGENVALUE PROBLEMS

119

Here, I ∈ Rn×n is the identity matrix. The inverse of A is uniquely determined, and we denote it by A−1 . Related to the inverse and a matrix norm  ·  is the condition number, which is defined for a general square matrix A as κ(A) = A A−1 . In general, if κ(A) is large1 , then A is said to be an ill-conditioned matrix. Useful matrix norms include the well-known p-norms or the Frobenius norm  · F , which is given as ( ) n ) n  |ai,j |2 , AF = * i=1 j=1

where ai,j is the (i, j) entry of A. Let us come to the heart of this paper. The algebraic eigenvalue problem has the following form: Definition 2.2 (Algebraic Eigenvalue Problem). λ ∈ C is called an eigenvalue of A if there exists a vector 0 = x ∈ Cn such that Ax = λx. The vector x is called a (right) eigenvector of A associated with λ. We call the pair (λ, x) an eigenpair of A. The set of all eigenvalues of A is called the spectrum of A and is denoted by λ(A). Note from the above definition that real matrices can have complex eigenpairs. Geometrically, the action of a matrix A expands or shrinks any vector lying in the direction of an eigenvector of A by a scalar factor. This scalar factor is given by the corresponding eigenvalue of A. Remark 2.3. Similarly, a left eigenvector of A associated with the eigenvalue λ is defined as a vector 0 = y ∈ Cn that satisfies y ∗ A = λy ∗ . Here, y ∗ = y¯T is the conjugate transpose of y. Throughout the survey, we use the term eigenvector for a right eigenvector. The eigenvalues of A can be used to determine the invertibility of A. Definition 2.4 (Determinant). If λ(A) = {λ1 , . . . , λn }, then the determinant of A is given as n  λi . det(A) = i=1

A is nonsingular if and only if det(A) = 0. Another way to define eigenvalues is the following: Definition 2.5 (Characteristic Polynomial). The polynomial pA (x) = det(A − xI) is called the characteristic polynomial of A. It is a polynomial of degree n. The roots of pA (x) are the eigenvalues of A. 1 Of

course, this depends on the definition of “large”; see, e.g., [22, Chap. 3.5].

120

JESSICA BOSCH AND CHEN GREIF

A useful concept for eigenvalues solvers is the Rayleigh quotient: Definition 2.6 (Rayleigh Quotient). Let 0 = z ∈ Cn . The Rayleigh quotient of A and z is defined by z ∗ Az RA (z) = ∗ . z z Note that if z is an eigenvector of A, then the Rayleigh quotient is the corresponding eigenvalue. Another way to express the eigenvalue problem in Definition 2.2 is that (λ, x) is an eigenpair of A if and only if 0 = x ∈ ker(A − λI). Based on this kernel, we can define the eigenspace of A: Definition 2.7 (Eigenspace). Eλ (A) = ker(A − λI) is the eigenspace of A corresponding to λ. The following concept of invariant subspaces bears similarities to the eigenvalue problem. Definition 2.8 (Invariant Subspace). A subspace S ⊂ Cn is said to be invariant under a matrix A ∈ Cn×n (A-invariant) if AS ⊂ S. Definition 2.9. Let A ∈ Cn×n , S ⊂ Cn , and S ∈ Cn×k with k = rank(S) = dim(S) ≤ n and S = ran(S). Then, S is A-invariant if and only if there exists a matrix B ∈ Ck×k such that AS = SB. Remark 2.10. Using Definition 2.9, it is easy to show the following relations: • If (λ, x) is an eigenpair of B, then (λ, Sx) is an eigenpair of A. Hence, λ(B) ⊂ λ(A). • If k = n, then S is invertible and hence A = SBS −1 . This means A and B are similar and λ(B) = λ(A). This concept is introduced next. Most of the presented algorithms will transform a matrix A into simpler forms, such as diagonal or triangular matrices, in order to simplify the original eigenvalue problem. Transformations that preserve the eigenvalues of matrices are called similarity transformations. Definition 2.11 (Similarity Transformation). Two matrices A, B ∈ Cn×n are said to be similar if there exists a nonsingular matrix C ∈ Cn×n such that A = CBC −1 . The mapping B → A is called a similarity transformation. The similarity of A and B implies that they have the same eigenvalues. If (λ, x) is an eigenpair of B, then (λ, Cx) is an eigenpair of A. The simplest form to which a matrix can be transformed is a diagonal matrix. But as we will see, this is not always possible. Definition 2.12 (Diagonalizability). If A ∈ Cn×n is similar to a diagonal matrix, then A is said to be diagonalizable.

NUMERICAL SOLUTION OF LINEAR EIGENVALUE PROBLEMS

121

A similarity transformation in which C is orthogonal (or unitary), i.e., C T C = I (or C ∗ C = I), is called orthogonal (or unitary) similarity transformation. Unitary/orthogonal similarity transformations play a key role in numerical computations since C2 = 1. Considering the calculation of similarity transformations, it can be shown (cf. [22, Chap. 7.1.5]) that the roundoff error E satisfies E ≈ machine κ2 (C)A2 . Here, machine is the machine precision 2 and κ2 (C) the condition number of C with respect to the 2-norm. In particular, κ2 (C) is the error gain. Therefore, if the similarity transformation is unitary, we get E ≈ machine A2 and hence no amplification of error. Theorem 2.13 (Unitary Diagonalizability; see [22, Cor. 7.1.4].). A ∈ Cn×n is unitarily diagonalizable if and only if it is normal (A∗ A = AA∗ ). Now, let us show the connection between a similarity transformation of a matrix A and its eigenpairs: It follows from Definition 2.5 and the Fundamental Theorem of Algebra that A has n (not necessarily distinct) eigenvalues. If we denote the n eigenpairs by (λ1 , x1 ), . . . , (λn , xn ), i.e. Axi = λi xi for i = 1, . . . , n, we can write (2.1)

AX = XΛ,

where Λ = diag(λi )i=1,...,n ∈ C is a diagonal matrix containing the eigenvalues, and X = [x1 | . . . |xn ] ∈ Cn×n is a matrix whose columns are formed by the eigenvectors. This looks almost as a similarity transformation. In fact, the “only” additional ingredient we need is the invertibility of X. Under the assumption that X is nonsingular, we obtain X −1 AX = Λ, and hence, A and Λ are similar. But when can we expect of X to be nonsingular? To discuss this, we introduce some terminology: n×n

Definition 2.14 (Multiplicity). Let λ be an eigenvalue of A. • λ has algebraic multiplicity ma , if it is a root of multiplicity ma of the characteristic polynomial pA . • If ma = 1, then λ is called simple. Otherwise, λ is said to be multiple. • The geometric multiplicity mg of λ is defined as the dimension of the associated eigenspace, i.e., mg = dim (Eλ (A)). It is the maximum number of independent eigenvectors associated with λ. • It holds mg ≤ ma . • If mg < ma , then λ and A are called defective or non-diagonalizable. Note that if all eigenvalues of A are simple, then they are distinct. Now, we can state a result about the nonsingularity of the eigenvector matrix X in (2.1): Theorem 2.15 (Diagonal Form; see [22, Cor. 7.1.8].). Let A ∈ Rn×n with eigenvalues λ1 , . . . , λn ∈ C. A is nondefective if and only if there exists a nonsingular matrix X ∈ Cn×n such that X −1 AX = diag(λi )i=1,...,n . 2 The machine precision is  −53 ≈ 1.11 · 10−16 in the double precision IEEE machine = 2 floating point format and machine = 2−24 ≈ 5.96 · 10−6 in the single precision IEEE floating point format. For more details, we refer to, e.g., [27, 37, 66].

122

JESSICA BOSCH AND CHEN GREIF

The similarity transformation given in Theorem 2.15 transforms A into a diagonal matrix whose entries reveal the eigenvalues of A. We have seen that a similarity transformation to a diagonal matrix is not always possible. Before we come to the next similarity transformation, we introduce the concept of deflation – the process of breaking down an eigenvalue problem into smaller eigenvalue problems. Theorem 2.16 (See [22, Lemma 7.1.3].). Let A ∈ Cn×n , S ∈ Cn×k with rank(S) = k < n and B ∈ Ck×k such that AS = SB, i.e., ran(S) is an A-invariant subspace. Then, there exists a unitary Q ∈ Cn×n such that + , T11 T12 Q∗ AQ = T = 0 T22 and λ(T ) = λ(T11 ) ∪ λ(T22 ), λ(T11 ) = λ(A) ∩ λ(B) with T11 ∈ Ck×k . From Theorem 2.16, we obtain a similarity transformation that transforms a matrix A into an upper triangular matrix whose diagonal entries reveal the eigenvalues of A. Such a decomposition always exists. Theorem 2.17 (Schur Decomposition; see [22, Theor. 7.1.3].). Given A ∈ Cn×n with eigenvalues λ1 , . . . , λn ∈ C. Then, there exists a unitary matrix Q ∈ Cn×n such that Q∗ AQ = T = D + N , where D = diag(λi )i=1,...,n , and N ∈ Cn×n is strictly upper triangular. Moreover, Q can be chosen such that the eigenvalues λi appear in any order in D. The transformation in Theorem 2.17 deals with a complex matrix Q even when A is real. A slight variation of the Schur decomposition shows that complex arithmetic can be avoided in this case. This is based on the fact that complex eigenvalues always occur in complex conjugate pairs, i.e., if (λ, x) is an eigenpair of A ∈ Rn×n , ¯ x) ¯ is an eigenpair of A. then (λ, Theorem 2.18 (Real Schur Decomposition; see [22, Theor. 7.4.1].). Let A ∈ Rn×n with eigenvalues λ1 , . . . , λn ∈ C. Then, there exists an orthogonal matrix Q ∈ Rn×n such that ⎤ ⎡ T1,1 · · · T1,m ⎥ ⎢ .. .. QT AQ = T = ⎣ ⎦, . . Tm,m is quasi-upper triangular. The diagonal blocks Ti,i are either 1 × 1 where T ∈ R or 2×2 matrices. A 1×1 block corresponds to a real eigenvalue λj ∈ R. A 2×2 block corresponds to a pair of complex conjugate eigenvalues. For a complex conjugate eigenvalue pair λk = μ + ıν, λl = μ − ıν, Ti,i has the form + , μ ν Ti,i = . −ν μ n×n

NUMERICAL SOLUTION OF LINEAR EIGENVALUE PROBLEMS

123

Moreover, Q can be chosen such that the diagonal blocks Ti,i appear in any order in T . The next similarity transformation we present transforms a matrix A into upper Hessenberg form. Such a decomposition always exists and will play an important role in eigenvalue solvers for nonsymmetric matrices. Theorem 2.19 (Hessenberg Decomposition). Let A ∈ Cn×n . Then, there exists a unitary matrix Q ∈ Cn×n such that ⎤ ⎡ ··· h1,n h1,1 h1,2 h1,3 ⎢ h2,1 h2,2 h2,3 ··· h2,n ⎥ ⎥ ⎢ ⎥ ⎢ .. .. ⎥. . . 0 h h Q∗ AQ = H = ⎢ 3,2 3,3 ⎥ ⎢ ⎥ ⎢ . . . . .. .. .. ⎣ .. hn−1,n ⎦ 0 ··· 0 hn,n−1 hn,n H is called an upper Hessenberg matrix. Further, H is said to be unreduced if hj+1,j = 0 for all j = 1, . . . , n − 1. From a theoretical point of view, one of the most important similarity transformations is the Jordan decomposition, or Jordan Canonical Form. Theorem 2.20 (Jordan Decomposition; see [22, Theor. 7.1.9].). Let A ∈ Cn×n with exactly p distinct eigenvalues λ1 , . . . , λp ∈ C for p ≤ n. Then, there exists a nonsingular matrix X ∈ Cn×n such that ⎤ ⎡ J1 (λ1 ) ⎥ ⎢ .. X −1 AX = ⎣ ⎦. . Jp (λp ) Each block Ji (λi ) has the block diagonal structure ⎤ ⎡ Ji,1 (λi ) a a ⎥ ⎢ .. Ji (λi ) = ⎣ ⎦ ∈ Cmi ×mi . Ji,mgi (λi ) with

⎡ ⎢ ⎢ Ji,k (λi ) = ⎢ ⎣

λi

1 .. .

⎤ ..

.

λi

⎥ ⎥ ⎥ ∈ Cmi,k ×mi,k , 1 ⎦ λi

where mai and mgi are the algebraic and geometric multiplicity of the eigenvalue λi . Each of the subblocks Ji,k (λi ) is referred to as a Jordan block. Unfortunately, from a computational point of view, the computation of the Jordan Canonical Form is numerically unstable. An important and practical factorization is the QR decomposition: Definition 2.21 (QR Decomposition; see [22, Theor. 5.2.1].). Let A ∈ Rm×n . Then, there exists an orthogonal Q ∈ Rm×m and an upper triangular R ∈ Rm×n such that A = QR.

124

JESSICA BOSCH AND CHEN GREIF

This concludes the theoretical part of the background study. Next, we are getting started with computational aspects.

2.2. First computational aspects. This section quickly reviews aspects of perturbation theory and illustrates possible difficulties in computing eigenvalues accurately. This is followed by a brief overview of different classes of methods for solving eigenvalue problems. Details about all mentioned methods are given in the upcoming sections. First of all, it should be clear that in general we must iterate to find eigenvalues of a matrix: According to Definition 2.5, the eigenvalues of a matrix A are the roots of the characteristic polynomial pA (x). In 1824, Abel proved that for polynomials of degree n ≥ 5, there is no formula for its roots in terms of its coefficients that uses only the operations of addition, subtraction, multiplication, division, and taking kth roots. Hence, even if we could work in exact arithmetic, no computer would produce the exact roots of an arbitrarily polynomial in a finite number of steps. (This is different than direct methods for solving systems of linear equations such as Gaussian elimination.) Hence, computing the eigenvalues of any n × n matrix A requires an iterative process if n ≥ 5. As already indicated in the previous section, many methods are based on repeatedly performing similarity transformations to bring A into a simpler equivalent form. This typically means generating as many zero entries in the matrix as possible. The goal is eventually to perform a Schur decomposition. If the matrix is normal, then the Schur decomposition simplifies to a diagonal matrix (not only an upper triangular matrix), and this has implications in terms of stability of numerical computations. As part of the process, we often aim to reduce the matrix into tridiagonal form (symmetric case) or upper Hessenberg form (nonsymmetric case). Deflation, projection, and other tools can be incorporated and are extremely valuable. Now, let us focus on the reduction of a matrix A to upper Hessenberg form. One way to accomplish this is the use of Householder reflectors (also called Householder transformations). They can be used to zero out selected components of a vector. Hence, by performing a sequence of Householder reflections on the columns of A, we can transform A into a simpler form. Householder reflectors are matrices of the form P =I−

2 vv ∗ , v∗ v

where v ∈ Cn \ {0}. Householder matrices are Hermitian (P = P ∗ ), unitary, and numerically stable. Geometrically, P applied to a vector x reflects it about the hyperplane span{v}⊥ . Assume we want to bring A into upper Hessenberg form. Then, the first step is to introduce zeros into all except the first two entries of the first column of A. Let us denote by x = [a2,1 , . . . , an,1 ]T the part of the first column of A under consideration. We are looking for a vector v ∈ Cn−1 \ {0} such that P x results in a multiple of the first unit vector e1 . This can be achieved with the ansatz v = x ± x2 e1 , since this yields P x = ∓x2 e1 .

NUMERICAL SOLUTION OF LINEAR EIGENVALUE PROBLEMS

Let us illustrate the action of ⎡ a1,1 a1,2 · · · ⎢ a2,1 a2,2 · · · ⎢ ⎢ a3,1 a3,2 · · · ⎢ ⎢ .. .. .. ⎣ . . . an,1

an,2

P to the first column ⎡ ⎤ a1,1 a1,n ⎢ ∓x2 a2,n ⎥ ⎢ ⎥ ⎢ a3,n ⎥ 0 ⎥→⎢ ⎢ .. ⎥ .. ⎣ . ⎦ . · · · an,n 0

of A: a1,2 a2,2 a3,2 .. .

··· ··· ··· .. .

an,2

· · · an,n

a1,n a2,n a3,n .. .

125

⎤ ⎥ ⎥ ⎥ ⎥. ⎥ ⎦

Note that the first step is not complete yet: Remember that, for a similarity transformation, we need to apply the Householder matrix twice, i.e., P ∗ AP . Note that the right multiplication with P does not destroy the zeros: ⎤ ⎤ ⎤ ⎡ ⎡ ⎡ ∗ ∗ ··· ∗ ∗ ∗ ··· ∗ ∗ ∗ ··· ∗ ⎢ ∗ ∗ ··· ∗ ⎥ ⎢ ∗ ∗ ··· ∗ ⎥ ⎢ ∗ ∗ ··· ∗ ⎥ ⎥ ⎥ ⎥ ⎢ ⎢ ⎢ ⎢ ∗ ∗ · · · ∗ ⎥ P ∗ · ⎢ 0 ∗ · · · ∗ ⎥ ·P ⎢ 0 ∗ · · · ∗ ⎥ ⎥ −→ ⎢ ⎥. ⎥ −→ ⎢ ⎢ ⎢ .. .. . . ⎢ .. .. . . ⎢ .. .. . . . ⎥ . ⎥ . ⎥ ⎣ . . ⎣ . . ⎣ . . . .. ⎦ . .. ⎦ . .. ⎦ ∗ ∗ ··· ∗ A

0 ∗ ··· ∗ P ∗A

0 ∗ ··· ∗ P ∗ AP

Let us denote the Householder matrix in the first step by P1 . The above procedure is repeated with the Householder matrix P2 to the second column of P1∗ AP1 , then to the third column of P2∗ P1∗ AP1 P2 with the Householder matrix P3 , and so on, until we end up with a matrix in upper Hessenberg form as given in Definition 2.19. Let us denote the Householder matrix in step i by Pi . After n − 2 steps, we obtain the upper Hessenberg form: ⎤ ⎡ ∗ ∗ ∗ ··· ∗ ⎢ ∗ ∗ ∗ ··· ∗ ⎥ ⎥ ⎢ ⎢ .. ⎥ .. ∗ ∗ ⎢ . ∗ . ⎥ Pn−2 · · · P1 A P1 · · · Pn−2 = H = ⎢ 0 ∗ ⎥. 45 6 3 45 6 3 ⎥ ⎢ . . . . . . . . ∗ ⎣ . P P . . . ∗ ⎦ 0 ··· 0 ∗ ∗ Remark 2.22. Here are a few additional comments about the process: • In practice, one choses v = x + sign(x1 )x2 e1 , where x1 is the first entry of the vector x under consideration. • Note that in each step i, the corresponding vector xi , and hence vi and Pi , shrink by one in size. • The reduction of an n × n matrix to upper Hessenberg form via Householder reflections requires O(n3 ) operations. One may ask why do we first bring A to upper Hessenberg form and not immediately to triangular form using Householder reflections? In that case, the right multiplication with P would destroy the zeros previously introduced: ⎡ ⎡ ⎡ ⎤ ⎤ ⎤ ∗ ∗ ··· ∗ ∗ ∗ ··· ∗ ∗ ∗ ··· ∗ ⎢ ⎢ ∗ ∗ ··· ∗ ⎥ ∗ ⎢ 0 ∗ ··· ∗ ⎥ ⎥ ⎢ ⎥ ·P ⎢ ∗ ∗ · · · ∗ ⎥ ⎥ P · ⎢ −→ −→ ⎢ ⎢ ⎢ .. .. . . ⎥ ⎥ . . . . . . ⎥. . ⎣ .. .. . . . .. ⎦ ⎣ .. .. . . . .. ⎦ ⎣ . . . .. ⎦ ∗ ∗ ··· ∗ A

0 ∗ ··· ∗ P ∗A

∗ ∗ ··· ∗ P ∗ AP

126

JESSICA BOSCH AND CHEN GREIF

This should not come as a surprise: we already knew from Abel (1824) that it is impossible to obtain a Schur form of A in a finite number of steps; see the beginning of this Section. Remark 2.23. If A is symmetric, the reduction to upper Hessenberg form turns into a tridiagonal matrix. That is because the right multiplication with Pi also introduces zeros above the diagonal: ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ∗ ∗ ∗ ··· ∗ ∗ ∗ ∗ ··· ∗ ∗ ∗ 0 ··· 0 ⎢ ∗ ∗ ∗ ··· ∗ ⎥ ⎢ ∗ ∗ ∗ ··· ∗ ⎥ ⎢ ∗ ∗ ∗ ··· ∗ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ∗ ∗ ∗ · · · ∗ ⎥ P1∗ · ⎢ 0 ∗ ∗ · · · ∗ ⎥ ·P1 ⎢ 0 ∗ ∗ · · · ∗ ⎥ ⎢ ⎥ −→ ⎢ ⎥ −→ ⎢ ⎥. ⎢ .. .. .. . . ⎢ .. .. .. . . ⎢ .. .. .. . . . ⎥ . ⎥ . ⎥ ⎣ . . . ⎣ . . . ⎣ . . . . .. ⎦ . .. ⎦ . .. ⎦ ∗ ∗ ∗ ··· ∗ A

0

∗ ∗ ··· ∗ P1∗ A

0 ∗ ∗ ··· ∗ P1∗ AP1

Later on, we will discuss fast algorithms for eigenvalue problems with symmetric tridiagonal matrices. Remark 2.24. The Hessenberg reduction via Householder reflections is backward stable, i.e., there exists a small perturbation δA of A such that ˆ = Pˆ ∗ (A + δA)Pˆ , δAF ≤ cn2 machine AF . H ˆ is the computed upper Hessenberg matrix, Pˆ = Pˆ1 · · · Pˆn−2 is a product Here, H of exactly unitary Householder matrices based on computed vectors v ˆi , and c > 0 a constant. For more details, we refer to [66, p. 351] and [27, Sec. 19.3]. During the next sections, we will see how the upper Hessenberg form (or the tridiagonal form in case of symmetric matrices) is used within eigenvalue solvers. Before we talk about algorithms, we need to understand when it is difficult to compute eigenvalues accurately. The following example shows that eigenvalues of a matrix are continuous (but not necessarily differentiable) functions of it. Example 2.25. Consider the perturbed Jordan block ⎡ ⎤ 0 1 ⎢ ⎥ .. .. ⎥ ⎢ . . ⎥ ∈ Rn×n . A(ε) = ⎢ ⎢ ⎥ . . ⎣ . 1 ⎦ ε 0 The characteristic polynomial is given as pA(ε) (x) = (−1)n (xn − ε). Hence, the 1 eigenvalues are λj (ε) = ε n exp( 2ıjπ n ) for j = 1, . . . , n. None of the eigenvalues is differentiable at ε = 0. Their rate of change at the origin is infinite. Consider for instance the case n = 20 and ε = 10−16 (machine precision), then λ1 (ε) = 0.1507 + 0.0490ı whereas λ(0) = 0. Let us quickly address the issue of estimating the quality of computed eigenvalues. The question here is: How do eigenvalues and eigenvectors vary when the original matrix undergoes small perturbations? We start with considering the sensitivity of simple eigenvalues. Theorem 2.26 (See, e.g. [22, Chap. 7.2.2].). Let A ∈ Cn×n with a simple eigenvalue λ, a right (unit norm) eigenvector x, and a left (unit norm) eigenvector y. Let

NUMERICAL SOLUTION OF LINEAR EIGENVALUE PROBLEMS

127

A + δA be a perturbation of A and λ + δλ the corresponding perturbed eigenvalue. Then

y ∗ δAx + O δA22 . δλ = ∗ y x The condition number of λ is defined as s(λ) = s(λ) =

1 |y ∗ x| .

It can be shown that

1 , cos (θ(x, y))

where θ(x, y) is the angle between x and y. ε In general, O(ε) perturbations in A can induce s(λ) changes in an eigenvalue. Thus, if s(λ) is small, then λ is ill-conditioned, and A is “close to” a matrix with multiple eigenvalues. If A is normal, then every simple eigenvalue satisfies s(λ) = 1, which means that these eigenvalues are well-conditioned. In the case of a multiple eigenvalue λ, s(λ) is not unique anymore. For a defective eigenvalue λ, it holds in  1 general that O(ε) perturbations in A can result in O ε p changes in λ, where p denotes the size of the largest Jordan block associated with λ. This is the effect we have observed in Example 2.25: A(0) has the effective eigenvalue zero with algebraic multiplicity n and multiplicity one. Hence, O(10−16 ) perturbations in A   geometric 16 can result in O 10− 20 = O (0.1585) changes in the eigenvalue. In other words, small perturbations in the input data caused a large perturbation in the output. This can lead to numerical instabilities of eigenvalue solvers. In the following, we start with algorithms for computing a few up to all eigenvalues for small to moderate-sized matrices. Then, we continue with large and sparse matrices.

3. Small to moderate-sized matrices In general, as previously stated, we may separate methods into ones that are based on matrix decompositions vs. ones that are based on matrix-vector products. The power method, which we start with in the sequel, is an important building block for both classes of methods. It is based on matrix-vector products, but it is invaluable for eigensolvers based on decompositions. We choose to include it in this section, noting that it is relevant also for eigensolvers for large and sparse matrices. 3.1. Power method. The power method is one of the oldest techniques for solving eigenvalue problems. It is used for computing a dominant eigenpair, i.e., the eigenvalue of maximum modulus of a matrix A and a corresponding eigenvector. The algorithm consists of generating a sequence of matrix-vector multiplications {Ak v0 }k=0,1,... , where v0 is some nonzero initial vector. Let A ∈ Rn×n with Axj = λj xj for j = 1, . . . , n. Assume that the eigenvectors xj , j = 1, . . . , n, are linearly independent, i.e., A is nondefective. Given 0 = v0 ∈ Cn , we can expand it using the eigenvectors of A to v0 =

n  j=1

βj xj ,

128

JESSICA BOSCH AND CHEN GREIF

where βj ∈ C for j = 1, . . . , n. Applying A to v0 yields Av0 =

n 

βj Axj =

j=1

n 

βj λj xj .

j=1

Hence, the eigenvectors corresponding to eigenvalues of larger modulus are favored. The above procedure can be repeated. In fact, for any k ∈ N, we have Ak v0 =

n 

βj Ak xj =

j=1

n 

βj λkj xj .

j=1

In order for the following algorithm to converge, we need the following assumptions: |λ1 | > |λ2 | ≥ |λ3 | ≥ . . . ≥ |λn |. Since we are interested in the eigenvalue of maximum modulus, we need some distance to the remaining eigenvalues. λ1 is called the dominant eigenvalue. We further need v0 to have a component in the direction of the eigenvector corresponding to λ1 , i.e., β1 = 0. Note that this assumption is less concerning in practice since rounding errors during the iteration typically introduce components in the direction of x1 . However, we need it for the following theoretical study. Due to β1 = 0, we can write ⎛ ⎞  k n n   λ β j j βj λkj xj = β1 λk1 ⎝x1 + xj ⎠ . Ak v0 = β1 λk1 x1 + β λ 1 1 j=2 j=2  Since λ1 is a dominant eigenvalue, we get

λj λ1

k

k→∞

−→ 0 for all j = 2, . . . , n. Hence,

k

it can be shown (cf. [47, Theor. 4.1]) that A v0 , as well as the scaled version k vk = AAk vv00 2 which is used in practice to avoid overflow/underflow, converges linearly to a multiple of x1 with a convergence rate proportional to |λ2 | |λ1 |

|λ2 | |λ1 | .

A value

≈ 1 indicates a slow convergence behavior. Algorithm 3.1 shows the power of method. The approximated eigenvalue in step k (k)

λ1 = vkT Avk is computed using the Rayleigh quotient; see Definition 2.6. This is based on the ˆ1 = following: Given a vector x ˆ1 that approximates the eigenvector x1 . Then, λ T x1 is the best eigenvalue approximation in the least-squares sense, i.e., x ˆ1 Aˆ (3.1)

ˆ 1 = arg min Aˆ λ x1 − μˆ x1 22 . μ

We can solve this minimization problem by solving the normal equation x ˆT1 x ˆ1 μ = x ˆT1 Aˆ x1 ⇔μ=

x1 x ˆT1 Aˆ ; x ˆT1 x ˆ1

see, e.g., [46, Chap. 5.3.3]. Since we normalize the computed eigenvectors in the power method, i.e., ˆ x1 2 = 1, we get the desired result. The cost for k iterations is O(2kn2 ) floating point operations (flops).

NUMERICAL SOLUTION OF LINEAR EIGENVALUE PROBLEMS

129

Algorithm 3.1: Power method v Choose v0 = v 2 2 for k = 1, 2, . . . , until termination do 3 v ˜ = Avk−1 4 vk = ˜vv˜ 2

1

(k)

λ1 = vkT Avk 6 end

5

The power method can be applied to large, sparse, or implicit matrices. It is simple and basic but can be slow. We assumed for the convergence that A is nondefective. For the case of a defective A, the power method can still be applied but converges even more slowly; see, e.g., [28]. Moreover, we want to emphasize again that the power method only works if the matrix under consideration has one dominant eigenvalue. This excludes the case of, e.g., a dominant complex eigenvalue3 or of dominant eigenvalues of opposite signs. The power method is rather used as a building block for other, more robust and general algorithms. We refer to [66, Chap. 10] for a detailed discussion of the power method. Next, we discuss a method that overcomes the mentioned difficulties. 3.2. Inverse power method. We have seen that the power method is in general slow. Moreover, it is good only for one well-separated dominant eigenvalue. How can we accelerate it, and what about the more general case of looking for a nondominant eigenpair? The inverse power method uses shift and invert techniques to overcome these limitations of the power method. It aims to compute the eigenvalue of A that is closest to a certain scalar (shift) and a corresponding eigenvector. It also enhances the convergence behavior. The price for these improvements is the solution of a linear system in each iteration. The idea is the following: Assume A ∈ Rn×n has eigenpairs (λj , xj )j=1,...,n with |λ1 | ≥ . . . ≥ |λn |. Let α ∈ R with α = λj for j = 1, . . . , n. This will be the shift in the inverse power method. In practice, we choose α ≈ λi for some i depending on which (real) eigenvalue λi we want to find. Hence, in order for the method to work, we need to know approximately the value of the eigenvalue we are interested in. Then, A − αI has eigenpairs (λj − α, xj )j=1,...,n , and (A − αI)−1 has eigenpairs (μj , xj )j=1,...,n with μj = (λj − α)−1 . Let λi and λj be the two eigenvalues that are closest to α with |λi − α| < |λj − α|. Then, the two largest eigenvalues μ1 and μ2 of (A − αI)−1 are 1 1 , μ2 = . μ1 = λi − α λj − α Hence, the power method applied to (A−αI)−1 converges to μ1 and an eigenvector of μ1 with convergence rate |μ2 | = |μ1 | 3 As

pairs.

1 |λj −α| 1 |λi −α|

=

|λi − α| . |λj − α|

noted in Section 2.1, eigenvalues of real matrices always occur in complex conjugate

130

JESSICA BOSCH AND CHEN GREIF

2| We know from the previous section that we need a small value of |μ |μ1 | in order to converge fast. Hence, we desire |λi − α| $ |λj − α|, which requires a “good” choice of the shift α. If we are interested for instance in the dominant eigenvalue, estimations based on norms of A can be used; see, e.g., [22, Chap. 2.3.2].

Algorithm 3.2: Inverse power method v Choose v0 = v 2 2 for k = 1, 2, . . . , until termination do 3 Solve (A − αI)˜ v = vk−1 4 vk = ˜vv˜ 2

1

λ(k) = vkT Avk 6 end

5

As already mentioned at the beginning of this section, the price for overcoming difficulties of the power method by using a shift and invert approach is the solution of a linear system in every iteration. If α is fixed, then we have to solve linear systems with one matrix and many right-hand sides: If a direct method can be applied, then we form an LU decomposition of A − αI once. The cost for solving the two triangular systems arising from the LU decomposition is O(n2 ). For huge problems, iterative methods have to be employed to solve the linear systems. This pays off only if the inverse iteration converges very fast. In summary, we have seen that we can apply the inverse power method to find different eigenvalues using different shifts. During the whole iteration, the inverse power method uses a fixed shift α. The next method involves a dynamic shift αk . 3.3. Rayleigh quotient iteration. The idea of the Rayleigh quotient iteration is to learn the shift as the iteration proceeds using the calculated eigenvalue λ(k−1) from the previous step k − 1. Now, each iteration is potentially more expensive since the linear systems involve different matrices in each step. However, the new algorithm may converge in many fewer iterations. In the Hermitian case, we potentially obtain a cubic convergence rate; see, e.g., [39]. Note that the matrix (A−λ(k−1) I) may be singular. This is the case when the shift hits an eigenvalue of A. The cost for solving the linear system with (A − λ(k−1) I) is O(n3 ) if A is full. For an upper Hessenberg matrix, it reduces to O(n2 ), and for a tridiagonal matrix even to O(n). Next, we discuss a technique that uses information of a computed dominant eigenpair for the approximation of a second-dominant eigenpair. 3.4. Deflation. Let A ∈ Rn×n have eigenvalues |λ1 | > |λ2 | ≥ . . . ≥ |λn |, right eigenvectors x1 , . . . , xn , and left eigenvectors y1 , . . . , yn . Note that (λ1 , x1 ) is a dominant eigenpair. Suppose we have approximated the dominant eigenvector ˆ1 with ˆ x1 2 = 1. Now, we are interested in approximating the next x1 of A by x eigenvalue λ2 . Deflation is based on a simple rank-one modification of A, as follows: Compute x1 w T , where α ∈ R is an appropriate shift and w ∈ Rn an arbitrary A1 = A − αˆ ˆ1 = 1. vector such that w T x

NUMERICAL SOLUTION OF LINEAR EIGENVALUE PROBLEMS

131

Theorem 3.1 (Wielandt; see [65].). In the case x ˆ1 = x1 , the eigenvalues of A1 are λ1 − α, λ2 , . . . , λn . Moreover, the right eigenvector x1 and the left eigenvectors y2 , . . . , yn are preserved. Proof.



A − αx1 w T x1 = Ax1 − αx1 w T x1 = λ1 x1 − αx1

since w T x1 = 1. For i = 2, . . . , n, we have

yi∗ A − αx1 w T = yi∗ A − αyi∗ x1 w T = yi∗ A = λi yi∗ since yi∗ x1 = 0 for i = 2, . . . , n.



Hence, a modification of A to A1 = A − αˆ x1 w T displaces the dominant eigenvalue of A. The rank-one modification should be chosen such that λ2 becomes the dominant eigenvalue of A1 . We can then proceed for instance with the power method applied to A1 in order to obtain an approximation of λ2 . This technique is called Wielandt deflation. There are many ways to choose w. A simple choice (due to Hotelling [29]) is to choose w = y1 the first left eigenvector (or an approximation of it) or w = x1 or rather its approximation w = x ˆ1 . It can be shown (cf. [47, Chap. 4.2.2]) that if x1 2 is nearly orthogonal to x2 or if λ1 −λ $ 1, the choice w = x1 is nearly optimal in α terms of eigenvalue conditioning. Note that we never need to form the matrix A1 explicitly. This is important since A1 is a dense matrix. For calculating the matrix-vector product y = A1 x, we just need to perform y ← Ax, β = αwT x, and y ← y − β x ˆ1 . We can apply this procedure recursively without difficulty. However, keep in mind that, for a long deflation process, errors accumulate. So far, the discussed algorithms compute only one eigenpair at once, i.e., a onedimensional invariant subspace. Next, we consider another generalization of the power and inverse power method that can be used to compute higher-dimensional invariant subspaces. 3.5. Orthogonal iteration. Let A ∈ Rn×n have eigenpairs (λj , xj )j=1,...,n with |λ1 | ≥ . . . ≥ |λn |. From the real Schur decomposition in Theorem 2.18, we know there exists an orthogonal Q ∈ Rn×n such that QT AQ = T , where the diagonal blocks of T correspond to the eigenvalues of A in real form. Assume that the eigenvalues λi , represented in T in real form, are ordered from λ1 to λn . Let 1 ≤ r < n. Then, we can do the following partitioning: + (r,r) , T T (r,n−r) , Q = [Q(r) , Q(n−r) ], T = 0 T (n−r,n−r) where Q(r) ∈ Rr×r and T (r,r) ∈ Rr×r . Note that r should be chosen such that the (r + 1, r) entry in T is zero, i.e., we do not split a complex conjugate eigenpair to T (r,r) and T (n−r,n−r) . Then, AQ(r) = Q(r) T (r,r) , i.e., ran(Q(r) ) is an A-invariant subspace corresponding to the r largest (in modulus) eigenvalues. Due to this property, this subspace is also called dominant.

132

JESSICA BOSCH AND CHEN GREIF

Now, we are interested in computing such a dominant r-dimensional invariant subspace. Hence, instead of dealing with a matrix-vector product as in the algorithms before, we go over to a matrix-matrix product, i.e., we apply A to a few vectors simultaneously. This can be achieved by the orthogonal iteration presented in Algorithm 3.3. Algorithm 3.3: Orthogonal iteration 1 2 3 4 5

Choose Q0 ∈ Rn×r with orthonormal columns for k = 1, 2, . . . , until termination do Zk = AQk−1 Qk Rk = Zk (QR factorization) end

The QR factorization in Line 4 refers to the QR decomposition in Definition 2.21. It

2 flops can be computed by, e.g., the modified Gram–Schmidt algorithm in O 2nr

[22, Chap. 5.2.8], Householder reflections in

O 2r 2 n − 3r flops [22, Chap. 5.2.2], 2 or Givens transformations in O 3r n − 3r flops [22, Chap. 5.2.5]. Note that the complexity can be reduced if the corresponding matrix is of upper Hessenberg form. We will discuss this further below. Note that Line 3 and 4 yield AQk−1 = Qk Rk , where Rk is upper triangular. Now, if |λr | > |λr+1 | and Q0 has components in the desired eigendirections, then we have k→∞

ran(Qk ) −→ ran(Q(r) ) with a convergence rate proportional to Chap. 7.3.2].

|λr+1 | |λr | .

For more details, we refer to [22,

Remark 3.2. By replacing the QR factorization in Line 4 of Algorithm 3.3 with Qk = Zk , we obtain the subspace iteration, also called simultaneous iteration. Under the same conditions as before, it holds k→∞

ran(Zk ) −→ ran(Q(r) ). However, the columns of Zk form an increasingly ill-conditioned basis for Ak ran(Q0 ) since each column of Zk converges to a multiple of the dominant eigenvector. The orthogonal iteration overcomes this difficulty by orthonormalizing the columns of Zk at each step. From the orthogonal iteration, we can derive the QR iteration, a method for finding all eigenvalues of a matrix A. 3.6. QR iteration. We obtain the QR iteration from the orthogonal iteration if we set r = n, i.e., we want to compute all eigenvalues, and Q0 = I. We can rewrite Algorithm 3.4 by using the following equivalence: (3.2)

Ak−1 Ak

Line 5

= QTk−1 AQk−1

Line 3

= QTk−1 Zk

Line 5

Line 4

˜ k Rk , = QTk−1 Qk Rk =: Q

= QTk AQk = QTk AQk−1 QTk−1 Qk (3.2) ˜k . = Rk QTk−1 Qk = Rk Q

Line 4

Line 3

= QTk Zk QTk−1 Qk

NUMERICAL SOLUTION OF LINEAR EIGENVALUE PROBLEMS

133

Algorithm 3.4: Prelude to QR iteration Choose Q0 = I (orthogonal) for k = 1, 2, . . . , until termination do Zk = AQk−1 4 Qk Rk = Zk (QR factorization) 5 Ak = QTk AQk 6 end 1

2 3

˜ k is Note that the product of two orthogonal matrices is orthogonal. Hence, Q orthogonal. Therefore, Ak is determined by a QR decomposition of Ak−1 . This form of the QR iteration is presented in Algorithm 3.5. Note that Q0 does not have to be the identity matrix. Algorithm 3.5: QR iteration 1 2 3 4 5

A0 = QT0 AQ0 (real Schur form, Q0 ∈ Rn×n orthogonal) for k = 1, 2, . . . , until termination do Qk Rk = Ak−1 (QR factorization) Ak = Rk Qk end

From Line 3 and 4 of Algorithm 3.5 we get ˆ Tk AQ ˆk , Ak = (Q0 · · · Qk )T A(Q0 · · · Qk ) =: Q ˆ k ) is an A-invariant subspace and λ(A) = ˆ k is orthogonal. Hence, ran(Q where Q λ(Ak ). If |λ1 | > . . . > |λn | and Q0 has components in the desired eigendirections, then we have k→∞

ˆ k (:, 1 : l)) −→ ran(Q(:, 1 : l)) ∀1 ≤ l ≤ n ran(Q | with a convergence rate proportional to |λ|λl+1 . Hence, Ak −→ T , where QT AQ = l| T is a real Schur decomposition of A. For more details, we refer to [22, Chap. 7.3.3]. Overall, the QR iteration computes the Schur form of a matrix. As in the previous section, we considered the real Schur form here. But note that if we allow complex arithmetic, we get the same results with a (complex) Schur form. For further readings on the QR iteration we refer to, e.g., [32, 55, 62]. As already mentioned in the previous section, in this form the cost of each step of the QR iteration is O(n3 ). But we can reduce the complexity if we start with A0 in upper Hessenberg form. Moreover, we can speed up the convergence using shifts. k→∞

3.7. QR iteration with shifts. If we choose Q0 such that A0 is in upper Hessenberg form, the cost of each step of the QR iteration reduces to O(n2 ). If A is symmetric, then the cost per step is O(n). It can be shown that each Ak is upper Hessenberg. This is the first modification. Second, shifts ζk ∈ R are introduced in order to accelerate the deflation process (see Theorem 2.16). Deflation occurs every time Ak is reduced, i.e., at least one of its subdiagonal entries is zero. In such a

134

JESSICA BOSCH AND CHEN GREIF

case, we continue with two smaller subproblems. The matrices Ak−1 and Ak in Algorithm 3.6 are orthogonally similar since Ak = Rk Qk + ζk I = QTk (Qk Rk + ζk I) Qk = QTk Ak−1 Qk , and Qk from the QR decomposition is orthogonal. Algorithm 3.6: QR iteration with shifts 1 2 3 4 5

A0 = QT0 AQ0 upper Hessenberg form (Tridiagonal if A is symmetric) for k = 1, 2, . . . , until termination do Qk Rk = Ak−1 − ζk I (QR factorization) Ak = Rk Qk + ζk I end

Why does the shift strategy work? If ζk is an eigenvalue of the unreduced Hessenberg matrix Ak−1 , then Ak−1 − ζk I is singular. This implies Rk is singular, where the (n, n) entry of Rk is zero. Then, the last row of the upper Hessenberg matrix Ak = Rk Qk + ζk I consists of zeros except for the (n, n) entry which is ζk . So we have converged to the form +  , A a Ak = , 0T ζk and can now work on a smaller matrix (deflate) and continue the QR iteration. We can accept the (n, n) entry as ζk as it is presumably a good approximation to the eigenvalue. In summary, we obtain deflation after one step in exact arithmetic if we shift by an exact eigenvalue. If ζ = ζk for all k = 1, 2, . . ., and we order the eigenvalues λi of A such that |λ1 − ζ| ≥ . . . ≥ |λn − ζ|, |λ

−ζ|

. Of then the pth subdiagonal entry in Ak converges to zero with rate |λp+1 p −ζ| course, we need |λp+1 − ζ| < |λp − ζ| in order to get any convergence result. (k) In practice, deflation occurs whenever a subdiagonal entry ap+1,p of Ak is small enough, e.g., if (k) (k) |ap+1,p | ≤ c machine (|a(k) p,p | + |ap+1,p+1 |) for a small constant c > 0. Let us quickly summarize some shift strategies: The single-shift strategy uses ζk = k→∞ (k−1) (k) an,n . It can be shown (cf. [22, Chap. 7.5.3]) that the convergence an,n−1 −→ 0 (k−1)

is even quadratic. When we deal with complex eigenvalues, then ζk = an,n tends to be a poor approximation. Then, the double-shift strategy is preferred which performs two single-shift steps in succession, i.e., Lines 3–4 in Algorithm 3.6 are repeated a second time with a second shift. Using implicit QR factorizations, one double-shift step can be implemented with O(n2 ) flops (O(n) flops in the symmetric case); see e.g., [22, Chap. 7.5.5]. This technique was first described by Francis [18, 19] and refers to a Francis QR step. The overall QR algorithm requires O(n3 ) flops. For more details about the QR iteration, we refer to, e.g., [33, 35, 42, 60, 63, 64]. Further readings concerning shift strategies include [15, 17, 61].

NUMERICAL SOLUTION OF LINEAR EIGENVALUE PROBLEMS

135

We know that the Hessenberg reduction of a symmetric matrix leads to a tridiagonal matrix. In the following, we review methods for this special case. 3.8. Algorithms for symmetric (tridiagonal) matrices. Before we consider eigenvalue problems for the special case of symmetric tridiagonal matrices, we review one of the oldest methods for symmetric matrices A — Jacobi’s method. For general symmetric eigenvalue problems, we refer the reader to [11, Chap. 5] — it contains important theoretic concepts, e.g., gaps of eigenvalues and the related perturbation theory, and gives a nice overview of direct eigenvalue solvers. 3.8.1. Jacobi’s method. Jacobi’s method is one of the oldest algorithms [30] for eigenvalue problems with a cost of O(cn3 ) flops with a large constant c. However, it is still of current interest due to its parallelizability and accuracy [12]. The method is based on a sequence of orthogonal similarity transformations (3.3)

. . . QT3 QT2 QT1 AQ1 Q2 Q3 . . . ,

such that each transformation becomes closer to diagonal form. We can write (3.3) as Ak+1 = QTk+1 Ak Qk+1 , k = 0, 1, 2, . . . with A0 = A. In particular, the orthogonal matrices Qi are chosen such that the Frobenius norm of the off-diagonal elements ( ) n  n  2 ) (k) ai,j off(Ak ) = ) * i=1 j=1 j=i

is reduced with each transformation. This rotations) ⎡ 1 ⎢ .. ⎢ . ⎢ ⎢ 1 ⎢ ⎢ c 0 ⎢ ⎢ 0 1 ⎢ ⎢ . . J (p, q, θ) = ⎢ . ⎢ ⎢ 0 ⎢ ⎢ −s 0 ⎢ ⎢ ⎢ ⎢ ⎣

is done using Jacobi rotations (Givens ⎤

··· 0 ..

.

1 ··· 0

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

s 0 .. . 0 c 1 ..

.

p

q

1 p

q

where 1 ≤ p < q ≤ n, c = cos(θ), and s = sin(θ). Givens rotations are orthogonal. The application of J (p, q, θ)T to a vector rotates the vector counterclockwise in the (p, q) coordinate plane by θ radians. In order to make Ak+1 iteratively more diagonal, Qk+1 = J (p, q, θ) is chosen to make one pair of off-diagonal entries of Ak+1 = QTk+1 Ak Qk+1 zero at a time. Thereby, θ is chosen such that the (p, q) and

136

JESSICA BOSCH AND CHEN GREIF

(q, p) entry of Ak+1 become zero. To determine θ, we can consider the corresponding 2 × 2 system 8+ 8 + ,T 7 (k) , 7 (k+1) (k) c s 0 ap,p ap,q c s ap,p = , (k) (k) (k+1) −s c −s c 0 aq,q aq,p aq,q (k+1)

where ap,p

(k+1)

and aq,q

are the eigenvalues of 7 8 (k) (k) ap,p ap,q . (k) (k) aq,p aq,q

One can show (cf. [11, Chap. 5.3.5]) that tan(2θ) =

2a(k) p,q (k) (k) aq,q −ap,p

and using this, we can

compute c and s. Using the fact that the Frobenius norm is preserved by orthogonal transformations, we obtain (cf. [11, Lemma 5.4]) 2  (3.4) off(Ak+1 )2 = off(Ak )2 − 2 a(k) , p,q i.e., Ak+1 moves closer to diagonal form with each Jacobi step. In order to maximize (k) the reduction in (3.4), p and q should be chosen such that |ap,q | is maximal. With this choice, we get after k Jacobi steps (cf. [11, Theor. 5.11]) k  2 2 off(A0 )2 , off(Ak ) ≤ 1 − n(n − 1) i.e., convergence at a linear rate. This scheme is the original version from Jacobi in 1846 and is referred to as classical Jacobi algorithm. It even can be shown that the asymptotic convergence rate is quadratic (cf. [11, Theor. 5.12]); see [49, 58]. While the cost for an update is O(n) flops, the search for the optimal (p, q) costs O(n2 ) flops. For a simpler method, we refer the reader to the cyclic Jacobi method ; see e.g. [66, p. 270] or [22, Chap. 8.5.3]. In general, the cost of the cyclic Jacobi method is considerably higher than the cost of the symmetric QR iteration. However, it is easily parallelizable. Again we want to emphasize that we do not need to tridiagonalize in Jacobi’s method. In the following, we discuss two methods that need to start by reducing a symmetric matrix A to tridiagonal form: bisection and divide-and-conquer. Another method is MR 3 or MRRR (Algorithm of Multiple Relatively Robust Representations) [13] — a sophisticated variant of the inverse iteration, which is efficient when eigenvalues are close to each other. 3.8.2. Bisection. For the rest of Section 3, we consider eigenvalue problems for symmetric tridiagonal matrices of the form ⎤ ⎡ a1 b1 ⎥ ⎢ b1 a2 b2 ⎥ ⎢ ⎥ ⎢ . . ⎥. ⎢ . b2 a3 (3.5) A=⎢ ⎥ ⎥ ⎢ .. .. ⎣ . . bn−1 ⎦ bn−1 an We begin with bisection, a method that can be used to find a subset of eigenvalues, e.g., the largest/smallest eigenvalue or eigenvalues within an interval. Let A(k) =

NUMERICAL SOLUTION OF LINEAR EIGENVALUE PROBLEMS

137

A(1 : k, 1 : k) be the leading k × k principal submatrix of A with characteristic polynomial p(k) (x) = det(A(k) − xI) for k = 1, . . . , n. If bi = 0 for i = 1, . . . , n − 1, then det(A(k) ) = ak det(A(k−1) ) − b2k−1 det(A(k−2) ), which yields p(k) (x) = (ak − x)p(k−1) (x) − b2k−1 p(k−2) (x) with p(−1) (x) = 0 and p(0) (x) = 1. Hence, p(n) (x) can be evaluated in O(n) flops. Given y < z ∈ R with p(n) (y)p(n) (z) < 0 (Hence, there exists a w ∈ (y, z) with p(n) (w) = 0.), we can use the method of bisection (see, e.g., [22, Chap. 8.4.1]) to find an approximate root of p(n) (x) and hence an approximate eigenvalue of A. The method of bisection converges linearly in the sense that the error is approximately halved at each step. Assume that the eigenvalues λj (A(k) ) of A(k) are ordered as λ1 (A(k) ) ≥ . . . ≥ λk (A(k) ). For computing, e.g., λk (A) for a given k or the largest eigenvalue that is smaller than a given μ ∈ R, then we need the following theorem: Theorem 3.3 (Sturm Sequence Property; see [22, Theor. 8.4.1].). If A is unreduced, i.e., bi = 0 for i = 1, . . . , n − 1, then the eigenvalues of A(k−1) strictly separate the eigenvalues of A(k) : λk (A(k) ) < λk−1 (A(k−1) ) < λk−1 (A(k) ) < λk−2 (A(k−1) ) < λk−2 (A(k) ) < . . . < λ2 (A(k) ) < λ1 (A(k−1) ) < λ1 (A(k) ). Moreover, if a(μ) denotes the number of sign changes in the sequence {p(0) (μ), p(1) (μ), . . . , p(n) (μ)}, where p(k) (μ) has the opposite sign from p(k−1) (μ) if p(k) (μ) = 0, then a(μ) equals the number of A’s eigenvalues that are less than μ. In order to find an initial interval for the method of bisection, we make use of the following simplified version of the Gershgorin theorem: Theorem 3.4 (Gershgorin). If A ∈ Rn×n is symmetric, then where ri =

n

j=1 j=i

λ(A) ⊆ ∪ni=1 [ai,i − ri , ai,i + ri ], |ai,j |.

The more general version of the Gershgorin theorem can be found, e.g., in [22, Theor. 8.1.3]. Suppose we want to compute λk (A), where A is symmetric tridiagonal as in (3.5). Then, from Theorem 3.4, we get λk (A) ∈ [y, z] with y = min ai − |bi | − |bi−1 |, 1≤i≤n

z = max ai + |bi | + |bi−1 | 1≤i≤n

and b0 = bn = 0. Hence, with this choice of y and z, we can reformulate the method of bisection to converge to λk (A); see, e.g., [22, Chap. 8.4.2]. Another version of this scheme can be used to compute subsets of eigenvalues of A; see [3]. For a variant that computes specific eigenvalues, we refer to [39, p. 46].

138

JESSICA BOSCH AND CHEN GREIF

The cost of bisection is O(nk) flops, where k is the number of desired eigenvalues. Hence, it can be much faster than the QR iteration if k $ n. Once the desired eigenvalues are found, we can use the inverse power method (Section 3.2) to find the corresponding eigenvectors. The inverse power method costs in the best case (wellseparated eigenvalues) O(nk) flops. In the worst case (many clustered eigenvalues), the cost is O(nk2 ) flops and the accuracy of the computed eigenvectors is not guaranteed. Next, we review a method that is better suited for finding all (or most) eigenvalues and eigenvectors, especially when the eigenvalues may be clustered. 3.8.3. Divide-and-conquer. The idea of divide-and-conquer is to recursively divide the eigenvalue problem into smaller subproblems until we reach matrices of dimension one, for which the eigenvalue problem is trivial. The method was first introduced in 1981 [9] while its parallel version was developed in 1987 [14]. The starting point is to write the symmetric tridiagonal matrix A in (3.5) as a sum of a block diagonal matrix of two tridiagonal matrices T1 and T2 , plus a rank-1 correction: , + T1 0 + bm vv T , A= 0 T2 where v ∈ Rn is a column vector whose mth and (m + 1)st entry is equal to one (1 ≤ m ≤ n − 1) and all remaining entries are zero. Suppose we have the real Schur decompositions of T1 and T2 , i.e., Ti = Qi Di QTi for i = 1, 2 with Q1 ∈ Rm×m and Q2 ∈ R(n−m)×(n−m) orthogonal. Then, + , + , + T , Q1 0 D1 0 Q1 0 A= + bm uuT , 0 Q2 0 D2 0 QT2 where

+

, last column of QT1 . first column of QT2 + , D1 0 Hence, λ(A) = λ(D + bm uuT ) where D = = diag(di )i=1,...,n is a 0 D2 diagonal matrix. Hence, the problem now reduces to finding the eigenvalues of a diagonal plus a rank-1 matrix. This can be further simplified to finding the eigenvalues of the identity matrix plus a rank-1 matrix, using simply the characteristic polynomial: In particular, under the assumption that D − λI is nonsingular, and using   det(D + bm uuT − λI) = det(D − λI)det I + bm (D − λI)−1 uuT , u=

we obtain

QT1 0

0 QT2

,

+

v=

  λ ∈ λ(A) ⇔ det I + bm (D − λI)−1 uuT = 0. −1

The matrix I + bm (D − λI) uuT is of special structure, and its determinant can be computed using the following lemma: Lemma 3.5 (See [11, Lemma 5.1].). Let x, y ∈ Rn . Then

det I + xy T = 1 + y T x. In our case, we get n    det I + bm (D − λI)−1 uuT = 1 + bm i=1

u2i ≡ f (λ), di − λ

NUMERICAL SOLUTION OF LINEAR EIGENVALUE PROBLEMS

139

and the eigenvalues of A are the roots of the secular equation f (λ) = 0. This can be solved, e.g., by Newton’s method, which converges in practice in a bounded number of steps per eigenvalue. Note that solving the secular equation needs caution due to possible deflations (di = di+1 or ui = 0) or small values of ui . For more details on the function f and on solving the secular equation, we refer to [11, Chap. 5.3.3]. The cost for computing all eigenvalues is O(n2 log(n)) flops by using the above strategy of recursively dividing the eigenvalue problem into smaller subproblems. Hence, the pure eigenvalue computation (without eigenvectors) is more expensive than in the QR iteration. However, the eigenvectors can be computed more cheaply: Lemma 3.6 (See [11, Lemma 5.2].). If λ ∈ λ(D + bm uuT ), then (D − λI)−1 u is a corresponding eigenvector. The cost for computing all eigenvectors is O(n2 ) flops. However, the eigenvector computation from Lemma 3.6 is not numerically stable. If λ is too close to a diagonal entry di , we obtain large roundoff errors since we divide by di − λ. If two eigenvalues λi and λj are very close, the orthogonality of the computed eigenvectors can get lost. For a numerical stable computation, we refer to [24], which requires in practice O(cn3 ) flops where c $ 1. Here, we finish the discussion of eigenvalue problems for small to moderate-sized matrices. We now move to discuss problems where the matrix is large and sparse. 4. Large and sparse matrices In this section, A ∈ Rn×n is considered to be large and sparse. Sparse matrices are matrices with very few nonzero entries. Sparse often means that there are O(1) nonzero entries per row. We note that matrices that are not necessarily sparse but give rise to very fast matrix-vector products (for example, via the Fast Fourier Transform) often also allow for applying the methods discussed in this section. The meaning of “large matrices” is relative. Let us say we consider matrices of size millions. In order to take advantage of the large number of zero entries, special storage schemes are required; see, e.g., [47, Chap. 2]. We will assume that it is not easy to form a matrix decomposition such as the QR factorization. In particular, similarity transformations would destroy the sparsity. Hence, we will mainly rely on matrix-vector products, which are often computable in O(n) flops instead of O(n2 ). In this chapter, we review methods for computing a few eigenpairs of A. In fact, in practice, one often needs the k smallest/largest eigenvalues or the k eigenvalues closest to μ ∈ C for a small k and their corresponding eigenvectors. In the following, we introduce orthogonal projection methods, from which we can derive the stateof-the-art Krylov methods, which make use of cheap matrix-vector products. Note that projection methods even play a role for the methods discussed in Section 3. 4.1. Orthogonal projection methods. Suppose we want to find an approxˆ x imation (λ, ˆ) of an eigenpair (λ, x) of A ∈ Rn×n . The idea of projection techniques is to extract x ˆ from some subspace K. This is called the subspace of approximants or the right subspace. The uniqueness of x ˆ is typically realized via the imposition of orthogonality conditions. We denote by ˆx r = Aˆ x − λˆ

140

JESSICA BOSCH AND CHEN GREIF

the residual vector. It is a measure for the quality of the approximate eigenpair ˆ x (λ, ˆ). The orthogonality conditions consist of constraining the residual r to be orthogonal to some subspace L, i.e., r ⊥ L. L is called the left subspace. This framework is commonly known as the PetrovGalerkin conditions in diverse areas of mathematics, e.g., the finite element method. The case L = K leads to the Galerkin conditions and gives an orthogonal projection, which we discuss next. The case where L is different from K is called oblique projection, and we quickly have a look into this framework at the end of this section. Let us assume that A is symmetric. Let K be a k-dimensional subspace of Rn . An ˆ x orthogonal projection technique onto K seeks an approximate eigenpair (λ, ˆ) such that x ˆ ∈ K and ˆ x ⊥ K, Aˆ x − λˆ or equivalently   ˆ x, v = 0 ∀v ∈ K. (4.1) Aˆ x − λˆ Let {q1 , . . . , qk } be an orthonormal basis of K and Qk = [q1 | . . . |qk ] ∈ Rn×k . Then, (4.1) becomes   ˆ x, qi = 0 ∀i = 1, . . . , k. Aˆ x − λˆ If we express x ˆ in terms of the basis of K, i.e., x ˆ = Qk y, we get   ˆ k y, qi = 0 ∀i = 1, . . . , k, AQk y − λQ and due to QTk Qk = I, we obtain ˆ QTk AQk y = λy. This is the basis for Krylov subspace methods, which we discuss in the next section. The matrix QTk AQk ∈ Rk×k will often be smaller than A and is either upper Hessenberg (nonsymmetric case) or tridiagonal (symmetric case). The Rayleigh– Ritz procedure presented in Algorithm 4.1 computes such a Galerkin approximation. Algorithm 4.1: Rayleigh–Ritz procedure Compute an orthonormal basis {q1 , . . . , qk } of the subspace K. Set Qk = [q1 | . . . |qk ]. 2 Compute Tk = QT k AQk . 3 Compute j eigenvalues of Tk , say θ1 , . . . , θj . 4 Compute the corresponding eigenvectors vj of Tk . Then, the corresponding approximate eigenvectors of A are x ˆj = Qk vj . 1

The θi are called Ritz values and x ˆi are the Ritz vectors. We will see that the Ritz values and Ritz vectors are the best approximate eigenpairs in the least-squares sense. But first, let us put the presented framework into a similarity transformation of A: Suppose Q = [Qk , Qu ] ∈ Rn×n is an orthogonal matrix with Qk ∈ Rn×k being the matrix above that spans the subspace K. We introduce + T , + , Qk AQk QTk AQu Tk Tuk T T := Q AQ = =: . QTu AQk QTu AQu Tku Tu

NUMERICAL SOLUTION OF LINEAR EIGENVALUE PROBLEMS

141

Let Tk = V ΘV T be the eigendecomposition of Tk . Note that for k = 1, T1 is just the Rayleigh quotient (see Definition 2.6). Now, we can answer the question on the ”best” approximation to an eigenvector in K. Similar to the observation in Section 3.1 that the Rayleigh quotient is the best eigenvalue approximation in the least-squares sense, we have the following useful result. Theorem 4.1 (See [11, Theor. 7.1].). The minimum of AQk − Qk R2 over all k × k symmetric matrices R is attained by R = Tk , in which case AQk − Qk R2 = Tku 2 . Let Tk = V ΘV T be the eigendecomposition of Tk . The minimum of APk − Pk D2 over all n × k orthogonal matrices Pk (PkT Pk = I) where span(Pk ) = span(Qk ) and over diagonal matrices D is also Tku 2 and is attained by Pk = Qk V and D = Λ. In practice, the columns of Qk will be computed by, e.g., the Lanczos algorithm or Arnoldi algorithm, which we discuss in Section 4.3 and 4.4. Now, let us have a quick look at the oblique projection technique in which L is different from K. Let K and L be k-dimensional subspaces of Rn . An oblique ˆ x projection technique onto K seeks an approximate eigenpair (λ, ˆ) such that x ˆ∈K and ˆ x ⊥ L, Aˆ x − λˆ or equivalently   ˆ x, v = 0 ∀v ∈ L. (4.2) Aˆ x − λˆ Let {q1 , . . . , qk } be an orthonormal basis of K, {p1 , . . . , pk } an orthonormal basis of L, Qk = [q1 | . . . |qk ] ∈ Rn×k , and Pk = [p1 | . . . |pk ] ∈ Rn×k . Further, we assume biorthogonality, i.e., PkT Qk = I. Then, (4.2) becomes   ˆ x, pi = 0 ∀i = 1, . . . , k. Aˆ x − λˆ If we express x ˆ in terms of the basis of K, i.e., x ˆ = Qk y, we get   ˆ k y, pi = 0 ∀i = 1, . . . , k, AQk y − λQ and due to PkT Qk = I, we obtain ˆ PkT AQk y = λy. Oblique projection techniques form the basis for the non-Hermitian Lanczos process [25, 26, 41], which belongs to the class of Krylov subspace solvers. Krylov subspace solvers form the topic of the next section. For a further discussion on the oblique projection technique, we refer to, e.g., [47, Chap. 4.3.3]. 4.2. Krylov subspace methods. Let A ∈ Rn×n . Krylov subspace methods are used to solve linear systems or eigenvalue problems of sparse matrices. They only require that A be accessible via a “black-box” subroutine which describes the application of A to a vector. A k-dimensional Krylov subspace associated with a matrix A and a vector v is the subspace given by Kk (A; v) = span{v, Av, A2 v, . . . , Ak−1 v}. The corresponding Krylov matrix is denoted by Kk (A; v) = [v|Av|A2 v| . . . |Ak−1 v].

142

JESSICA BOSCH AND CHEN GREIF

Using a Krylov subspace as right subspace K in projection methods has proven to be efficient. Various Krylov subspace methods arose from different choices of the left subspaces L. The Krylov subspace Kk (A; v) arises naturally if we refer to it as the subspace generated by k − 1 steps of the power iteration (see Section 3.1) with initial guess v. Similarly, for the inverse power iteration (see Section 3.2), we obtain the subspace

Kk (A − αI)−1 ; v . Both iterations produce a sequence of vectors v1 , . . . , vk that span a Krylov subspace and take vk as the approximate eigenvector. Now, rather than taking vk , it is natural to use the whole sequence v1 , . . . , vk in searching for the eigenvector. In fact, we saw in the previous section (Theorem 4.1 for the symmetric case) that we can even use Kk to compute the k best approximate eigenvalues and eigenvectors. There are three basic algorithms for generating a basis for the Krylov subspace: the Lanczos process for symmetric matrices, which we discuss next, the Arnoldi process for nonsymmetric matrices (Section 4.4), and the nonsymmetric Lanczos process. The latter computes matrices Q and P with P T Q = I such that P T AQ is tridiagonal; see, e.g., [25, 26, 41]. Moreover, there exist block versions of the Arnoldi and Lanczos process; see, e.g., [8, 48], which may exploit the block structure of a matrix in some situations. They are basically an acceleration technique of the subspace iteration, similar to the way the subspace iteration generalizes the power methods. 4.3. The Lanczos process. Let A ∈ Rn×n be symmetric. The Lanczos process computes an orthogonal basis for the Krylov subspace Kk (A; v) for some initial vector v, and approximates the eigenvalues of A by the Ritz values. Recall that the Hessenberg reduction of a symmetric matrix A reduces to a tridiagonal matrix, i.e., there exists an orthogonal Q ∈ Rn×n such that ⎤ ⎡ α1 β1 ⎥ ⎢ β1 α2 β2 ⎥ ⎢ ⎥ ⎢ . T . ⎥. ⎢ . β2 α3 (4.3) T = Q AQ = ⎢ ⎥ ⎥ ⎢ .. .. ⎣ . . βn−1 ⎦ βn−1 αn The connection between the tridiagonalization of A and the QR factorization of Kk (A; q1 ), where q1 = Qe1 is given as follows: Theorem 4.2 (See [22, Theor. 8.3.1].). Let ( 4.3) be the tridiagonal decomposition of a symmetric matrix A ∈ Rn×n with q1 = Qe1 . Then: (1) QT Kn (A; q1 ) = R is upper triangular. (2) If R is nonsingular, then T is unreduced. (3) If k = arg minj=1,...,n {rj,j = 0}, then k − 1 = arg minj=1,...,n−1 {βj = 0}. It follows from (1) in Theorem 4.2 that QR is the QR factorization of Kn (A; q1 ). In order to preserve the sparsity, we need an alternative to similarity transformations in order to compute the tridiagonalization. Let us write Q = [q1 | . . . |qn ]. Considering the kth column of AQ = QT , we obtain the following three-term recurrence: (4.4)

Aqk = βk−1 qk−1 + αk qk + βk qk+1 .

NUMERICAL SOLUTION OF LINEAR EIGENVALUE PROBLEMS

143

Since the columns of Q are orthonormal, multiplying (4.4) from the left by qk yields αk = qkT Aqk . This leads to the method in Algorithm 4.2 developed by Lanczos in 1950 [34]. Algorithm 4.2: Lanczos process 1 2 3 4 5 6 7 8 9 10 11

v Given q0 = 0, q1 = v , β0 = 0 2 for k = 1, 2, . . . do zk = Aqk αk = qkT zk zk = zk − βk−1 qk−1 − αk qk βk = zk 2 if βk = 0 then quit end qk+1 = zβkk end

The vectors qk computed by the Lanczos algorithm are called Lanczos vectors. The Lanczos process stops before the complete tridiagonalization if q1 is contained in an exact A-invariant subspace: Theorem 4.3 (See [22, Theor. 10.1.1].). The Lanczos Algorithm 4.2 runs until k = m, where m = rank(Kn (A, q1 )). Moreover, for k = 1, . . . , m, we have (4.5)

AQk = Qk Tk + βk qk+1 eTk ,

where Tk = T (1 : k, 1 : k), Qk = [q1 | . . . |qk ] has orthonormal columns with span{q1 , . . . , qk } = Kk (A, q1 ). In particular, βm = 0, and hence AQm = Qm Tm . The eigenvalues of the tridiagonal Tm can then be computed via, e.g., the QR iteration. A corresponding eigenvector can be obtained by using the inverse power iteration with the approximated eigenvalue as shift. We can show that the quality of the approximation after k Lanczos steps depends on βk and on parts of the eigenvectors of Tk (cf. [22, Chap. 10.1.4]): Therefore, let (θ, y) be an eigenpair of Tk . Applying (4.5) to y yields AQk y = Qk Tk y + βk qk+1 eTk y = θQk y + βk qk+1 eTk y and hence the following error estimation (cf. Theorem 4.1) AQk y − θQk y2 = |βk | |eTk y|. Hence, we want to accomplish βk = 0 fast. Regarding the convergence theory, we refer to [31, 38, 43] and [22, Chap. 10.1.5]. In summary, the Ritz values converge

144

JESSICA BOSCH AND CHEN GREIF

fast to the extreme eigenvalues. Using shift and invert strategies (as in the inverse power method in Section 3.2), we can obtain convergence to interior eigenvalues. In practice, rounding errors have a significant effect on the behavior of the Lanczos iteration. If the computed βk are close to zero, then the Lanczos vectors lose their orthogonality. Reorthogonalization strategies provide a remedy; see, e.g., [6, 21, 38, 40, 50, 67]. Nevertheless, we know from the last section that the Ritz values and vectors are good approximations. So far, we have assumed that A is symmetric. Next, we consider the nonsymmetric case. 4.4. The Arnoldi process. For a nonsymmetric A ∈ Rn×n , we know that there exists a Hessenberg decomposition, i.e., there exists an orthogonal Q ∈ Rn×n such that ⎤ ⎡ h1,1 h1,2 h1,3 ··· h1,n ⎢ h2,1 h2,2 h2,3 ··· h2,n ⎥ ⎥ ⎢ ⎥ ⎢ .. . .. ⎥. . 0 h h (4.6) H = QT AQ = ⎢ 3,2 3,3 ⎥ ⎢ ⎥ ⎢ . . . . .. .. .. ⎣ .. hn−1,n ⎦ 0 ··· 0 hn,n−1 hn,n The connection between the Hessenberg reduction of A and the QR factorization of Kk (A; q1 ), where q1 = Qe1 is given as follows (cf. the symmetric case in Theorem 4.2) Theorem 4.4 (See [22, Theor. 7.4.3].). Suppose Q ∈ Rn×n is orthogonal and let q1 = Qe1 and A ∈ Rn×n . Then, QT AQ = H is an unreduced upper Hessenberg matrix if and only if QT Kn (A; q1 ) = R is nonsingular and upper triangular. It follows from Theorem 4.4 that QR is the QR factorization of Kn (A; q1 ). As before, in order to preserve the sparsity, we need an alternative to similarity transformations in order to compute the Hessenberg reduction. Let us write Q = [q1 | . . . |qn ]. Considering the kth column of AQ = QH, we obtain the following recurrence: k+1  hi,k qi . (4.7) Aqk = i=1

Since the columns of Q are orthonormal, multiplying (4.7) from the left by qi yields hi,k = qiT Aqk for i = 1, . . . , k. This leads to the method in Algorithm 4.3 developed by Arnoldi in 1951 [1]. It can be viewed as an extension of the Lanczos process to nonsymmetric matrices. Note that, in contrast to the symmetric case, we have no three-term recurrence anymore. Hence, we have to store all computed vectors qk . These vectors are called Arnoldi vectors. The Arnoldi process can be seen as a modified GramSchmidt orthogonalization process (cf. [22, Chap. 5.2.8]) since in each step k we orthogonalize Aqk against all previous qi — ths requires O(kn) flops. Hence, the computational cost grows rapidly with the number of steps. After k steps of the Arnoldi Algorithm 4.3, we have (4.8)

AQk = Qk Hk + hk+1,k qk+1 eTk ,

NUMERICAL SOLUTION OF LINEAR EIGENVALUE PROBLEMS

145

Algorithm 4.3: Arnoldi process 1 2 3 4 5 6 7 8 9 10 11 12 13

v Given q1 = v 2 for k = 1, 2, . . . do zk = Aqk for i = 1, . . . , k do hi,k = qiT zk zk = zk − hi,k qi end hk+1,k = zk 2 if hk+1,k = 0 then quit end zk qk+1 = hk+1,k end

where Hk = H(1 : k, 1 : k) and (4.9)

span{q1 , . . . , qk } = Kk (A, q1 ).

We can show that the quality of the approximation depends on the magnitude of hk+1,k and on parts of the eigenvectors of Hk (cf. [22, Chap. 10.5.1]): Therefore, let (θ, y) be an eigenpair of Hk . Applying (4.8) to y yields AQk y = Qk Hk y + hk+1,k qk+1 eTk y = θQk y + hk+1,k qk+1 eTk y and hence the following error estimation (cf. Theorem 4.1) AQk y − θQk y2 = |hk+1,k | |eTk y|. Hence, we want hk+1,k = 0 fast. As the Lanczos process, the Arnoldi process has a (lucky) breakdown at step k = m if hm+1,m = 0 since Km (A, q1 ) is an A-invariant subspace in this case. In the following, we discuss accelerating techniques for the Arnoldi and Lanczos process. 4.5. Restarted Arnoldi and Lanczos. Note that with each Arnoldi step we have to store one additional Arnoldi vector. A remedy is restarting the Arnoldi process with carefully chosen restarts after a certain maximum of steps is reached. Acceleration techniques (mainly of a polynomial nature) generate an initial guess with small components in the unwanted parts of the spectrum. The strategies we present are called polynomial acceleration or filtering techniques. They exploit the powers of a matrix similar as the power method in the sense that they generate iterations of the form zr = pr (A)z0 , where pr is a polynomial of degree r. In the case of the power method, we have pr (t) = tr . Filtering methods have been successfully combined with the subspace iteration. When combined with the Arnoldi process, they are often called implicitly restarted methods, which we discuss next. Selecting a good polynomial often relies on some knowledge of the eigenvalues or related quantities (e.g., Ritz values).

146

JESSICA BOSCH AND CHEN GREIF

Suppose A ∈ Rn×n is diagonalizable and has eigenpairs {(λi , xi )}i=1,...,n with λ1 ≥ . . . ≥ λn . Let n  αi xi q1 = i=1

be an initial guess for the Arnoldi process. After running r steps of the Arnoldi process, we do a restart. We may seek a new initial vector from the span of the Arnoldi vectors q1 , . . . , qr , which has, due to (4.9), the form q+ = =

r  j=1 n 

βj Aj−1 q1 =

r 

βj

j=1

n 

αi Aj−1 xi =

i=1

r 

βj

n 

j=1

αi λj−1 xi i

i=1

αi pr−1 (λi )xi .

i=1

Suppose we are interested in the eigenvalue λj . If |αj pr−1 (λj )| ' |αl pr−1 (λl )| for all l = j, then q+ has large components in the eigendirection xj . Note that the αi are unknown. Hence, with an appropriate constructed polynomial, we can amplify the components in the desired parts of the spectrum. For instance, we are seeking for a polynomial that satisfies pr−1 (λj ) = 1 and |pr−1 (λl )| $ 1 for all l = j. However, the eigenvalues λi are unknown as well. Hence we need some approximation. Let Ω be a domain (e.g., an ellipse) that contains λ(A) \ {λj }, and suppose we have an estimate of λj . Then, we can aim to solve min

max |pr−1 (t)|.

pr−1 ∈Pr−1 , t∈Ω pr−1 (λj )=1

Suitable polynomials include the shifted and scaled Chebyshev polynomials, and in the symmetric case, we can exploit the three-term recurrence for fast computation; see, e.g., [45]. An alternative to Chebyshev polynomials is the following: Given {θi }i=1,...,r−1 , then one natural idea is to set (4.10)

pr−1 (t) = (t − θ1 )(t − θ2 ) · · · (t − θr−1 );

see [22, Chap. 10.5.2]. If λi ≈ θl for some l, then q+ has small components in the eigendirection xi . Hence, the θi are all unwanted values. For θi we can use the Ritz values, which presumably approximate the eigenvalues of A. For further heuristics, we refer to [44]. The above strategies are explicit restarting techniques, which use only one vector for the restart. The following implicit restarting strategy uses k vectors from the previous Arnoldi process for the new restarted Arnoldi process and throws away the remaining r − k =: p vectors. The procedure was developed in 1992 [53]. It implicitly determines a polynomial of the form (4.10) using the QR iteration with shifts. Suppose we have performed r steps of the Arnoldi iteration with starting vector q1 . Due to (4.8), we have (4.11)

AQr = Qr Hr + hr+1,r qr+1 eTr ,

where Hr ∈ Rr×r is upper Hessenberg, Qr ∈ Rn×r has orthonormal columns, and Qr e1 = q1 . Next, we apply p steps of the QR iteration with shifts θ1 , . . . , θp

NUMERICAL SOLUTION OF LINEAR EIGENVALUE PROBLEMS

147

(Algorithm 3.6), i.e., in step i we compute (4.12)

Vi Ri = H (i−1) − θi I,

(4.13)

H (i) = Ri Vi + θi I,

where H (0) = Hr . After p steps, we have H (p) = Rp Vp + θp I = VpT (Vp Rp + θp I) Vp = VpT H (p−1) Vp = . . . = V T H (0) V with V = V1 · · · Vp . We use the notation (4.14)

H+ := H (p) = V T H (0) V = V T Hr V .

The relationship to a polynomial of the form (4.10) is the following: Theorem 4.5 (See [22, Theor. 10.5.1].). If V = V1 · · · Vp and R = Rp · · · R1 are defined by ( 4.12)–( 4.13), then V R = (Hr − θ1 I) · · · (Hr − θp I). Using (4.14), we get in (4.11) (4.15)

AQr = Qr V H+ V T + hr+1,r qr+1 eTr .

Multiplying (4.15) from the right by V yields (4.16)

AQ+ = Q+ H+ + hr+1,r qr+1 eTr V

with Q+ = Qr V . It can be shown that V1 , . . . , Vp from the shifted QR iteration are upper Hessenberg. Hence, V (r, 1 : r − p − 1) = 0T and therefore eTr V = [0 · · · 0 α ∗ · · · ∗] is a row vector of length r whose first r − p − 1 entries are zero. ˆ r−p , Q ˆ p ] with Q ˆ r−p ∈ Rn×(r−p) we can write Now, using the notation Q+ = [Q (4.16) as + , ˆ r−p H ∗ ˆ r−p , Q ˆ p ] = [Q ˆ r−p , Q ˆp] (4.17) A[Q · · 06 α ∗ · · · ∗]. + hr+1,r qr+1 [03 ·45 βe1 eTr−p ∗ r−p−1

Now, we throw away the last p columns in (4.17) and obtain an (r − p)-step Arnoldi decomposition ˆ r−p + β Q ˆ r−p = Q ˆ r−p H ˆ p e1 eTr−p + hr+1,r qr+1 [0 · · · 0 α] AQ 3 45 6 r−p−1

  ˆ p e1 + αhr+1,r qr+1 eTr−p ˆ r−p + β Q ˆ r−p H =Q ˆ r−p H ˆ r−p + v =: Q ˆr+1 eTr−p .

This is the Arnoldi recursion we would have obtained by restarting the Arnoldi process with the starting vector q+ = Q+ e1 . Hence, we do not need to restart the Arnoldi process from step one but rather from step r − p + 1. For further details, we refer to [22, Chap. 10.5.3] and the references therein. Remark 4.6. It can be shown (cf. [22, Chap. 10.5.3]) that q+ = c (A − θ1 I) · · · (A − θp I) Qr e1 for some scalar c and is hence of the form (4.10). For further reading on the Arnoldi process we refer to, e.g., [47, 54, 55, 62]. Next we present another acceleration technique which is very popular for solving linear systems.

148

JESSICA BOSCH AND CHEN GREIF

4.6. Preconditioning. In the following, we quickly review the preconditioning concept for solving large and sparse systems of linear equations of the general form (4.18)

Az = b.

Here, A ∈ Rn×n is the given coefficient matrix, z ∈ Rn is the unknown solution vector, and b ∈ Rn is the given right-hand side vector. In order for the Equation (4.18) to have a unique solution, we assume that A is nonsingular. Systems of the form (4.18) arise after the discretization of a continuous problem like partial differential equations such as the time-harmonic Maxwell equations. Other applications arise in incompressible magnetohydrodynamics as well as constrained optimization. As already mentioned in Section 4.2, Krylov subspace solvers are also used for solving linear systems. In fact, they are state-of-the-art iterative solvers. However, they are usually only efficient in combination with an accelerator, which is called a preconditioner. The aim of a preconditioner is to enhance the convergence of the iterative solver. In our case, we want to accelerate the speed of convergence of Krylov subspace solvers. The basic idea is to construct a nonsingular matrix P ∈ Rn×n and solve (4.19)

P −1 Az = P −1 b

instead of Az = b. In order for P to be efficient, it should approximate A, and at the same time, the action of P −1 should require little work. The construction process of P should incorporate the goal of eigenvalue clustering. That means, P −1 A is aimed to have a few number of eigenvalues or eigenvalue clusters. This is bases on the following: In a nutshell, for linear systems the residual of a Krylov subspace solver rk = b − Azk satisfies rk = pk (A)r0 , and one approach would be to minimize the norm of the residual, which amounts to requiring that pk (λi )vi 2 be as small as possible for all i = 1, . . . , n. Here, {(λi , vi )}i=1,...,n are the eigenpairs of A. Therefore, replacing A by P −1 A such that P −1 A has more clustered eigenvalues is one way to go. This typically results in outstanding performances of Krylov subspace solvers. For an overview of iterative solvers and preconditioning techniques, we refer to [2, 4, 5, 16, 20, 23, 46, 59]. Preconditioning plays an important role in eigenvalue problems as well. Taken in the same spirit as seeking an operator that improves the spectrum, we can think of the inverse power iteration (see Section 3.2) as a preconditioning approach: The operator (A − θI)−1 has a much better spectrum than A for a suitable chosen shift θ. So, we can run Arnoldi on (A − θI)−1 rather than A since the eigenvectors of A and (A − θI)−1 are identical. Another idea is to incorporate polynomial preconditioning, i.e., replace A by pk (A). As a guideline, we want to transform the k wanted eigenvalues of A to k eigenvalues of pk (A) that are much larger than the other eigenvalues, so as to accelerate convergence. Preconditioning also plays a role in solving generalized eigenvalue problems Ax = λBx. They can be solved, e.g., by the Jacobi–Davidson method, whose idea we briefly discuss in Section 4.8 for solving the standard algebraic eigenvalue problem Ax = λx. The discussion of generalized eigenproblems is out of the scope of this survey. We refer the reader to [56, 57, 62] for a background to these problems.

NUMERICAL SOLUTION OF LINEAR EIGENVALUE PROBLEMS

149

4.7. Davidson method. Davidson’s method is basically a preconditioned version of the Lanczos process, but the amount of work increases similarly to Arnoldi, due to increased orthogonalization requirements. Let A ∈ Rn×n and Kk = Kk (A; v) be a Krylov subspace with respect to some vector v. Let {q1 , . . . , qk } be an orthonormal basis of Kk . In the orthogonal projection technique, we are seeking for an x ˆ ∈ Kk such that   ˆ x, qi = 0 ∀i = 1, . . . , k, Aˆ x − λˆ Suppose, we have a Ritz pair (θi , ui ). Then, the residual is given by ri = Aui − θi ui = (A − θi I)ui . Now, we can improve the eigenpair approximation by precondition the residual, i.e., by solving (P − θi I) t = ri , and define t as a new search direction, enriching the subspace. That is, t is orthogonalized against all basis vectors q1 , . . . , qk , and the resulting vector qk+1 enriches Kk to Kk+1 . Davidson [10] originally proposed to precondition with the diagonal matrix of A, i.e., P = diag(A), since he dealt with a diagonal dominant matrix A. Additionally, diagonal preconditioning offers a computationally cheap iteration. For the use of other preconditioners, we refer to [7]. Further references on Davidson’s method include [36, 54, 55]. Next, we consider an extension of Davdison’s method, which has the potential of working better for matrices that are not diagonally dominant. 4.8. Jacobi–Davidson method. The idea is to extend the strategy of preˆ x conditioning the residual. If (λ, ˆ) with ˆ x2 = 1 is an approximate eigenpair of ˆ x. Now, we look for (λ ˆ + δ λ, ˆ x A, then the residual is r = Aˆ x − λˆ ˆ + δx ˆ) to improve the eigenpair. We write ˆ + δ λ)(ˆ ˆ x + δx A(ˆ x + δx ˆ) = (λ ˆ), which is equivalent to ˆ ˆ x = −r + δ λδ ˆ x (A − λI)δ x ˆ − δ λˆ ˆ. By neglecting the second-order term, we obtain ˆ ˆ x = −r. (A − λI)δ x ˆ − δ λˆ This is an underdetermined system and a constraint must be added, e.g., ˆ x+ x2 = 1 and neglecting the second-order term, this condition δx ˆ2 = 1. With ˆ becomes x ˆT δ x ˆ = 0. T ˆ x, then we obtain δ x ˆ by solving the projected system If λ = x ˆ Aˆ





T ˆ ˆ x) ˆx ˆT δ x ˆ=− I −x ˆx ˆT (r − δ λˆ I −x ˆx ˆ (A − λI) I − x

=− I −x ˆx ˆT r

ˆ x) =− I −x ˆx ˆT (Aˆ x − λˆ

x =− I −x ˆx ˆT Aˆ ˆ x) = −r = −(Aˆ x − λˆ

150

JESSICA BOSCH AND CHEN GREIF

subject to the constraint x ˆT δ x ˆ = 0. As in the previous section, we replace A by a preconditioner P , such that we have to solve an approximate projected system



ˆ I −x ˆx ˆT (P − λI) I −x ˆx ˆT δ x ˆ = −r ˆ = 0. subject to the constraint x ˆT δ x The connection of the described method to Jacobi is given in Remark 4.7. ˆ x Remark 4.7. Given an approximate eigenpair (λ, ˆ) of A, Jacobi [30] proposed to solve an eigenvalue problem Ax = λx by finding a correction t such that A(ˆ x + t) = λ(ˆ x + t),

x ˆ ⊥ t.

This is called the Jacobi Orthogonal Component Correction (JOCC). The Jacobi–Davidson framework can also be connected with Newton’s method; see, e.g., [47, Chap. 8.4]. The debate over the advantaged and disadvantages of Jacobi–Davidson versus other approaches such as the Arnoldi process (with shift and invert) is delicate. Sleijpen and van der Vorst [51] relate it to whether the new direction has a strong component in previous directions. It is a fairly technical argument, and not much theory is available. For more details about the Jacobi–Davidson method, we refer to [51, 52, 54]. 5. Conclusions The numerical solution of eigenvalue problems is an extremely active area of research. Eigenvalues are very important in many areas of applications, and challenges keep arising. The survey covers only some basic principles, which have established themselves as the fundamental building blocks of eigenvalue solvers. We have left out some important recent advances, which are extremely important but also rather technical. Generalized eigenvalue problems are also very important, but there is not enough room to cover them in this survey. One of the main messages of this survey is the distinction between important mathematical observations about eigenvalues, and practical computational considerations. Objects such as the Jordan Canonical Form or determinants are classical mathematical tools, but they cannot be easily utilized in practical computations. On the other hand, sparsity of the matrix and the availability of matrix decompositions are often overlooked when a pure mathematical discussion of the problem ensues, but they are absolutely essential in the design of numerical methods. Altogether, this topic is satisfyingly rich and challenging. Efficiently and accurately computing eigenvalues and eigenvectors of matrices continues to be one of the most important problems in mathematical sciences. References [1] W. E. Arnoldi, The principle of minimized iteration in the solution of the matrix eigenvalue problem, Quart. Appl. Math. 9 (1951), 17–29, DOI 10.1090/qam/42792. MR0042792 [2] O. Axelsson, A survey of preconditioned iterative methods for linear systems of algebraic equations, BIT 25 (1985), no. 1, 166–187, DOI 10.1007/BF01934996. MR785811 [3] W. Barth, R. S. Martin, and J. H. Wilkinson, Handbook Series Linear Algebra: Calculation of the eigenvalues of a symmetric tridiagonal matrix by the method of bisection, Numer. Math. 9 (1967), no. 5, 386–393, DOI 10.1007/BF02162154. MR1553954 [4] M. Benzi, Preconditioning techniques for large linear systems: a survey, J. Comput. Phys. 182 (2002), no. 2, 418–477, DOI 10.1006/jcph.2002.7176. MR1941848

NUMERICAL SOLUTION OF LINEAR EIGENVALUE PROBLEMS

151

[5] M. Benzi, G. H. Golub, and J. Liesen, Numerical solution of saddle point problems, Acta Numer. 14 (2005), 1–137, DOI 10.1017/S0962492904000212. MR2168342 [6] D. Calvetti, L. Reichel, and D. C. Sorensen, An implicitly restarted Lanczos method for large symmetric eigenvalue problems, Electron. Trans. Numer. Anal. 2 (1994), no. March, 1–21. MR1270124 [7] M. Crouzeix, B. Philippe, and M. Sadkane, The Davidson method, SIAM J. Sci. Comput. 15 (1994), no. 1, 62–76, DOI 10.1137/0915004. MR1257154 [8] J. Cullum and W. E. Donath, A block Lanczos algorithm for computing the q algebraically largest eigenvalues and a corresponding eigenspace of large, sparse, real symmetric matrices, 1974 IEEE Conference on Decision and Control, 1974, pp. 505–509. [9] J. J. M. Cuppen, A divide and conquer method for the symmetric tridiagonal eigenproblem, Numer. Math. 36 (1980/81), no. 2, 177–195, DOI 10.1007/BF01396757. MR611491 [10] E. R. Davidson, The iterative calculation of a few of the lowest eigenvalues and corresponding eigenvectors of large real-symmetric matrices, J. Computational Phys. 17 (1975), 87–94. MR0381271 [11] J. W. Demmel, Applied numerical linear algebra, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1997. MR1463942 [12] J. Demmel and K. Veseli´ c, Jacobi’s method is more accurate than QR, SIAM J. Matrix Anal. Appl. 13 (1992), no. 4, 1204–1245, DOI 10.1137/0613074. MR1182723 [13] I. S. Dhillon and B. N. Parlett, Multiple representations to compute orthogonal eigenvectors of symmetric tridiagonal matrices, Linear Algebra Appl. 387 (2004), 1–28, DOI 10.1016/j.laa.2003.12.028. MR2069265 [14] J. J. Dongarra and D. C. Sorensen, A fully parallel algorithm for the symmetric eigenvalue problem, SIAM J. Sci. Statist. Comput. 8 (1987), no. 2, S139–S154, DOI 10.1137/0908018. Parallel processing for scientific computing (Norfolk, Va., 1985). MR879400 [15] A. A. Dubrulle and G. H. Golub, A multishift QR iteration without computation of the shifts, Numer. Algorithms 7 (1994), no. 2-4, 173–181, DOI 10.1007/BF02140681. MR1283096 [16] H. C. Elman, D. J. Silvester, and A. J. Wathen, Finite elements and fast iterative solvers: with applications in incompressible fluid dynamics, Numerical Mathematics and Scientific Computation, Oxford University Press, New York, 2005. MR2155549 [17] E. X. Jiang, A note on the double-shift QL algorithm, Linear Algebra Appl. 171 (1992), 121–132, DOI 10.1016/0024-3795(92)90254-8. MR1165449 [18] J. G. F. Francis, The QR transformation: a unitary analogue to the LR transformation. I, Comput. J. 4 (1961/1962), 265–271, DOI 10.1093/comjnl/4.3.265. MR0130111 [19] J. G. F. Francis, The QR transformation. II, Comput. J. 4 (1961/1962), 332–345, DOI 10.1093/comjnl/4.4.332. MR0137289 [20] R. W. Freund, G. H. Golub, and N. M. Nachtigal, Iterative solution of linear systems, Acta numerica, 1992, Acta Numer., Cambridge Univ. Press, Cambridge, 1992, pp. 57–100. MR1165723 [21] G. H. Golub, R. R. Underwood, and J. H. Wilkinson, The Lanczos algorithm for the symmetric Ax = λBx problem, Tech. report, Dep. Comput. Sci., Stanford Univ., Stanford, CA, 1972. [22] G. H. Golub and C. F. Van Loan, Matrix computations, 4th ed., Johns Hopkins Studies in the Mathematical Sciences, Johns Hopkins University Press, Baltimore, MD, 2013. MR3024913 [23] A. Greenbaum, Iterative methods for solving linear systems, Frontiers in Applied Mathematics, vol. 17, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1997. MR1474725 [24] M. Gu and S. C. Eisenstat, A divide-and-conquer algorithm for the symmetric tridiagonal eigenproblem, SIAM J. Matrix Anal. Appl. 16 (1995), no. 1, 172–191, DOI 10.1137/S0895479892241287. MR1311425 [25] M. H. Gutknecht, A completed theory of the unsymmetric Lanczos process and related algorithms. I, SIAM J. Matrix Anal. Appl. 13 (1992), no. 2, 594–639, DOI 10.1137/0613037. MR1152770 [26] M. H. Gutknecht, A completed theory of the unsymmetric Lanczos process and related algorithms. II, SIAM J. Matrix Anal. Appl. 15 (1994), no. 1, 15–58, DOI 10.1137/S0895479890188803. MR1257615 [27] N. J. Higham, Accuracy and stability of numerical algorithms, 2nd ed., Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2002. MR1927606

152

JESSICA BOSCH AND CHEN GREIF

[28] L. Hogben, Elementary linear algebra, West, St. Paul, MN, 1987. [29] H. Hotelling, Analysis of a complex of statistical variables into principal components, J. Educ. Psychol. 24 (1933), 417–441. ¨ [30] C. G. J. Jacobi, Uber ein leichtes Verfahren die in der Theorie der S¨ acularst¨ orungen vorkommenden Gleichungen numerisch aufzul¨ osen (German), J. Reine Angew. Math. 30 (1846), 51–94, DOI 10.1515/crll.1846.30.51. MR1578455 [31] S. Kaniel, Estimates for some computational techniques in linear algebra, Math. Comp. 20 (1966), 369–378, DOI 10.2307/2003590. MR0234618 [32] D. Kressner, Numerical methods for general and structured eigenvalue problems, Lecture Notes in Computational Science and Engineering, vol. 46, Springer-Verlag, Berlin, 2005. MR2164298 [33] V. N. Kublanovskaja, Some algorithms for the solution of the complete problem of eigenvalues ˇ Vyˇ (Russian), Z. cisl. Mat. i Mat. Fiz. 1 (1961), 555–570. MR0136058 [34] C. Lanczos, An iteration method for the solution of the eigenvalue problem of linear differential and integral operators, J. Research Nat. Bur. Standards 45 (1950), 255–282. MR0042791 [35] R. S. Martin, G. Peters, and J. H. Wilkinson, Handbook Series Linear Algebra: The QR algorithm for real hessenberg matrices, Numer. Math. 14 (1970), no. 3, 219–231, DOI 10.1007/BF02163331. MR1553971 [36] R. B. Morgan and D. S. Scott, Generalizations of Davidson’s method for computing eigenvalues of sparse symmetric matrices, SIAM J. Sci. Statist. Comput. 7 (1986), no. 3, 817–825, DOI 10.1137/0907054. MR848565 [37] M. L. Overton, Numerical computing with IEEE floating point arithmetic, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2001. Including one theorem, one rule of thumb, and one hundred and one exercises. MR1828597 [38] C. C. Paige, The computation of eigenvalues and eigenvectors of very large sparse matrices, PhD thesis, Univ. London, 1971. [39] B. N. Parlett, The symmetric eigenvalue problem, Classics in Applied Mathematics, vol. 20, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1998. Corrected reprint of the 1980 original. MR1490034 [40] B. N. Parlett and D. S. Scott, The Lanczos algorithm with selective orthogonalization, Math. Comp. 33 (1979), no. 145, 217–238, DOI 10.2307/2006037. MR514820 [41] B. N. Parlett, D. R. Taylor, and Z. A. Liu, A look-ahead L´ anczos algorithm for unsymmetric matrices, Math. Comp. 44 (1985), no. 169, 105–124, DOI 10.2307/2007796. MR771034 [42] H. Rutishauser, Solution of eigenvalue problems with the LR-transformation, Nat. Bur. Standards Appl. Math. Ser. 1958 (1958), no. 49, 47–81. MR0090118 [43] Y. Saad, On the rates of convergence of the Lanczos and the block-Lanczos methods, SIAM J. Numer. Anal. 17 (1980), no. 5, 687–706, DOI 10.1137/0717059. MR588755 [44] Y. Saad, Variations on Arnoldi’s method for computing eigenelements of large unsymmetric matrices, Linear Algebra Appl. 34 (1980), 269–295, DOI 10.1016/0024-3795(80)90169-X. MR591435 [45] Y. Saad, Chebyshev acceleration techniques for solving nonsymmetric eigenvalue problems, Math. Comp. 42 (1984), no. 166, 567–588, DOI 10.2307/2007602. MR736453 [46] Y. Saad, Iterative methods for sparse linear systems, 2nd ed., Society for Industrial and Applied Mathematics, Philadelphia, PA, 2003. MR1990645 [47] Y. Saad, Numerical methods for large eigenvalue problems, Classics in Applied Mathematics, vol. 66, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2011. Revised edition of the 1992 original [ 1177405]. MR3396212 [48] M. Sadkane, A block Arnoldi-Chebyshev method for computing the leading eigenpairs of large sparse unsymmetric matrices, Numer. Math. 64 (1993), no. 2, 181–193, DOI 10.1007/BF01388686. MR1199284 [49] A. Sch¨ onhage, Zur quadratischen Konvergenz des Jacobi-Verfahrens (German), Numer. Math. 6 (1964), 410–412, DOI 10.1007/BF01386091. MR0174171 [50] H. D. Simon, Analysis of the symmetric Lanczos algorithm with reorthogonalization methods, Linear Algebra Appl. 61 (1984), 101–131, DOI 10.1016/0024-3795(84)90025-9. MR755252 [51] G. L. G. Sleijpen and H. A. Van der Vorst, A Jacobi-Davidson iteration method for linear eigenvalue problems, SIAM J. Matrix Anal. Appl. 17 (1996), no. 2, 401–425, DOI 10.1137/S0895479894270427. MR1384515

NUMERICAL SOLUTION OF LINEAR EIGENVALUE PROBLEMS

153

[52] G. L. G. Sleijpen and H. A. Van der Vorst, A Jacobi-Davidson iteration method for linear eigenvalue problems, SIAM Rev. 42 (2000), no. 2, 267–293, DOI 10.1137/S0036144599363084. MR1778354 [53] D. C. Sorensen, Implicit application of polynomial filters in a k-step Arnoldi method, SIAM J. Matrix Anal. Appl. 13 (1992), no. 1, 357–385, DOI 10.1137/0613025. MR1146670 [54] D. C. Sorensen, Numerical methods for large eigenvalue problems, Acta Numer. 11 (2002), 519–584, DOI 10.1017/S0962492902000089. MR2009379 [55] G. W. Stewart, Matrix algorithms. Vol. II, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2001. Eigensystems. MR1853468 [56] G. W. Stewart, Introduction to matrix computations, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1973. Computer Science and Applied Mathematics. MR0458818 [57] G. W. Stewart and J. G. Sun, Matrix perturbation theory, Computer Science and Scientific Computing, Academic Press, Inc., Boston, MA, 1990. MR1061154 [58] H. P. M. van Kempen, On the quadratic convergence of the special cyclic Jacobi method, Numer. Math. 9 (1966), 19–22, DOI 10.1007/BF02165225. MR0202291 [59] A. J. Wathen, Preconditioning, Acta Numer. 24 (2015), 329–376, DOI 10.1017/S0962492915000021. MR3349311 [60] D. S. Watkins, Understanding the QR algorithm, SIAM Rev. 24 (1982), no. 4, 427–440, DOI 10.1137/1024100. MR678561 [61] D. S. Watkins, The transmission of shifts and shift blurring in the QR algorithm, Proceedings of the Fourth Conference of the International Linear Algebra Society (Rotterdam, 1994), Linear Algebra Appl. 241/243 (1996), 877–896, DOI 10.1016/0024-3795(95)00545-5. MR1400468 [62] D. S. Watkins, The matrix eigenvalue problem, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2007. GR and Krylov subspace methods. MR2383888 [63] D. S. Watkins, The QR algorithm revisited, SIAM Rev. 50 (2008), no. 1, 133–145, DOI 10.1137/060659454. MR2403061 [64] D. S. Watkins, Francis’s algorithm, Amer. Math. Monthly 118 (2011), no. 5, 387–403, DOI 10.4169/amer.math.monthly.118.05.387. MR2805025 [65] H. Wielandt, Das Iterationsverfahren bei nicht selbstadjungierten linearen Eigenwertaufgaben (German), Math. Z. 50 (1944), 93–143, DOI 10.1007/BF01312438. MR0017460 [66] J. H. Wilkinson, The algebraic eigenvalue problem, Clarendon Press, Oxford, 1965. MR0184422 [67] K. Wu and H. Simon, Thick-restart Lanczos method for large symmetric eigenvalue problems, SIAM J. Matrix Anal. Appl. 22 (2000), no. 2, 602–616, DOI 10.1137/S0895479898334605. MR1781506 Department of Computer Science, The University of British Columbia, Vancouver, BC, V6T 1Z4, Canada E-mail address: [email protected], [email protected] Department of Computer Science, The University of British Columbia, Vancouver, BC, V6T 1Z4, Canada E-mail address: [email protected]

Contemporary Mathematics Volume 700, 2017 http://dx.doi.org/10.1090/conm/700/14186

Finite element methods for variational eigenvalue problems Guido Kanschat

Contents 1. Introduction 2. Problem setting 3. Galerkin approximation 4. The finite element method 5. Saddle point problems Appendix A. Programs for experiments Appendix B. Selected problems References

1. Introduction The discretization of the eigenvalue problem through finite elements relies on two independent concepts: first, the abstract idea of a Galerkin discretization in Section 3, which describes projections to general finite dimensional subspaces of Hilbert spaces, and second, the choice of these subspaces by piecewise polynomials and the resulting concrete approximation error estimates in Section 4. These notes are only a short overview over the theory of finite element approximation of eigenvalue problems, and to a large extend a compilation of results found in [Bof10, BO87, BO89]. They are also content with discussing the approximation theory and giving a pointer to a software package for experiments. After discretization, we always obtain a matrix eigenvalue problem of the form Ax = λM x, where A is the matrix of the discretized differential operator and M is the so-called mass matrix, which generates the bilinear form of the L2 -inner product on the finite element space. Methods for the computation of eigenvalues and eigenvectors of the discretized problem are discussed in the chapter by Chen Greif in this volume. 2010 Mathematics Subject Classification. Primary 65N25, 65N30. c 2017 American Mathematical Society

155

156

GUIDO KANSCHAT

2. Problem setting 2.1. Source and eigenvalue problems for the Laplacian. We begin by shortly reviewing the most simple differential operator in this framework and its setting in Hilbert spaces. We are beginning with the so-called source problem, the Dirichlet problem for Poisson’s equation. Let Ω be a domain in Rd , where the dimension d is typically 2 or 3, but not restricted to those values. The boundary ∂Ω is assumed to be Lipschitz in the sense that there is a finite covering of ∂Ω of open sets Bi and a family of Lipschitz continuous mappings Φi , which map the open unit ball B ⊂ Rd to Ωi , such that Φ−1 (Ω ∩ Bi ) ⊂ {x ∈ B|x0 > 0} with x0 being the first component of the vector x. On such a domain, we search for the solution of −Δu = f in Ω, u = 0 on ∂Ω. In order to develop a reasonable solution theory for this problem, we multiply with a test function v and integrate by parts to obtain the weak formulation   (1) ∇u · ∇v dx = f v dx ∀v ∈ V, Ω

Ω

where V is a suitable test function space, here V = H01 (Ω), the Sobolev space obtained by completing the space C0∞ (Ω) of infinitely often differentiable functions with compact support in Ω with respect to the norm 9 |∇v|2 dx. vV = Ω

The Riesz representation theorem guarantees that this equation has a unique solution in u ∈ V for any suitable right hand side f , see for instance ‘¡[Eva98, GT98]. Indeed, while the right hand side in (1) suggests f ∈ L2 (Ω), we can actually relax this requirement to f being a continuous linear functional in V  = H −1 (Ω), such that the right hand side of (1) becomes f (v). Then, the weak form redefines the Laplacian as an operator −Δ : H01 (Ω) → H −1 (Ω). Now, we tend to the eigenvalue problem of finding u ∈ V and λ ∈ C such that −Δu = λu. With the above understanding of the operator −Δ, this equation does not make sense, since the object on the right is in V , the object on the left in V  . Thus, we transform this problem into weak form as well. First, we require a second space H, which in basic examples is typically chosen L2 (Ω). Then, invoke the Riesz representation theorem again, this time in H, yielding the variational eigenvalue problem   (2) ∇u · ∇v dx = λ uv dx ∀v ∈ V. Ω

Since fined.

H01 (Ω)

Ω

is continuously embedded into L2 (Ω), the equation above is well de-

FINITE ELEMENT METHODS FOR VARIATIONAL EIGENVALUE PROBLEMS

157

Due to the existence and uniqueness of solutions, we can define the inverse of the Laplacian as an operator (−Δ)−1 : L2 (Ω) → H01 (Ω). Indeed, since the Laplacian contains two derivatives, it might be suggested that the range is contained in H 2 (Ω). Nevertheless, the regularity theory of solutions on Lipschitz domains does not admit this conclusion [Gri85]. On the other hand, H01 (Ω) as the domain of the Laplacian is just what we need. Since the embedding of H01 (Ω) into L2 (Ω) is not only continuous, but also compact, see for instance [AF03], (−Δ)−1 is a compact operator. The spectral theorem for compact operators says that (−Δ)−1 has a point spectrum of at most countably many eigenvalues λ−1 k , which may not have any accumulation point in C except zero. Consequently, we deduce that the spectrum of the Laplacian as an operator on H01 (Ω) consists of at most countably many eigenvalues λk = 0 which do not have an accumulation point in C. Furthermore, the Laplacian in its weak form is self-adjoint, see for instance [GT98], such that its spectrum is contained in R+ . Thus, we conclude Theorem 1. The spectrum of the Laplacian defined as the values λ such that there is a function u ∈ V such that equation (2) holds contains at most countably many real values 0 < λ1 ≤ λ2 ≤ · · · ≤ λk ≤ . . . 2.2. Abstraction from the Laplacian. The eigenvalue problem for the Laplacian is only a model problem. Therefore, we generalize the ideas from the preceding paragraphs and introduce some notation. First, we introduce Definition 1. A bilinear form a(., .) is bounded and elliptic on a vector space V if there are positive constants C and C such that a(u, v) ≤ CuV vV a(u, u) ≥

Cu2V

∀u, v ∈ V

(boundedness)

∀u ∈ V

(ellipticity).

If in addition the bilinear form is symmetric, it defines a norm on V by  v = a(v, v). We introduce the oprator A associated to this bilinear form defined by ∀v, w ∈ V.

Av, w V ∗ ×V = a(v, w) If the form a(., .) is bounded, then A : V → V

∗

is a bounded operator.

Definition 2 (Source problem). Given a bilinear form a(., .) on V and f in the normed dual V ∗ , find a function u ∈ V such that (4)

a(u, v) = f (v)

∀v ∈ V.

For the class of bilinear forms introduced above, the well-posedness1 is established in the elliptic case by 1 Well-posedness in the context of numerical approximation follows the definition of Hadamard [Had02], namely that the solution shall exist, be unique, and shall depend continuously on the parameters.

158

GUIDO KANSCHAT

Lemma 1 (Lax-Milgram). Let a(., .) be symmetric, bounded and elliptic on V . Then a unique solution u ∈ V of equation (4) exists with the estimate uV ≤

(5)

C f V ∗ , C

where f V ∗ = sup v∈V

|f (v)| vV

is the operator norm of f on the space V . In the general case, we use a version of Banach’s closed range theorem which is particularly amenable to the study of numerical approximation Lemma 2 (Inf-sup condition). Let a(., .) be bounded and let there be a constant C > 0 such that (6)

inf

sup |a(u, v)| ≥ C,

u∈V v∈V

u =1 v =1

inf

sup |a(u, v)| ≥ C.

v∈V u∈V

v =1 u =1

Then, equation (4) has a unique solution u ∈ V admitting the estimate (5). Obviously, Lemma 2 implies Lemma 1 for symmetric bilinear forms. We proceed by generalizing (2) to Definition 3 (Variational eigenvalue problem). A pair (λ, u) ∈ C × V with u = 0 is an eigenpair of the operator A associated with the bilinear form a(., .), if there holds (7)

a(u, v) = λ u, v

∀v ∈ V.

Here, ., . is the inner product in a Hilbert space H with norm . and V ⊂ H is the domain of A. If the bilinear form a(., .) is nonsymmetric, we also introduce the Definition 4 (Adjoint variational eigenvalue problem). A pair (λ∗ , u∗ ) ∈ C×V with u = 0 is an adjoint eigenpair of the operator A associated with the bilinear form a(., .), if there holds (8)

a(v, u∗ ) = λ∗ v, u∗

∀v ∈ V.

Here, ., . is the inner product in a Hilbert space H with norm . and V ⊂ H is the domain of A. It is a well known fact, that the eigenvalues λ in (7) and λ∗ in (8) coincide (note that u∗ is on the right hand side of the sesquilinear form ., . ). Thus, we speak of the eigenfunction u and the adjoint eigenfunction u∗ with eigenvalue λ. We point out that while the source problem was solely defined on the space V , typically the domain of the operator associated with the bilinear form, this definition of an eigenvalue problem makes use of an additional space H containing V. An immediate consequence of ellipticity is the fact that the real parts of all eigenvalues of the variational eigenvalue problem (7) are positive and bounded from below by C. If additionally the bilinear form is symmetric (Hermitean in the complex case), the eigenvalues are even real and positive, and a(., .) forms an inner product on its domain.

FINITE ELEMENT METHODS FOR VARIATIONAL EIGENVALUE PROBLEMS

159

Let a(., .) be such that either Lemma 1 or Lemma 2 holds. Then, we define a solution operator T : H → V for the source problem such that for any w ∈ H there holds a(T w, v) = w, v

∀v ∈ V.

Then, by entering T u into (7), the eigenvalue problem is equivalent to λT u = u. Throughout the remainder of this chapter, we impose Assumption 1. The solution operator T is compact, that is, the embedding V → H is compact. As a consequence, the spectral theorem for compact operators holds. We denote that by this assumption, we exclude bilinear forms with zero eigenvalues, a problem which can be easily fixed by adding a shift of the form μ u, v to a(u, v) and computing the eigenvalues λ + μ of the new form. Let a(., .) be an inner product on its domain V and V compactly embedded in H. Then, Theorem 1 holds for this variational eigenvalue problem, and thus all eigenvalues are real and positive. Thus, the spectrum can be ordered and we have two important results Theorem 2 (Minimum principle). The eigenpairs (λk , uk ) of the symmetric variational eigenvalue problem (7) are recursively characterized by uk =

argmin u∈V a(u,uj )=0 ∀j 0 such that for any n there holds (12)

inf

sup |a(un , vn )| ≥ C,

un ∈Vn vn ∈Vn

un =1 vn =1

inf

sup |a(un , vn )| ≥ C.

vn ∈Vn un ∈Vn

vn =1 un =1

Then, equation (10) has a unique solution un ∈ Vn admitting the estimate (5).

FINITE ELEMENT METHODS FOR VARIATIONAL EIGENVALUE PROBLEMS

163

In principle, the constants C and C in Lemmas 2 and 4, respectively may be different. By taking the worse pair, we can keep the presentation simple and consider them equal. Indeed, Lemma4 would even hold with constants depending on n. We do not consider this case because it leads to suboptimal convergence estimates. We assume unique solvability of the discrete problem either by Lemma 3 or by Lemma 4. Thus, let u ∈ V and un ∈ Vn be the unique solutions of a(u, v) = f (v)

∀v ∈ V

a(un , vn ) = f (vn )

∀vn ∈ Vn .

An immediate consequence obtained by testing with v ∈ Vn ⊂ V in the first equation and subtracting the second is Lemma 5 (Galerkin orthogonality). There holds a(u − un , vn ) = 0

∀v ∈ Vn .

From this result, we immediately obtain Lemma 6 (C´ea’s lemma). Let a(., .) be bounded and elliptic such that equations (3) hold. Let furthermore u and un be the solutions in equations (13). Then, there holds the quasi-best approximation result u − un V ≤

C inf u − vV . C v∈Vn

Proof. From the assumptions there holds for arbitrary wn ∈ Vn : 1 u − un 2V ≤ a(u − un , u − un ) ellipticity C = a(u − un , u − wn ) Galerkin orthogonality ≤ Cu − un V u − wn V

boundedness.

Dividing by u − un V and making use of the fact that wn was chosen arbitrarily  in Vn yields the lemma. Thus, the error between the continuous solution u ∈ V and the discrete solution un ∈ Vn is reduced to the approximability of the function u ∈ V by a function in Vn . This is now simply a property of the approximation space and independent of the bilinear form a(., .). Thus, we introduce the notation η u (Vn ) = inf u − vV , v∈Vn

which describes the approximation accuracy of u by elements of Vn . The estimate in C´ea’s lemma thus becomes C u − un V ≤ η u (Vn ). C 3.2. The discrete eigenvalue problem. The results on eigenvectors and eigenspaces in this section will all be of the same type. The choice of finite element spaces for Vn and their approximation properties are deferred to Section 4. Here, we assume that the sequence of spaces {Vn } is exhausting V , that is, (14)

∀u ∈ V

inf u − vV → 0

v∈Vn

as n → ∞.

164

GUIDO KANSCHAT

We introduce two operators important for the analysis of eigenvalue problems. First, the Ritz projection Pn : V → Vn is defined such that for any w ∈ V there holds a(Pn w, vn ) = a(w, vn )

∀vn ∈ Vn .

In addition, we define the discrete solution operator Tn , such that for any wn ∈ Vn there holds a(Tn wn , vn ) = (wn , vn ) Pn∗

be the a(., .)-adjoint of Pn , Pn = Let any v ∈ V there holds

Pn∗

∀vn ∈ Vn . for a symmetric bilinear form. For

a(Tn wn , v) = a(Tn wn , Pn∗ v) = (wn , Pn∗ v) = a(T wn , Pn∗ v) = a(Pn wn , v). Thus, we obtain the relation Tn = Pn T. Entering Tn un into (11), we obtain the equivalent eigenvalue problem λ n P n T un = un . In this context, the assumption that the sequence of discrete spaces exhausts V in (14) translates into T − Pn T  = sup T u − Pn T uV → 0

(15)

u V =1

as n → ∞.

3.3. A single eigenpair. In this section, we derive error estimates for one solution of equation (11) in comparison to those of equation (7). We first follow [Fix73] for the general, non-selfadjoint problem4 . We assume that (6) holds for the space V as well as for the finite dimensional subspace Vn with the same constant C independent of n. For an eigenvalue λ of the continuous problem (7), its eigenspace Eλ , and its adjoint eigenspace Eλ∗ , we characterize approximation by Vn by η λ (Vn ) = sup

inf u − vV ,

u∈Eλ v∈Vn

u =1

η ∗λ (Vn ) = sup inf u∗ − vV , ∗ v∈Vn u∗ ∈Eλ

u∗ =1

that is, we measure the best approximation of the unit vector worst approximated in this eigenspace. From [Fix73, Theorem 1], we obtain the following estimates Theorem 6. Let the inf-sup condition (6), the discrete inf-sup condition (12), and the uniform convergence (15) hold. Let λ be an eigenvalue of (7). Then, there is an eigenvalue λn of (11), such that |λn − λ| ≤ Cηλ (Vn )η ∗λ (Vn ).

(17)

Furthermore, if un ∈ Vn with un  = 1, there is a function u ∈ Eλ such that un − uV ≤ Cη λ (Vn ).

(18)

In both estimates, the constant C depends on the bilinear forms, but not on n. 4 We

only consider the case of diagonalizable operators here. See [Fix73] for the general case.

FINITE ELEMENT METHODS FOR VARIATIONAL EIGENVALUE PROBLEMS

165

From estimate (18), we deduce that eigenvectors are approximated with the same accuracy as solutions to the source problem. Eigenvalues, as (17) shows, typically converge twice as fast as n → ∞. 3.4. Multiple eigenvalues and their eigenspaces. The result in the previous section covers multiple eigenvalues, but it does not make any predictions how a multidimensional eigenspace is approximated. Such a theory was developed in [BO87, BO89]. It makes much stronger statements on the approximation of eigenvectors, but it assumes an order relation on the spectrum. Thus, here we assume a real symmetric bilinear form and real eigenvalues. In the case of a multiple eigenvalue of the continuous problem, the approximating eigenvalues may be either simple or or multiple. Obviously, we have the Theorem 7 (Discrete min-max condition). (19)

λn,k =

min max

W ⊂Vn v∈W dim E=k

a(v, v) . v, v

Since the maximum in this equation only depends on the choice of the space W and the minimum is taken over a smaller set than in equation (9), we always have λk,n ≥ λk . We now consider the case of a q-fold eigenvalue of the continuous problem, that is, λk−1 < λk = · · · = λk+q−1 < λk+q . Due to the operator approximation (15), each discrete eigenvector converges to its continuous counterpart. Thus, if n is sufficiently large, we have the situation λk ≤ λn,k ≤ · · · ≤ λn,k+q−1 < λk+q . We have (see e.g. [BO89]) the following quasi-orthogonality condition, which relates the error of the eigenvalue to the residual of the eigenvalue problem: Lemma 7. Assume that (λ, u) is an eigenpair according to Definition 3 and w ∈ V arbitrary with wH = 1. Furthermore, let μ = a(w, w). Then, μ − λ = a(w − u, w − u) − λw − u2 . Proof. From the variational eigenvalue problem, we obtain a(u, u) = λu2 and by symmetry a(v, u) = λ v, u

∀v ∈ V.

Thus, entering the assumptions, we obtain a(w − u, w − u) − λw − u2 =a(w, w) − 2a(w, u) + a(u, u) − λw2 + 2λ w, u − λu2 =μw2 + λu2 − λw2 − λu2 =μ − λ. 

166

GUIDO KANSCHAT

Next we refine the notion of approximability of eigenvectors. In (15) and consequently in (17) and (18), only the worst approximation within the eigenspace Eλ was used in the estimate. Here, we define recursively and begin with ηk,0 (Vn ) = inf

inf u − vV .

u∈Eλk v∈Vn

u =1

Let u0 be a vector for which this infimum is achieved, and similarly uj below. Define for j = 1, . . . , q − 1 :   Eλj k = u ∈ Eλk : u, ui = 0, i = 0, . . . , j − 1 . Then, define ηk,j (Vn ) = infj

inf u − vV .

u∈Eλ v∈Vn k

u =1

Clearly, ηk,j (Vn ) ≤ η λk (Vn ), such that estimates involving these quantities are sharper than the previous ones. We obtain the estimate [BO87, Theorem 3.1]: Theorem 8. For every eigenvalue λk of (9) of multiplicity q, there is an n0 and a constant C such that for any n ≥ n0 and any discrete eigenvalue λn,k+j of (19) with 0 ≤ j < q there holds 2 (Vn ). λn,k+j − λk ≤ Cηk,j

Furthermore, for each discrete eigenfunction un,k+j solving (11) for λn,k+j , there is a continuous eigenfunction uk+j , such that un,k+j − uk+j V ≤ Cηk,j (Vn ). First, this theorem shows that the eigenvalue estimate in the previous section may be pessimistic and it should be possible to obtain an error bounded by ηλ2 k ,0 instead of η 2λk . But the consequences of this theorem are reaching farther. It predicts, that the kth discrete eigenvalue approximates the kth continuous eigenvalue, therefore preserving the structure of the spectrum. Furthermore, in the case that a continuous multiple eigenvalue is “split” into different discrete eigenvalues, the order of these is determined by the approximability of the corresponding eigenfunction. Thus, the method will in fact compute approximations for the functions u0 , . . . , uq−1 in the definition of ηk,j above in exactly this order. See Problem 5 in Section B.1. 4. The finite element method In this section, we give a cursory introduction to finite element spaces and their approximation properties. We will not present the method in all detail and refer the reader to the textbooks [Bra97, BS02, Cia78, GRS07, SF73]. The fundamental property of the finite element method as a Galerkin scheme consists in defining a discrete subspace for approximation. Thus, in this section we construct the spaces and provide estimates for the right hand sides of the quasi best approximation results of the previous section.

FINITE ELEMENT METHODS FOR VARIATIONAL EIGENVALUE PROBLEMS

167

4.1. Finite element spaces. Finite element spaces consist of three components: (1) A mesh consisting of mesh cells (2) A shape function space for each mesh cell (3) Node functionals 4.1.1. Meshes and mesh cells. A mesh5 Th is a covering6 of the computational domain Ω by nonoverlapping cells of simple geometry. Here, simple typically means simplices7 or smooth images of hypercubes in any dimension. We assume that for each topology type of a mesh cell T there is a reference cell Tˆ. For simplices, this might be the simplex with the origin and the unit vectors ei as its corners. For hypercubes, this might be the reference hypercube [0, 1]d . For each cell of the mesh Th , there is a one-to-one mapping Φ : Tˆ → T . The inverse of this mapping can be decomposed into the following parts: (translation and rotation) such that the (1) A fixed body movement Φ−1 F longest edge of T is a subset of the positive x1 -axis and the barycenter has only positive coordinates. (2) An isotropic scaling Φ−1 S by the scaling factor 1/hT such that this longest edge is mapped to [0, 1]. (3) A warping operation Φ−1 W . We compose Φ = ΦF ◦ ΦS ◦ ΦW ,

∇Φ = ∇ΦF ∇ΦS ∇ΦW .

By construction, ∇ΦF is orthogonal and ΦS = hT I. The “shape” of T is encoded in ΦW . We call a family of meshes {Th } shape regular, if there is a constant C such that for any cell T of any of the meshes we can find such a decomposition such that x) of ΦW (ˆ x) holds for the singular values σi (ˆ x) ≤ C min min σi (ˆ x). max max σi (ˆ x ˆ∈Tˆ i=1,...,d

x ˆ∈Tˆ i=1,...,d

Several sufficient geometric conditions for this have been introduced in the literature, namely for simplices, that the circumference is bounded uniformly by the radius of the inscribed sphere. For quadrilaterals, all angles should be uniformly bounded away from π and two vertices should not get too close to each other. 4.1.2. Shape functions. Shape functions form the local function spaces on each mesh cell. In almost all finite element methods, they are polynomial spaces or derived from such. It is convenient to derive them on the reference cell Tˆ and define them on the mesh cell T by pull-back through the mapping Φ. Standard spaces are the space Pk of multivariate polynomials of degree k, namely (for example in three dimensions) :   Pk = span xα y β z γ :0 ≤ α, β, γ ∧ α + β + γ ≤ k , 5 The index h is used to be able to denote sequences of meshes. It is loosely understood to mean mesh size. On the other hand, modern finite element methods do not use a fixed mesh size, such that the meaning of h is somewhat diffuse. 6 A favorite discussion in finite element expositions is whether cells should be open or closed. When we consider a covering or the intersection of cells, they should be closed. When we write nonoverlapping, they should be open. In this text we decide to be undecided, they may be open or closed, whatever is befitting at the moment. The context and a second of thought will provide the right meaning. 7 deal.II and thus Amandus do not provide simplicial meshes. Other software does.

168

GUIDO KANSCHAT

and the space of tensor product polynomials of degree k, namely :   Qk = span xα y β z γ :0 ≤ α, β, γ ≤ k . 4.1.3. Node functionals. Node functionals establish continuity over cell boundaries. For instance, in order that a piecewise polynomial function be in the space H 1 (Ω), it must be continuous. This can be achieved by the following mechanism: if the mesh cell is a cube, the trace of a polynomial in Qk on one of its faces is again in Qk , just of one dimension lower. The trace on one of its edges is again Qk , but only in dimension one. Thus, in order to establish continuity of functions between sharing a face, it is sufficient to establish interpolation conditions on the face and require that they are equal for the functions on both cells. In order to achieve continuity, we have to do this for vertices, edges, and faces. A very instructive graphical representation can be found in [AL14]. Node functionals define the continuity between cells, but they also determine the topology of the global (in the sense of “on the whole mesh”) finite element space Vn . In particular, the number of node functionals scattered over the mesh is the dimension of the finite element space. 4.1.4. Approximation properties. Error estimates for finite element functions can be derived either by averaged Taylor expansion of Sobolev functions [BS02] or by an abstract argument [Cia78]. They use an interpolation operator based on node functionals and the degree of the shape function spaces to Lemma 8. Let u ∈ H s (Ω), and let Tn be a mesh of mesh size h = max hT . Let Vn be a finite element space on Tn based on a shape function space containing Pk . Then, there is a function vn ∈ Vn , such that |u − vn |H m ≤ Chmin s−1,k+1−m |u|H s . From this estimate follows immediately that the quantities η λ (Vn ) and ηk,j (Vn ) defined in (16) and (20), respectively, are of order k if the eigenfunctions are sufficiently smooth. On the other hand, since eigenfunctions become more and more oscillatory for larger eigenvalues, this estimate also implies, that these values grow for higher eigenvalues. The largest eigenvalues of the discrete problem correspond to functions oscillating with a period corresponding to the grid spacing, and are thus determined more by mesh geometry and shape function spaces than by the shape of continuous eigenfunctions. Thus, only smaller eigenvalues and their eigenfunctions are approximated reliably. 5. Saddle point problems So far we have only considered elliptic problems. In this section, we are going to extend our theory to problems with constraints. The two model problems we consider with increasing difficulty are the Stokes and the Maxwell eigenvalue problem. In both cases, the mathematical formulation is based on Lagrange multipliers and leads to saddle point problems. We provide their basic theory first and then study the two applications. The material in this section can be found for instance in [BFB13]. 5.1. Saddle point problems. Consider a symmetric, positive definite bilinear form a(., .) on the space V . Consider a second bilinear form b(v, q) with the

FINITE ELEMENT METHODS FOR VARIATIONAL EIGENVALUE PROBLEMS

169

arguments v ∈ V and q from a second space Q. Now, we formulate the minimization problem: find u ∈ V , such that % & subject to b(v, q) = g, q Q , ∀q ∈ Q, a(u, u) = min a(v, v) − f, v V , v∈V

that is, the bilinear form b(., .) constrains the original minimization problem in V . The Lagrange multiplier rule says, that the solution u of this constrained minimization problem is a stationary point of the Lagrange functional % & % & L(u, p) = a(u, u) − f, u V − b(u, p) − g, p Q . Closer analysis reveals that locally it is the minimum with respect to variations of u and the maximum with respect to variations of p, hence the term saddle point problem. The Euler-Lagrange equations of this system can be written as: find (u, p) ∈ V × Q such that for all v ∈ V and q ∈ Q there holds (21)

a(u, v) − b(v, p) − b(u, q) = f, v V + g, q Q .

Here, we already have the immediate analogue of the weak formulation (6), the source problem for a saddle point formulation. It is customary to consider (21) more in the form of a system of equations for the vector (u, p)T , yielding the equivalent formulation ∀v ∈ V a(u, v) − b(v, p) = f, v V (22) ∀q ∈ Q. b(u, q) = g, q Q In order to simplify the presentation, we will assume g = 0, which is also the physically relevant situation in both our examples. Since we have already established that the solution is a saddle point, the system must be indefinite and therefore cannot be elliptic. Thus, well-posedness must be derived by the inf-sup condition. For saddle point problems, this condition takes a special form. To this end, let us first define the space of functions which obey the constraint, namely :   V 0 = v ∈ V :b(v, q) = 0 ∀q ∈ Q . Then, u is a solution of either (21) or (22) if and only if it is a solution to the reduced problem: find u ∈ V 0 such that (23)

a(u, v) = f, v

∀v ∈ V 0 .

Thus, we can derive unique existence of u without considering p. In a second step, we can determine p from the equation b(v, p) = f, v V − a(u, v)

∀v ∈ V.

Thus, we need an inf-sup condition for u on the subspace V 0 and one for p with respect to the form b(., .). Lemma 9 (Mixed inf-sup condition). Let a(., .) be bounded on V × V and b(., .) be bounded on V × Q. Let a(., .) admit the inf-sup condition (6) on the space V 0 . Assume furthermore that for b(., .) there holds (24)

inf

sup |b(v, q)| ≥ B.

q∈Q v∈V

u =1 v =1

Then, there exists a unique solution (u, p) ∈ V × Q of equations (21) and (22), respectively.

170

GUIDO KANSCHAT

Considering eigenvalue problems of equations in saddle point form we will adopt the variational setting of Section 2. But, here we are faced with two options, depending on whether we consider eigenvalues of the whole operator on the left of (22) with associated eigenvalues in V × Q or eigenvalues of the constrained problem (23). Accordingly, we define Definition 10 (Eigenvalue problem for the saddle point operator). A triplet (λ, u, p) ∈ C × V × Q with (u, p) = 0 is an eigenpair of the operator associated with the saddle point problem (22), if there holds & % (25) ∀v ∈ V, q ∈ Q. a(u, v) − b(v, p) − b(u, q) = λ u, v H + p, q Q , Definition 11 (Constrained eigenvalue problem). A pair (λ, u) ∈ C × V with u = 0 is an eigenpair of the operator A associated with the bilinear form a(., .) on the subspace V 0 , if there is a function p ∈ Q such that there holds (26)

a(u, v) − b(v, p) − b(u, q) = λ u, v H ,

∀v ∈ V, q ∈ Q.

Note that in the first definition (u, p) ∈ V × Q is considered the eigenfunction, while in the second definition, u ∈ V is the eigenfunction and p ∈ Q is just the Lagrange multiplier for the constraint. Note also that the two variational formulations only differ by the inner products on the right hand side. 5.2. The Stokes equations. The Stokes equations for incompressible flow are formulated for a vector valued velocity u and a scalar pressure. We choose the simplest setting here and refer to [BFB13] for more details: the boundary condition is no-slip, that is, all velocities vanish at the boundary of the domain. Thus, the velocity space is V = H01 (Ω; Rd ). The matching pressure space in the sense of (24) is :   : q dx = 0 . Q = q ∈ L2 (Ω):: Ω

The incompressibility constraint is ∇ · u = 0, such that we choose  b(v, q) = q∇ · v dx. Ω

Finally, the bilinear form a(., .) of our abstract framework has to be defined. While the form should involve the strain tensor for reasons of frame invariance, it turns out that no-slip boundary conditions allow for a simpler form: let ui be the components of the velocity vector, then d   a(u, v) = ∇ui · ∇vi dx. i=1

Ω

The next step involves finding discrete spaces. Like in Section 2, we have to require that the inf-sup condition holds in the discrete setting. The operator a(., .) is elliptic on V , thus, the inf-sup condition (6) holds for any subspace. It remains to guarantee (24). In order to find conditions on the spaces Vn and Qn , we rephrase (24) to ∀q ∈ Qh ∃v ∈ Vh : vV = qQ ∧ b(v, q) ≥ Bq2Q . In this form, we see that (24) is among others a condition that the velocity is “big enough” to control the pressure. Many pairs of spaces have been proposed over the

FINITE ELEMENT METHODS FOR VARIATIONAL EIGENVALUE PROBLEMS

171

last decades. Here, we point out the Hood/Taylor pair Qk+1 /Qk and the divergence conforming discontinuous Galerkin method, see [CKS07, KS14]. Both of them are implemented in the programs described in Section A. From the subspace property V 0 ⊂ V , it is clear that constrained eigenvalues of an elliptic bilinear form are greater than unconstrained ones. Otherwise, after obeying the mixed inf-sup condition, the Stokes problem does not pose any more challenges than the problems we considered before. In particular, we do not have to expect spurious eigenvalues inserted into the spectrum and the theory of subsection 3.4 holds. 5.3. The Maxwell equations. Maxwell’s equations have a similar setup as Stokes equations in the way that they also compute a vector field, here the magnetic field, under a divergence constraint. But, the bilinear form a(., .) is changed to one based on the curl of the vector field.  ∇ × u · ∇ × v dx. a(u, v) = Ω

The natural space for solutions is the graph space :   V = H0curl (Ω) = v ∈ L2 (Ω; Rd ):∇ × v ∈ L2 (Ω; Rd ) ∧ v × n = 0 on ∂Ω . The difficulties arising are two-fold: first, the divergence of a vector field in H0curl (Ω) is not well-defined. Thus, we have to integrate the divergence condition by parts to obtain  (27) v · ∇q dx. b(v, q) = Ω

The natural space for the so-called pseudo pressure p is thus Q = H01 (Ω). The second difficulty results from the fact that the curl operator has a big kernel, namely all gradients, plus additional functions if the domain is not simply connected. For a thorough study of the involved topics of cochain complexes of Hilbert- and finite element spaces, we refer the reader to [AFW06, AFW10], where the analytical framework is derived. Here, we just note that we have the Hodge decomposition of H0curl (Ω) into V = H0curl (Ω) = ∇Q ⊕ H ⊕V ⊥ . 3 45 6 =ker ∇×

Here, H are the harmonic forms, which are computed in Problem 4 in Appendix B. This decomposition is in fact orthogonal in L2 (Ω; Rd ), and the bilinear form a(., .) is elliptic just on V ⊥ . We surely do not want to compute a basis for the kernel of the curl operator. What we are interested in is the dimension of H, since it is equal to the Betti numbers and determines the topology of the domain. Furthermore, we are interested in the nonzero eigenvalues when we constrain the problem to V ⊥ . Thus, we are in the framework of the constrained eigenvalue problem (26) and the space V0 is defined by the form b(., .) in equation (27). When we investigated the Stokes problem, we concluded that the discrete velocity space had to be big enough to control the discrete pressure. Here now, we also need the opposite mechanism: the discrete space for the pseudo pressure must be big enough to guarantee that for any v ∈ Vh there holds (28)

b(v, q) = 0 ∀q ∈ Qh

⇒

b(v, q) = 0 ∀q ∈ Q.

172

GUIDO KANSCHAT

Thus, the spaces Vh and Qh must match exactly, which can be achieved by the mechanisms in [AFW06, AFW10]. But, does it actually matter? The results of problem 3 in Appendix B.2 indicate that the spectrum may be completely destroyed and spurious eigenvalues may be inserted at any point of the spectrum. How can this happen? Let un ∈ Vn be an eigenfunction with eigenvalue λn =

a(un , un ) , un , un

such that the left hand side of (28) holds, but not the right. Then, we can split 0 orthogonally un = u0n + u⊥ n , where the right hand side of (28) holds for un . Thus λn =

a(u0n , u0n ) . ⊥ u0n , u0n + u⊥ n , un

We see that even a high frequency function can produce a small eigenfunction, if its part in V 0 is sufficiently small. Appendix A. Programs for experiments Basic experiments with the finite element method for solving source and aigenvalue problems can be conducted with our sofware package Amandus. Amandus in turn is based on the finite element library deal.II available at www.dealii.org. Note that the instructions below require a Unix system like Linux or MAC OSX. Neither deal.II nor Amandus work reliably on native Windows as of now. If you only have a Windows computer available, either use a virtual box or consider a dual boot installation with Linux. A.1. The Amandus program package. Amandus is a collection of a few classes which simplify the usage of the quite complex finite element libary deal.II combined with a set of example applications. It can be cloned from the git archive at https://bitbucket.org/guidokanschat/amandus. A.1.1. Installation. First, you have to install a recent version of deal.II. When configuring with cmake, make sure that the Arpack library is picked up. The configuration output lists all configured options at the end. if you see a line DEAL_II_WITH_ARPACK set up with external dependencies you have it. If you see instead ( DEAL_II_WITH_ARPACK = OFF ) check your system installation. For instance, on Debian and related system, you have to install the developer package of libarpack as well. cmake and deal.II use three different directory: the source directory which you downloaded, the build directory in which you configure and build the library, and an install directory into which you install the library. A reasonable layout for these might be $HOME/dealii/deal.II (source) $HOME/dealii/build $HOME/dealii/install How do we get there? First, in your home directory, make a subdirectory dealii. Change into this directory, create a subdirectory build, and unpack deal.II, either

FINITE ELEMENT METHODS FOR VARIATIONAL EIGENVALUE PROBLEMS

173

the download of a release or the clone from github into the subdirectory deal.II. Change to the directory build created before and run cmake ../deal. II. Note: Amandus is currently under co-development with some functionality of deal.II. Therefore, we recommend to clone the developer version of deal.II (see the deal.II web site for this). You are free to delete the build directory after make install, and even the source directory, if you are not planning to rebuild deal.II. Installation of Amandus follows the same concept, except that you might skip the installation and run code in the build directory right away. Given the above directories for deal.II and the structure $HOME/amandus/amandus (source) $HOME/amandus/build for Amandus, you can configure Amandus in the build directory with cmake -DDEAL_II_DIR=$HOME/dealii/install ../amandus In order to try if everything is installed correctly, do in the build directory make laplace_eigen ./ laplace / eigen You should get approximations of the eigenvalues of the Laplacian on the square [−1, 1]2 . A.1.2. Experimenting with eigenvalue problems. The Amandus source directory contains subdirectories for different partial differential equations and some of them contain code for eigenvalue problems. Check for files eigen.h and eigen.cc. By the time of writing these notes, they exist in the subdirectories laplace, stokes, and maxwell. Replace XXX below for any of these. All of these directories contain a file eigen.prm, which is copied by cmake into the corresponding subdirectory of the build directory. In the file amandus/build/XXX/eigen.prm, change the number of eigenvalues or the number of refinement iterations (Steps). Use the output section to output eigenfunctions in gnuplot or VTK format; visualize them with gnuplot or a VTK viewer (check out paraview or visit). Experiment with different finite elements (FE). All these can be tried without recompiling the program. Try different domains and coarse meshes. In order to do this, you must change the file amandus/amandus/XXX/eigen.cc and recompile afterwards. Compilation can always be done in the build directory with make XXX_eigen Find the line which contains the word GridGenerator and change the function, for instance to GridGenerator::simplex. More ideas to get started are in Appendix B. Appendix B. Selected problems B.1. The Laplacian. As a basis for these problems, use the programs in amandus/laplace referring to eigenvalue problems, in particular eigen.cc. See Appendix A for compiling, running, and modifying them. (1) Verify the theoretical convergence estimates by computing on a sequence of meshes, where each mesh is obtained by global refinement of the previous one.

174

GUIDO KANSCHAT

(a) If the exact eigenvalue λ is known, by computing the error λh − λ on each mesh. Use the assumption λh − λ = Chp

(2) (3) (4) (5)

(6)

(7)

on consecutively refined meshes to estimate C and p. How do they depend on the mesh size and the eigenvalue? (b) If the exact eigenvalue is not known, use the “intrinsic convergence rate” generated by terms of the form λh − λh/2  instead of λh − λ and conclude with properties of the geometric series. Change the polynomial order (build/laplace/eigen.prm) and check how the results of the previous exercise change. Create your own program file my.cc in the subdirectory laplace and run cmake . in the build directory to add it to the compilation list. Use GridTools::distort random() to break the symmetry of the mesh and see how multiple eigenvalues separate. The code in eigen.cc uses regular divisions of a mesh for a square with a single cell obtained by GridGenerator::hyper cube(). Change this to an anisotropic mesh by using GridGenerator::subdivided hyper rectangle() and see how the approximations of eigenvalues 2 and 3 and their eigenfunctions changes. Change the domain from a square to (a) an L-shaped and a slit domain, a triangle (b) a circle (Hint: dealii tutorial step 6) (c) a cube (Hint: dealii tutorial step 4) and experiment with paraview visualization of functions in 3D Introduce a variable coefficient (a) First, select cells with a certain criterion and add a factor to the call of the Laplace:: cell matrix (). Example: double factor = 1.; if ( dinfo . cell - > center ()(0) < 0.5 && dinfo . cell - > center ()(0) < 0.5) factor = 10.; Laplace :: cell_matrix ( dinfo . matrix (0 , false ). matrix , info . fe_values (0) , factor );

(b) Find out how to use continuously varying coefficients (requires some more programming) (8) Implement error estimation and adaptive refinement (big exercise) B.2. Mixed problems. (1) Change the code for the Stokes eigenvalue problem such that it solves the eigenvalue problem (25) of the saddle point operator instead of the constrained eigenvalue problem (26). (2) Compute eigenvalues of the Oseen equation by adding an advection term (namespace LocalIntegrators :: Advection) to the bilinear form a(u, v). (3) For the Maxwell eigenvalue problem, check how the eigenvalues change if you replace FE Nedelec in the parameter file by FE System[FE Q(1)ˆd]

FINITE ELEMENT METHODS FOR VARIATIONAL EIGENVALUE PROBLEMS

175

(4) Compute the zero eigenvalues of the Maxwell operator and of the Stokes operator on a domain with holes generated with GridGenerator::cheese() and compare. References [AF03]

[AFW06]

[AFW10]

[AL14] [BFB13]

[BO87]

[BO89]

[Bof10] [Bra97]

[BS02]

[Cia78]

[CKS07]

[Eva98] [Fix73] [Gri85]

[GRS07]

[GT98]

[GW76]

Robert A. Adams and John J. F. Fournier, Sobolev spaces, 2nd ed., Pure and Applied Mathematics (Amsterdam), vol. 140, Elsevier/Academic Press, Amsterdam, 2003. MR2424078 Douglas N. Arnold, Richard S. Falk, and Ragnar Winther, Finite element exterior calculus, homological techniques, and applications, Acta Numer. 15 (2006), 1–155, DOI 10.1017/S0962492906210018. MR2269741 Douglas N. Arnold, Richard S. Falk, and Ragnar Winther, Finite element exterior calculus: from Hodge theory to numerical stability, Bull. Amer. Math. Soc. (N.S.) 47 (2010), no. 2, 281–354, DOI 10.1090/S0273-0979-10-01278-4. MR2594630 D. N. Arnold and A. Logg. Periodic table of the finite elements. SIAM News, 47(9), 2014. www.femtable.org. Daniele Boffi, Franco Brezzi, and Michel Fortin, Mixed finite element methods and applications, Springer Series in Computational Mathematics, vol. 44, Springer, Heidelberg, 2013. MR3097958 I. Babuˇska and J. E. Osborn, Estimates for the errors in eigenvalue and eigenvector approximation by Galerkin methods, with particular attention to the case of multiple eigenvalues, SIAM J. Numer. Anal. 24 (1987), no. 6, 1249–1276, DOI 10.1137/0724082. MR917451 I. Babuˇska and J. E. Osborn, Finite element-Galerkin approximation of the eigenvalues and eigenvectors of selfadjoint problems, Math. Comp. 52 (1989), no. 186, 275–297, DOI 10.2307/2008468. MR962210 Daniele Boffi, Finite element approximation of eigenvalue problems, Acta Numer. 19 (2010), 1–120, DOI 10.1017/S0962492910000012. MR2652780 Dietrich Braess, Finite elements, Cambridge University Press, Cambridge, 1997. Theory, fast solvers, and applications in solid mechanics; Translated from the 1992 German original by Larry L. Schumaker. MR1463151 Susanne C. Brenner and L. Ridgway Scott, The mathematical theory of finite element methods, 2nd ed., Texts in Applied Mathematics, vol. 15, Springer-Verlag, New York, 2002. MR1894376 Philippe G. Ciarlet, The finite element method for elliptic problems, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. Studies in Mathematics and its Applications, Vol. 4. MR0520174 Bernardo Cockburn, Guido Kanschat, and Dominik Sch¨ otzau, A note on discontinuous Galerkin divergence-free solutions of the Navier-Stokes equations, J. Sci. Comput. 31 (2007), no. 1-2, 61–73, DOI 10.1007/s10915-006-9107-7. MR2304270 Lawrence C. Evans, Partial differential equations, Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 1998. MR1625845 George J. Fix, Eigenvalue approximation by the finite element method, Advances in Math. 10 (1973), 300–316, DOI 10.1016/0001-8708(73)90113-8. MR0341900 P. Grisvard, Elliptic problems in nonsmooth domains, Monographs and Studies in Mathematics, vol. 24, Pitman (Advanced Publishing Program), Boston, MA, 1985. MR775683 Christian Grossmann and Hans-G¨ org Roos, Numerical treatment of partial differential equations, Universitext, Springer, Berlin, 2007. Translated and revised from the 3rd (2005) German edition by Martin Stynes. MR2362757 David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, Springer-Verlag, Berlin-New York, 1977. Grundlehren der Mathematischen Wissenschaften, Vol. 224. MR0473443 G. H. Golub and J. H. Wilkinson, Ill-conditioned eigensystems and the computation of the Jordan canonical form, SIAM Rev. 18 (1976), no. 4, 578–619, DOI 10.1137/1018113. MR0413456

176

[Had02] [KS14]

[SF73]

[Wil72]

GUIDO KANSCHAT

J. Hadamard. Sur les problemes aux derivees partielles et leur signification physique. Princeton University Bulletin, 1902. Guido Kanschat and Natasha Sharma, Divergence-conforming discontinuous Galerkin methods and C 0 interior penalty methods, SIAM J. Numer. Anal. 52 (2014), no. 4, 1822–1842, DOI 10.1137/120902975. MR3240852 Gilbert Strang and George J. Fix, An analysis of the finite element method, PrenticeHall, Inc., Englewood Cliffs, N. J., 1973. Prentice-Hall Series in Automatic Computation. MR0443377 J. H. Wilkinson, Note on matrices with a very ill-conditioned eigenproblem, Numer. Math. 19 (1972), 176–178, DOI 10.1007/BF01402528. MR0311092

Interdisciplinary Center for Scientific Computing, Heidelberg University, Mathematikon, Klaus-Tschira-Platz 1, 69120 Heidelberg, Germany E-mail address: [email protected]

Contemporary Mathematics Volume 700, 2017 http://dx.doi.org/10.1090/conm/700/14187

Computation of eigenvalues, spectral zeta functions and zeta-determinants on hyperbolic surfaces Alexander Strohmaier Abstract. These are lecture notes from a series of three lectures given at the summer school “Geometric and Computational Spectral Theory” in Montreal in June 2015. The aim of the lecture was to explain the mathematical theory behind computations of eigenvalues and spectral determinants in geometrically non-trivial contexts.

1. The method of particular solutions The method of particular solutions is a method to find eigenvalues for domains with Dirichlet boundary conditions. It goes back to an idea by Fox-Henrici-Moler from 1967 ([11]) and was revived by Betcke and Trefethen [6] essentially by modifying it to make it numerically stable. 1.1. A high accuracy eigenvalue solver in one dimension. In order to illustrate the method, let us look at it in the simple case of a differential operator ∂2 on an interval. Let [−L, L] ⊂ R be a compact interval. As usual, let −Δ = − ∂x 2 be the Laplace operator and assume that V ∈ C ∞ ([−L, L]) is a potential. Then the operator −Δ + V subject to Dirichlet boundary conditions has discrete spectrum. This means there exists a discrete set of values (λi )i∈N such that the equation (−Δ + V − λ)u = 0,

u(−L) = 0,

u(L) = 0.

admits a non-trivial solution u = φi . The eigenvalues (λi ) can be computed as follows. Step 1. Solve the initial value problem. For each λ ∈ C we can solve the initial value problem d (−Δ + V − λ)uλ = 0, uλ (−L) = 0, uλ (−L) = 1. dx This can be done either analytically or numerically depending on the type of differential equation. Then uλ (+L) as a function of λ is entire in λ. The function does not vanish identically as for example can be shown using integration by parts at λ = i. The eigenvalues are precisely the zeros of this function. This provides a direct proof that the eigenvalues form a discrete set. Step 2. Find the zeros of the function λ → uλ (+L) for example using the secant method or Newton’s method. This will converge rather fast because the function is analytic. c 2017 American Mathematical Society

177

178

A. STROHMAIER

This algorithm is implemented in the following Mathematica script in the case V (x) = 5(1 − x2 ) on the interval [−1, 1].

MPS AND GEOMETRY

179

180

A. STROHMAIER

0.5 0.4 0.3 0.2 0.1 -1.0

-0.5

0.5

1.0

0.5

1.0

0.5

1.0

0.3 0.2 0.1 -1.0

-0.5 -0.1 -0.2 -0.3 0.2 0.1

-1.0

-0.5 -0.1 -0.2

MPS AND GEOMETRY

181

1.2. Dirichlet eigenvalues for domains in Rn . The following is a classical result by Fox-Henrici-Moler from 1967 ([11]). Suppose that Ω ⊂ Rn is an open bounded domain in Rn . Then, the Laplace operator −Δ with Dirichlet boundary conditions can be defined as the self-adjoint operator obtained from the quadratic form q(f, f ) = ∇f, ∇f L2 (Ω) with form domain H01 (Ω). Since the space H01 (Ω), by Rellich’s theorem, is compactly embedded in L2 (Ω) the spectrum of this operator is purely discrete and has ∞ as its only possible accumulation point. Hence, there exists an orthonormal basis (uj )j∈N in L2 (Ω) consisting of eigenfunctions with eigenvalues λj , which we assume to be ordered, i.e. −Δuj = λj uj , (1)

uj L2 (Ω) = 1, uj |∂Ω = 0, 0 < λ1 ≤ λ2 ≤ · · ·

Suppose that u ∈ C ∞ (Ω) is a smooth function on the closure Ω of Ω satisfying −Δu = λu,

182

A. STROHMAIER

and assume that

(2)

uL2 (Ω) = 1,  = |Ω| · u|∂Ω ∞ < 1.

Then the theorem of Fox-Henrici-Moler states that there exists an eigenvalue λj of the Dirichlet Laplace operator −ΔD such that √ 2 + 2 |λ − λj | ≤ (3) . λ 1 − 2 This estimate can be used to obtain eigenvalue inclusions as follows. Choose a suitable set of functions (φj )j=1,...,N satisfying −Δφj = λφj . Such functions could for example be chosen to be plane waves φj = exp(ikj · x), where kj ∈ Rn are vectors such that kj  = λ. Then one tries to find a linear N combination u = j=1 vj φj , vj ∈ R such that u|∂Ω ∞ is very small. If one approximates the boundary by a finite set of points this reduces to a linear algebra problem. This strategy was quite successful to find low lying eigenvalues for domains in R2 , but was thought to be unstable for higher eigenvalues and for greater precision when more functions were used. The reason for this unstable behavior is that with too many functions being used, i.e. N being very large, there might be more linear combinations of the functions φj whose L2 -norm is rather small, despite the fact that the 2 -norm of the coefficient vector aj is not small. Betcke and Trefethen ([6]) managed to stabilize the method of particular solutions by preventing the function u from becoming small in the interior. A simple way to implement a stable method of particular solutions is as follows. Let (φk )k=1,...,N be functions as before. Let (xj )j=1,...,M be a family of points on the boundary ∂Ω, and let (yj )j=1,...,Q be a sufficiently large family of internal points in Ω, say randomly distributed. N We are looking for a linear combination u = k=1 vk φk that is small at the boundary, but that does not vanish in the interior of Ω. Thus, roughly, we are Q M seeking to minimize j=1 |u(xj )|2 whilst keeping j=1 |u(yj )|2 constant. Using the matrices A = (aij ), aij = φj (xi ), B = (bij ),

bij = φj (yi ),

Av we are thus looking for a vector v = (v1 , . . . , vN ) ∈ CN such that the quotient Bv is minimal. Minimizing this quotient is the same as finding the smallest generalized singular vector of the pair (A, B). The minimal quotient is the smallest singular value of the pair (A, B). This value can then be plotted as a function of λ.

The following simple Mathematica code implements this for in the interior of an ellipse. This is done for the interior 1 Ω = {(x, y) ∈ R2 | x2 + y 2 < 1}. 4 The code illustrates that the first Dirichlet eigenvalues can be computed with a remarkable precision.

MPS AND GEOMETRY

183

1.0

0.5

-2

-1

1

2

-0.5

-1.0

0.7 0.5 0.5 0.4 0.3 0.2 0.1 5

10

15

20

184

A. STROHMAIER

6.×10−6 5.×10−6 4.×10−6 3.×10−6 2.×10−6 1.×10−6 3.56671

3.56672

3.56673

3.56674

3.56675

Once the numerical part is successful and we have a singular vector for the smallest singular value, we are left with two analytical challenges to establish an eigenvalue inclusion in an interval [λ − , λ + ]: (1) Prove that the function u is small on the boundary, i.e. estimate u|∂Ω ∞ . (2) Prove that the L2 -norm of the function u is not too small, i.e. estimate uL2 (Ω) .

MPS AND GEOMETRY

185

The first point is easy to deal with, for example by Taylor expanding the function u at the boundary (in case the boundary is smooth) and using Taylor’s remainder estimate. The second point is more tricky. Since however even any bad bound from below will do the job, numerical integration with a remainder term can be used to check directly that the L2 -norm is not very small. Once a list of eigenvalues is established there is another analytical challenge. (3) Prove that the method does not miss any eigenvalue if the step-size is chosen small enough. This point is the most difficult one. It requires a proof that the set of functions is sufficiently large in a quantified sense. It is often easier to first compute a list of eigenvalues and then check afterwards, using other methods, that this list is complete. The method of particular solutions for domains has been further improved beyond what is presented here (see for example [4] and references therein) and a software package MPS-pack ([3]) exists that makes it possible to compute eigenvalues with very high accuracy for domains in R2 . 2. The Method of Particular Solutions in a Geometric Context Instead of the Dirichlet problem for a domain, we will now consider the problem of finding the spectral resolution of the Laplace operator on a closed Riemannian manifold M with metric g and dimension n. Then the metric Laplace operator −Δ : C ∞ (M ) → C ∞ (M ) is given in local coordinates by −Δ = −

(4)

n 

1 ∂  ∂  |g|g ik k . i ∂x ∂x |g| i,k=1

The space C ∞ (M ) is equipped with the metric inner product   f1 (x)f2 (x) |g|dx. f1 , f2 = M ∞

The completion of C (M ) is the space L2 (M ). The Laplace operator is essentially self-adjoint as an unbounded operator in L2 (M ) and the domain of the closure is equal to the second Sobolev space H 2 (M ). By Rellich’s theorem this space is compactly embedded in L2 (M ) and therefore the Laplace operator has compact resolvent, i.e. its spectrum is purely discrete with ∞ as the only possible accumulation point. Moreover, −Δ is a non-negative operator, and the zero eigenspace consists of locally constant functions. Because of elliptic regularity the eigenfunctions are smooth on M . Summarizing, we therefore know that there exists an orthonormal basis (uj ) in L2 (M ) such that −Δuj = λj uj , (5)

uj ∈ C ∞ (M ), 0 ≤ λ1 ≤ λ2 ≤ · · ·

We will be applying the idea of the method of particular solutions to manifolds (see [18]). We start by describing this in a very general setting. Suppose that M is a compact Riemannian manifold and suppose this manifold is glued from a finite number of closed subsets Mj along their boundaries so that M = ∪qj=1 Mj .

186

A. STROHMAIER

We assume here that Mj are manifolds that have a piecewise smooth Lipschitz boundary. Example 2.1. The n-torus T n can be obtained from the cube [0, 1]n by identifying opposite boundary components. In this case we have only one component M1 and its boundary ∂M1 . Example 2.2. A surface of genus 2 can be glued from two pair of pants, or alternatively, from 4 hexagons. This will be discussed in detail in Section 3. If f ∈ C ∞ (M ) is a function on M then we can of course restrict this function to each of the components Mj and we thus obtain a natural map R : C ∞ (M ) → C ∞ (*j Mj ).

(6)

Since the interior of Mj is naturally a subset in M , and its boundary has zero measure, we can also understand functions in C 1 (*j Mj ) as (equivalence classes of) functions on M that have jump type discontinuities along the boundaries of Mj . In this way we obtain a map E : C ∞ (*j Mj ) → L∞ (M ).

(7)

By construction, we have E ◦ R = 1. Given a function in C ∞ (*j Mj ), we can also measure its jump behavior as follows. After gluing the boundaries *j ∂Mj form a piecewise smooth Lipschitz hypersurface Σ in M . Suppose x is a point in Σ. Then x arises from gluing points in *j ∂Mj . We will assume that there are precisely two such points x+ ∈ ∂Mj1 and x− ∈ ∂Mj2 that form the point x after gluing. We will also assume that the normal outward derivatives ∂n(x+ ) and ∂n(x− ) are well defined at these points. These two assumption are satisfied on a set of full measure in Σ. Note that there is freedom in the choice of x+ and x− for a given x. We assume here that such a choice has been made and that this choice is piecewise continuous. Given f ∈ C ∞ (*j Mj ) we define Df (x) = f (x+ ) − f (x− ), Dn f (x) = ∂n(x+ ) f + ∂n(x− ) f.

(8)

These functions are then functions in L∞ (Σ). Df measures the extent to which f fails to be continuous and Dn f measures the extent to which f fails to be differentiable. The significance of the functions Df and Dn f is in the fact that they naturally appear in Green’s identity as follows. Suppose that (fj ) is a collection of smooth functions on Mj and f is the assembled function f = E(fj ). Then, by Green’s formula, for any test function g ∈ C0∞ (M ) we have < ;    f (x)(Δg)(x)dx = (Δfj )(x)g(x)dx M

(9)

;   + − j

∂Mj

j

Mj



(∂n f )(x)g(x)dx +

< f (x)(∂n g)(x)dx .

∂Mj

MPS AND GEOMETRY

The last two terms can be re-written as ;    − (∂n f )(x)g(x)dx + j

(10)



∂Mj

=−



(Dn f )(x)g(x)dx +

187

< f (x)(∂n g)(x)dx

∂Mj

(Df )(x)(∂n g)(x)dx

Σ

Σ

if the normal vector field ∂n at the point x is chosen to be ∂n(x+ ) . In other words, in the sense of distributional derivatives −Δf is the distribution  E(−Δfj ) + (Dn f ) ⊗ δΣ + (Df ) ⊗ δΣ .

(11)

 are the Dirac delta masses and the corresponding Here the distributions δΣ and δΣ normal derivative along the hypersurface Σ. The tensor product here is understood in the sense that pairing with test functions is defined as follows  (12) h(x)g(x)dx (h ⊗ δΣ )(g) := Σ

and  (h ⊗ δΣ )(g) := −

(13)

 h(x)(∂n g)(x)dx. Σ

In particular, if the functions fj satisfy the eigenvalue equation (Δ + λ)fj = 0 on each component Mj then we have in the sense of distributions  (−Δ − λ)f = (Dn f ) ⊗ δΣ + (Df ) ⊗ δΣ .

(14)

Since Σ was assumed to be piecewise smooth and Lipschitz, the Sobolev spaces H s (Σ) are well defined for any s ∈ R. Theorem 2.3. There exists a constant C > 0 which can be obtained explicitly for a given Riemannian manifold M and decomposition (Mj ) once the Sobolev norms are defined in local coordinates, such that the following statement holds. Suppose that (φj ) is a collection of smooth functions on Mj , and denote by φ the corresponding function E(φj ) on M . Suppose furthermore that (1) φL2 (M ) = 1, (2) −Δφ − λφ = χ on M \Σ, (3) χL2 (M ) = η,   12 (4) C Dφ2 − 1 + Dn φ2 − 3 = < 1. H

2

(Σ)

H

2

(Σ)

Then there exists an eigenvalue λj of −Δ in the interval [λ −

(1 + λ) + η (1 + λ) + η ,λ + ]. 1− 1−

Proof. By the Sobolev restriction theorems the distributions (Dn f ) ⊗ δ∂Σ as  are in H −2 (M ) and we have well as (Df ) ⊗ δ∂Σ  Dφ ⊗ δΣ H −2 (M ) ≤ C1 DφH −1/2 (Σ) ,

Dn φ ⊗ δΣ H −2 (M ) ≤ C2 Dn φH −3/2 (Σ) . Loosely speaking this follows since restriction to a co-dimension one Lipschitz hy1 persurface is continuous as a map from H s to H s− 2 for s > 12 and the corresponding dual statement. These estimates can also be obtained in local coordinates using

188

A. STROHMAIER

the Fourier transform. The constants C1 and C2 can therefore be estimated once local charts are fixed.  ). Then, Let us define the distribution g := (−Δ+1)−1 ((Dn f ) ⊗ δΣ + (Df ) ⊗ δΣ 2 by elliptic regularity, g ∈ L (M ) and   12 2 2 gL2 (M ) = ≤ C Dφ − 1 + Dn φ − 3 . H

2

(Σ)

H

2

(Σ)

One checks by direct computation that (−Δ − λ)(φ − g) = χ + (1 + λ)g. Using χ + (1 + λ)gL2 (M ) ≤ η + |1 + λ|gL2 (M ) , φ − g ≥ 1 − gL2 (M ) , one obtains

1 − gL2 (M ) . η + |1 + λ|gL2 (M ) This implies the statement as the resolvent norm is bounded by the distance to the spectrum.  (−Δ − λ)−1 L2 (M ) ≥

Of course, g2H s (Σ) ≤ g2L2 (Σ) for any s ≤ 0 so, one also obtains a bound  12  in terms of Dφ2L2 (Σ) + Dn φ2L2 (Σ) , although this bound does not take into account the different microlocal properties of Dn φ and Dφ, i.e. their behaviour for large frequencies. 3. Hyperbolic Surfaces and Teichm¨ uller Space The following section is a brief description of the construction and theory of hyperbolic surfaces. In the same way as the sphere S 2 admits a round metric and the torus T 2 admits a two dimensional family of flat metrics, a two dimensional compact manifold M of genus g ≥ 2 admits a family of metrics of constant negative curvature −1. By the theorem of Gauss-Bonnet all these metrics yield the same volume Vol(M ) = 4π(g − 1). For an introduction into hyperbolic surfaces and their spectral theory, we would like to refer to the reader to the excellent monograph [8]. We start by describing some two dimensional spaces of constant curvature −1. • The upper half space The hyperbolic upper half space H is defined as H := {x + iy ∈ C | y > 0} with metric g = y −2 (dx2 + dy 2 ). The Laplace operator with respect to this metric is then given by  2  ∂ ∂2 2 + 2 . −Δ = −y ∂x2 ∂y The geodesics in this space are circles that are perpendicular to the real line. The group of isometries of the space is the group P SL(2, R). The

MPS AND GEOMETRY

189

Figure 1. Fundamental domain for a hyperbolic cylinder action of P SL(2, R) derives from the action of SL(2, R) on H by fractional linear transformations as follows.   az + b a b . z= c d cz + d   −1 0 Since acts trivially, this factors to an action of P SL(2, R) = 0 −1 SL(2, R)/{−1, 1}. It is easy to check that this acts as a group of isometries. • The Poincar´ e disc The Poincare disc D is defined as D := {x + iy ∈ C | x2 + y 2 < 1} with metric 4 (dx2 + dy 2 ). g= (1 − x2 − y 2 )2 Geodesics in this model are circles perpendicular to the unit circle and straight lines through the origin. This space has constant negative curvature −1 and is simply connected. It therefore is isometric to the hyperbolic plane. An isometry from D to H is for example the Moebius transformation 1+z . z → i 1−z • Hyperbolic cylinders Let  > 0. Then the hyperbolic cylinder can be defined as the quotient Z := Γ\H of H by the group Γ ⊂ SL(2, R) defined by   k/2    /2 0 0 e e >= | k ∈ Z . Γ =< 0 e− /2 0 e− k/2 A fundamental domain is depicted in the Figure 1. Using the angle ϕ = arctan(x/y) and t = 12 log(x2 + y 2 ) as coordinates the metric becomes g=

1 (dϕ2 + dt2 ). cos2 ϕ

We can also use Fermi coordinates (ρ, t), where t is as before and cosh ρ = 1 cos ϕ . The coordinate ρ is the oriented hyperbolic distance from the yaxis in H. On the quotient Z the y-axis projects to a closed geodesic of

190

A. STROHMAIER

length . This is the unique simple closed geodesic on Z . Using Fermi coordinates we can see that the hyperbolic cylinder Z is isometric to R × (R/Z) with metric dρ2 + cosh2 ρ dt2 . The Laplace operator in these coordinates −

(15)

1 ∂2 ∂ ∂ 1 cosh ρ − . 2 cosh ρ ∂ρ ∂ρ cosh ρ ∂t2

A large set of solutions of the eigenvalue equation (−Δ − λ)Φ = 0 can then be obtained by separation of variables. Namely, if we assume that t Φ(ρ, t) = Φk (ρ) exp(2πi )  for some k ∈ Z then the eigenvalue equation is equivalent to (16)

(−

1 d d 4π 2 k2 cosh ρ + 2 − λ)Φk (ρ) = 0 cosh ρ dρ dρ  cosh2 ρ

A fundamental system of (non-normalized) solutions of this equation, consisting of an even and an odd function, can be given explicitly for each k ∈ Z in terms of hypergeometric functions (17)

s πik 1 − s πik 1 + , + ; ; − sinh2 ρ), 2  2  2 2πik 1 + s πik 2 − s πik 3  + , + ; ; − sinh2 ρ), Φodd 2 F1 ( k (ρ) = sinh ρ(cosh ρ) 2  2  2 where λ = s(1 − s) (see [7], where these functions are analysed). Normalization gives the corresponding solutions to the initial value problems. • Hyperbolic pair of pants (ρ) = (cosh ρ) Φeven k

2πik 

2 F1 (

2

1 3

Figure 2. Y -piece with boundary geodesics For any given 1 , 2 , 3 > 0 one can construct a right angled geodesic hexagon in the hyperbolic plane such that the length of every second side is 1 /2, 2 /2 and 3 /2. Two such hexagons can then be glued along the other sides to form a hyperbolic surface with three geodesic boundary components of lengths 1 , 2 , 3 . A hyperbolic pair of pants can also be glued from a subset of a hyperbolic cylinder as depicted in the figure.

191

Im z

MPS AND GEOMETRY

b

a

d

h

f g

c

e Re z

Figure 3. Two hyperbolic hexagons together form an octagon which can be glued into a pair of pants

• General surfaces of genus g Let g > 2 be an integer. Suppose we are given 2g − 2 pairs of pants, and a three-valent graph together with a map that associates with each vertex a pair of pants, and with each edge associated with that vertex a boundary component of that pair of pants. So each edge of the graph will connect two vertices and will therefore correspond to two different boundary components of that pair of pants. Suppose that these boundary components have the same length. So each edge of the graph will have a length j associated to it. There are 3g − 3 such edges. We can then glue the hyperbolic pair of pants together along the boundary components using a gluing scheme that identifies each collar neighborhood of the boundary component with a subset of the corresponding hyperbolic cylinder. Such a gluing is unique up to a twist angle αj ∈ S 1 . Once such a twist angle is fixed we obtain a surface of genus g equipped with a hyperbolic metric. It is known that each oriented hyperbolic surface can be obtained in this way. The parameters j and αj then constitute the Fenchel-Nielsen parameters of that construction. For each given threevalent graph and 6g − 6 Fenchel-Nielsen parameters there is a hyperbolic surface constructed. Of course, it may happen that different FenchelNielsen parameters yield an isometric surface. It can be shown that there is a discrete group, the mapping class group, acting on the Teichm¨ uller space R6g−6 such that the quotient coincides with the set of hyperbolic metrics on a given two dimensional oriented surface.

192

A. STROHMAIER

1

3 2

Figure 4. Genus two hyperbolic surface glued from two pairs of pants 3 2 1

6 5

4

Figure 5. Genus three hyperbolic surface glued from four pair of pants 4. The Method of Particular Solutions for Hyperbolic Surfaces In the following, we will describe a very efficient way to implement the method of particular solutions for hyperbolic surfaces. Each surface can be decomposed into 2g − 2 pairs of pants. Each pair of pants can then be cut open along one geodesic connecting two boundary components to obtain a subset of a hyperbolic cylinder. Our surface M can therefore be glued from 2g−2 subsets Mj of hyperbolic cylinders. M = ∪j Mj . This gives a decomposition of M as discussed before and the hypersurface Σ will consist of geodesic segments. On each piece Mj we have a large set of functions satisfying the eigenvalue equation (−Δ − λ)Φ = 0 by restricting the functions constructed on the hyperbolic cylinder to Mj . If we let k vary between −N and +N we obtain a 2(2N + 1)-dimensional space of functions with a canonical basis. (λ) We can assemble these into a 2(2N + 1)(2g − 2)-dimensional subspace WN in ∞ L (M ). Basis elements in this subspace are indexed by j ∈ {1, . . . , 2g − 2}, by k ∈ {−N, −N + 1, . . . , N − 1, N } and by {e, o} where the last index distinguishes between even and odd solutions of the ODE. We will assemble all these indices into (λ) a larger index α. So we have a set of basis function Φα on *j Mj and we would like to apply the estimate MPS in order to find eigenvalues. A simple strategy is as follows. Discretize the geodesic segments of Σ into a finite set of Q points (xj )j=1,...,Q . In order to keep things simple let us avoid corners. So every point xj will be contained in the boundary of precisely two components, so there are exactly two points yj and y˜j in *j ∂Mj that correspond to this point. A simple strategy of MPS for these surfaces is therefore to form the matrices Aλ = (ajα ), A˜λ = (˜ ajα ),

(λ) ajα = Φα (yj ), (λ) a ˜jα = Φα (˜ yj ),

(λ) Bλ = (bjα ), bjα = ∂n Φα (yj ) (λ) ˜λ = (˜bjα ), ˜bjα = ∂n Φα (˜ B yj ).

MPS AND GEOMETRY

193

˜λ ) and also Rλ := Aλ ⊕ A˜λ ⊕ Bλ ⊕ B ˜λ . We assemble Qλ := (Aλ − A˜λ ) ⊕ (Bλ + B Then the smallest singular value Qλ v v=0 Rλ v

sλ = inf

of the pair (Qλ , Rλ ) is then a measure of how close we are to an eigenvalue. For a quantitative statement see [18] where this method is described and analysed in great detail. The idea behind this is easily explained as follows. Suppose that λ is an eigenvalue. Then there exists a corresponding eigenfunction φ. This eigenfunction can be restricted to each piece Mj and can then be expanded in our basis functions. Since the eigenfunction is analytic, the Fourier series with respect to the circle action on the hyperbolic cylinder converges exponentially fast. This means the eigenfunction is approximated exponentially well by the chosen basis functions Φα . Cutting off at a Fourier mode will produce an error in the C 1 -norm that is exponentially small as N becomes large. Since the actual eigenfunction satisfies Dφ = 0 and Dn φ = 0 its approximation by our basis functions φN will satisfy the same equation up to an exponentially small error. Therefore, if v is the coefficient vector of φN with respect to our basis Φα , the norm of Qλ v is very small. On the other hand, by Green’s formula, the boundary data of φ does not vanish on ∂Mj but merely gives a measure for its L2 -norm. So the norm of Rλ v will be comparable to the L2 -norm of φ. We conclude that sλ is exponentially small as N gets large if λ is an eigenvalue. Conversely, since Qλ v roughly approximates the L2 -norm of Dφ ⊕ Dn φ and Rλ v roughly approximates the L2 norm of φ, the quotient will not be small if λ is not a eigenvalue. Hence, if we plot sλ as a function of λ we will be able to find the eigenvalues. In a similar way, multiplicities can be found by looking at higher singular values. 4.1. The Bolza surface. In the following, we would like to illustrate this method and some results for the case of the Bolza surface. The Bolza surface is the unique oriented hyperbolic surface of genus 2 with maximal group of orientation preserving isometries of order 48. It can be described in several different ways. The easiest way uses the Poincare disk model. Define the regular geodesic 1 octagon with corner points 2− 4 exp( πik 4 ). In order to obtain the Bolza surface, opposite sides are identified by means of hyperbolic isometries using the identification scheme as in the figure. The group of orientation preserving isomtries is GL(2, Z3 ) which is a double cover of S4 . The full isometry group GL(2, Z3 )  Z2 has 13 isomorphism classes of irreducible representations: four one-dimensional, two two-dimensional, four threedimensional, and three four-dimensional ones. The representation theory of this group and its connection to boundary conditions on subdomains has been worked out in detail by Joe Cook in his thesis ([10]). It was claimed by Jenni in his PhD thesis that the first non-zero eigenspace is a three dimensional irreducible representation. The proof seems to rely on some numerical input as well. Jenni also gives the bound for the first non-zero eigenvalue 3.83 < λ1 < 3.85. The Bolza surface was also investigated by Aurich and Steiner in the context of quantum chaos (see for example [1, 2]), where it was referred to as the HadamardGutzwiller model. A finite element method was applied to the surface and the first non-zero eigenvalue was indeed found to be of multiplicity three and was given by

194

A. STROHMAIER

Figure 6. The Bolza surface obtained from a regular octagon in the hyperbolic plane λ1 = 3.838. Nowadays, it is not difficult to code the Bolza surface in the available finite element frameworks. It can be done rather quickly in the freely available FreeFEM++ ([12]). Its Fenchel-Nielsen m-w-coordinates can be worked out to be (1 , t1 ; 2 , t2 ; 3 , t3 ) = √ √ √ 1 = (2 arccosh(3 + 2 2), ; 2 arccosh(1 + 2), 0; 2 arccosh(1 + 2), 0). 2 Another more symmetric decomposition of the Bolza surface into pairs of pants is one with Fenchel Nielsen paramaters given by

1

(1 , t1 ; 2 , t2 ; 3 , t3 ) = (s , t; s , t; s , t), √ s = 2 arccosh(1 + 2),  = √ 2 2 arccosh 7 3+ √ . t= arccosh 1 + 2 Note that the Bolza surface is also extremal in the sense that it is the unique maximizer for the length of the systole. The method of particular solutions can now be applied to the Bolza surface as well. The general code for genus 2 surfaces was written by Ville Uski (see [18]). Based on our paper, with high precision, one finds a multiplicity three eigenvalue at λ1 = 3.8388872588421995185866224504354645970819150157. The programme as well as further computed eigenvalues can be found at http:// www-staff.lboro.ac.uk/~maas3/publications/eigdata/datafile.html. Numerical evidence suggests that this is the global maximum for constant negative curvature genus 2 surfaces. The reason for it being locally maximal is however its degeneracy. For an analytic one parameter family of perturbations in Teichm¨ uller 1 derived

by Lucy McCarthy in a project

MPS AND GEOMETRY

195

Figure 7. Smallest singular value as a function of λ

0.1

σ1 (λ) σ2 (λ) σ3 (λ)

0.09 0.08 0.07

y

0.06 0.05 0.04 0.03 0.02 0.01 0 3.835

3.836

3.837

3.838

3.839

3.84

3.841

3.842

λ

Figure 8. Smallest three singular value as a function of λ

space one can choose the eigenvalues λ1 , λ2 and λ3 to depend analytically on the perturbation parameter. Numerically one can see that no matter what perturbation one chooses, none of the eigenvalues λ1 , λ2 and λ3 has an extremal value at the Bolza surface. The Bolza surface is also the unique global maximum of the length of the systole. This was shown by Schmutz-Schaller in [17], where more properties of the Bolza surface are discussed.

196

A. STROHMAIER

The following is a list of the first 38 non-zero eigenvalues computed using the method of particular solutions in the implementation described in the paper by Uski and the author in [18]. λn 3.83888725884219951858662245043546 5.35360134118905041091804831103144 8.24955481520065812189010645068245 14.7262167877888320412893184421848 15.0489161332670487461815843402588 18.6588196272601938062962346613409 20.5198597341420020011497712606420 23.0785584813816351550752062995745 28.0796057376777290815622079450011 30.8330427379325496742439575604701 32.6736496160788080248358817081014 36.2383916821530902525410974752583 38.9618157624049544290078974084124

multiplicity 3 4 2 4 3 3 4 1 3 4 1 2 4

5. Heat Kernels, Spectral Asymptotics, and Zeta functions Let us start again with general statements. Let M be a n-dimensional closed Riemannian manifold and let −Δ be the Laplace operator acting on functions on M . Assume that M is connected. Then the zero eigenspace is one-dimensional and we can arrange the eigenvalues such that 0 = λ0 < λ1 ≤ λ2 ≤ . . . The fundamental solution kt (x, y) of the heat equation, i.e. the integral kernel of the operator etΔ is well known to be a smoothing operator for all t > 0. It is hence of trace class and, by Mercer’s theorem, we have  ∞  (18) e−tλj = kt (x, x)dx. tr(etΔ ) = M

j=0

For large t one obtains (19)

tr(etΔ ) − 1 = O(e−ct ),

for some c > 0. From the construction of a short time parametrix for the heat equation (see for example [9] ) one obtains that as t → 0+ : (20)

tr(etΔ ) = t− 2

n

N 

aj tj + O(tN −n/2+1 ),

j=0

for any natural number N . The coefficients aj are integrals of functions aj (x) that are locally computable from the metric, i.e.  (21) aj (x)dx. aj = M

MPS AND GEOMETRY

197

The first couple of terms are well known 1 , (4π)n/2 1 a1 (x) = r(x)/6, (4π)n/2 a0 (x) =

where r is the scalar curvature. In two dimensions, the scalar curvature is twice the 1 in the case of a hyperbolic surface, Gauss curvature so that we have a1 (x) = − 12π g−1 and by Gauss-Bonnet a1 = 3 . An application of Ikehara’s Tauberian theorem to the heat expansion yields Weyl’s law that the counting function N (λ) = #{λj ≤ λ} satisfies N (λ) ∼ Cn Vol(M )λn/2 , where Cn depends only on n. 5.1. Zeta functions. Because of Weyl’s asymptotic formula, the following zeta function is well defined and holomorphic in s for Re(s) > n2 : ζΔ (s) :=

∞ 

λ−s j .

j=1

This can easily be rewritten as ζΔ (s) =

1 Γ(s)



∞



ts−1 tr(etΔ ) − 1 dt.

0

We can now split this integral into two parts to obtain  1  ∞

Γ(s)ζΔ (s) = ts−1 tr(etΔ ) − 1 dt + ts−1 tr(etΔ ) − 1 dt = I1 (s) + I2 (s). 0

1

Note that I2 (s) is entire in s. The integral I1 (s) can be rewritten using the asymptotic expansion ⎛ ⎞  1  1 N N  1   n n I1 (s) = ts−1 ⎝tr(etΔ ) − t− 2 aj tj ⎠ dt + aj tj+s−1− 2 dt − ts−1 dt. 0

j=0

j=0

0

0

The last two terms together yield aj 1  − + s j=0 s + j − N

n 2

,

and the first integral is holomorphic for Re s > n2 − N . This can be done for any natural number N . Therefore, I1 (s) has a meromorphic extension to the entire complex plane with simple poles at n2 − j and at 0. Hence, we showed that ζ admits a meromorphic extension to the complex plane. Since Γ(s) has a pole at the non-positive integers this shows that ζ is regular at all the non-positive integers. In particular zero is not a pole of ζ. The above shows that ζΔ (0) = −1 if n is odd  (0) is therefore well defined and is and ζΔ (0) = −1 + a n2 if n is even. The value ζΔ used to define the zeta-regularized determinant detζ (−Δ) of −Δ as follows  ζΔ (0) = − log detζ (−Δ).

198

A. STROHMAIER

The motivation for this definition is the formula ⎛ ⎞ N N   d ⎠ |s=0 , log λj = ⎝− λ−s log det(A) = j ds j=1 j=1 for a non-singular Hermitian N × N -matrix with eigenvalues λ1 , . . . , λN . The computation of this spectral determinant is quite a challenge. The method of meromorphic continuation for the zeta function also is a method of computation for the spectral determinant. 6. The Selberg Trace Formula Suppose that M is an connected oriented hyperbolic surface. Then there is an intriguing formula connecting the spectrum of the Laplace operator to the length spectrum. Suppose that g ∈ C0∞ (R) is an even real valued test function. Then its Fourier transform h = gˆ is an entire function defined on the entire complex plane. It is also in the Schwartz space S(R) and real valued on the real axis. As usual, we use the notation λj = rj2 + 14 , where for eigenvalues smaller than 14 we choose rj to have positive imaginary part. Hence, by Weyl’s law, the sum ;> <   1 h λj − h(rj ) = 4 j λj

converges and depends continuously on g. It therefore defines an even distribution ;> < 1 Tr cos t Δ − 4 in D (R). Selberg’s trace formula reads ∞ 

h(rn ) =

n=0

Vol(M ) 4π



∞

rh(r) tanh(πr)dr + −∞

∞   k=1 γ

(γ) g(k(γ)), 2 sinh(k(γ)/2)

where the second sum in the second term is over the set of primitive closed geodesics γ, whose length is denoted by (γ). We would like to refer to Iwaniec’s monograph [14] for an introduction and a derivation. In the sense of distributions this reads as follows. < ;> Vol(M ) cosh(t/2) 1 =− Tr cos t Δ − 4 8π sinh2 (t/2) +

∞   k=1 γ

(γ) (δ(|t| − k(γ))). 4 sinh(k(γ)/2)

Note that this is not a tempered distribution. Therefore, we may not pair either side with a general Schwartz functions. One can however still apply it to the function 2 h(x) = e−tx and obtain Vol(M )e− 4 )= 4πt t

tr(e

Δt

 0

∞

2

2

n (γ) 2 ∞  −t/4  (γ)e− 4t πe−r t e √ . dr + cosh2 (πr) 4πt 2 sinh n (γ) n=1 γ 2

MPS AND GEOMETRY

199

2 0

Note that the second term is of order O(e− 4t ) as t → 0+ , where 0 is the length of the shortest closed geodesic (the systole length). The first term can therefore be thought of as a much more refined version of the heat asymptotics. Exercise 6.1. Derive the heat asymptotics from the first term in Selberg’s trace formula by asymptotic analysis. Derive the first three heat coefficients.  ∞

1 ts−1 tr(eΔt ) − 1 dt Γ(s) 0 can now directly be used with the Selberg trace formula. In order to perform the analytic continuation, one can again split the integral into integrals over (0, 1] and over (1, ∞). For numerical purposes it is however convenient to instead split into (0, ] and ( , ∞) for a suitably chosen > 0. This means    ∞

1 1 ts−1 tr(eΔt ) − 1 dt + ts−1 tr(eΔt ) − 1 dt. ζΔ (s) = Γ(s) 0 Γ(s)  We now compute the first term from the Selberg trace formula and the second term from the spectrum. Using the same unique continuation process as described earlier, one obtains the following representation of the spectral zeta function for Re(s) > −N : 1 (T  (s) + T2,N (s) + T3,N (s) + T4,N (s)), ζΔ (s) = Γ(s) 1 where ∞  T1 (s) = λ−s i Γ(s, λi ), The formula

ζΔ (s) =

i=1 N  ak s+k−1 , s+k−1 k=0  Vol(M ) ∞  T3,N (s) = IN (r)dr, 4π 0 n2 (γ)2 ∞   −t/4  (γ)e− 4t ,N s−1 e √ t dt. T4 (s) = 4πt 2 sinh n (γ) 0 n=1 γ 2

T2,N (s) =

Here

  (r) IN

;



=

t

s−2

e

−(r 2 + 14 )t

0

−

N  (−1)k k=0

1 (r + )k tk k! 4 2

< dt,

and the coefficients ak are the heat coefficients of the expansion of tr(eΔt )−1, which are given by  Vol(M ) ∞ (−1)k π(r 2 + 1/4)k ak = dr − δ1,k . 4π k! cosh2 (πr) 0 As usual Γ(x, y) denotes the incomplete Gamma function  ∞ Γ(x, y) = tx−1 e−t dt. y

Differentiation gives the following formula for the spectral determinant.  − log detζ Δ = ζΔ (0) = L1 + L2 + L3

200

A. STROHMAIER

where L1 =

∞ 

Γ(0, λi ),

i=1

  Vol(M ) Vol(M ) Vol(M ) − + 1 (γ + log( )) + × =− 4π 12π 4 ; <

 ∞

1 − E2 (r 2 + 14 ) 1 2 + (r 2 + ) γ − 1 + log( (r 2 + 1/4)) dr, sech (πr) 4 0 L2

L3

=

∞  

n2 (γ)2



e

−t/4

0

n=1 γ

i e− 4t

dt, √ 3/2 4 πt sinh 12 n(γ)

and E2 (x) is the generalized exponential integral which equals x Γ(−1, x). All the integrals have analytic integrands and can be truncated with exponentially small error. They can therefore be evaluated to high accuracy using numerical integration. For fixed s and > 0 not too small the sums over the eigenvalues converge very quickly and therefore T1 (s) and L1 can be computed accurately from the first eigenvalues only. If is small compared to 20 the terms T4,N (s) and L3 are very small. The terms L1 and T1 (s) involve the spectrum but the sums converge rapidly, so that only a finite proportion of the spectrum is needed to numerically approximate these values. A detailed error analysis of these terms is carried out in [15]. In order to illustrate the idea behind this method, let us look at the function 2 t  N  Vol(M )e− 4 ∞ πe−r t e−λj t − RN (t) = dr. 4πt cosh2 (πr) 0 j=0 By Selberg’s trace formula we have RN (t) = −

2

∞ 

−λj t

e

j=N +1

2

n (γ) ∞  −t/4  e (γ)e− 4t √ + . 4πt 2 sinh n (γ) n=1 γ 2

The first term is negative and dominant when t is small. The second term is positive and dominates when t is large. Figure 9 shows this function for the Bolza surface. Here the first 500 eigenvalues were computed numerically using the method outlined in the previous paragraphs. The integral in the zero term of the Selberg trace formula is computed numerically. One can now clearly see the regions in which each term dominates. There is a clearly visible region between t = 0.05 and t = 0.2 where the function is very small. In fact its value at t = 0.1 is of order smaller than 10−9 . In order to compute the spectral zeta function one can therefore choose = 0.1 and estimate the errors of the contributions of T4 and L3 , as well as the error from cutting off the spectrum and considering only the first 500 eigenvalues. One obtains for example for the Bolza surface detζ (Δ) ≈ 4.72273, ζΔ (−1/2) ≈ −0.650006. To compute the first 500 eigenvalues of the Bolza surface to a precision of 12 digits, about 10000 λ-evaluations of generalized singular value decomposition were needed.

MPS AND GEOMETRY

201

0.4

0.3

0.2

0.1

0.2

0.4

0.6

0.8

1.0

-0.1

-0.2

Figure 9. The function RN for the Bolza surface with N = 500

ζ(s) 1

-3

-2

s

-1

-1

-2

-3

-4

Figure 10. ζΔ (s) as a function of s for the Bolza surface

This took about 10 minutes on a 2.5 GHz Intel Core i5 quad core processor (where parallelization was used). Numerical evidence suggests that the spectral determinant is maximized in genus 2 for the Bolza surface. One can see quite clearly from perturbing in Teichm¨ uller space that the Bolza surface is indeed a local maximum for the spectral determinant. Note that the Bolza surface is known to be a critical point by symmetry considerations.

202

A. STROHMAIER

7. Completeness of a Set of Eigenvalues The method of particular solution on oriented hyperbolic surfaces is able to produce quite quickly a list of eigenvalues. Once such a list is computed and error bounds are established, one would like to check that this list is complete and one has not missed an eigenvalue, for example because the step-size in the search algorithm was chosen too small, or an eigenvalue had a higher multiplicity. In [18] it was proved that the step size can always be chosen small enough so that no eigenvalues are missed. Choosing the step-size according to these bounds does however slow down the speed of computation significantly. In this section we discuss two methods by which completeness of a set of eigenvalues can be checked. 7.1. Using the heat kernel and Selberg’s trace formula. Suppose that {μ0 , . . . , μN } is a list of computed eigenvalues. We would like to use this list and check that there are no additional eigenvalues in an interval [0, λ], where λ is possibly smaller than μN . As before consider the function 2 t  N  Vol(M )e− 4 ∞ πe−r t −λj t e − RN (t) = dr, 4πt cosh2 (πr) 0 j=0 and recall that RN (t) = −

∞ 

2

−λj t

e

j=N +1

2

n (γ) ∞  −t/4  e (γ)e− 4t √ + . 4πt 2 sinh n (γ) n=1 γ 2

 20 + 1 − 1 the second term is bounded by > −l2 T l2 T FT (t) = tr(e−ΔT )e 4 + 4T e 4t , t In [15] Fourier Tauberian theorems were used to establish the bound  3   2 2 4ν + 2ν 2 π 1 2ν 2 + νπ √ Vol(M ) 1 T4 + 4T 0 − 0 4t √ e √ + √ FT (t) ≤ + T , 4π π20 π0 t T For t < T
FT (t). For the Bolza surface we have 0 ≈ 3.05714 and we can choose for instance T = 2. Using the list of the first 200 eigenvalues one can see from Fig. 11 that choosing ˜N R t near 0.1 maximizes the function − log FT (t)− . For t = 0.095 one gets that there t

MPS AND GEOMETRY

203

200

150

100

50

0.2

0.1

0.3

0.4

˜

RN Figure 11. − log FT (t)− as a function of t for the Bolza surface, t N = 200

are no additional eigenvalues smaller than 172. Note that λ200 ≈ 200.787. So we had to compute roughly 30 more eigenvalues to make sure our list is complete. This method in principle can be made rigorous by using interval arithmetics. Its disadvantage is that for larger lists it requires the low lying eigenvalues to be known with very high accuracy. 7.2. Using the Riesz mean of the counting function. It is sometimes convenient to reparametrize in terms of square roots of eigenvalues. Let us define the local counting function  ˜ (t) = N (t2 ) = #{λj ≤ t2 } = #{ λj ≤ t}. N For a general negatively curved two dimensional compact Riemannian manifold one has (see [5]) ˜ (t) ∼ Vol(M ) t2 + O( t ), N 4π log(t) as t → ∞. Because of the growing error term this is unsuitable to detect missed eigenvalues from the spectrum. However, the so-called Riesz means of the counting functions are known to have improved asymptotic expansions. In our case define the first Riesz mean as  1 t ˜ ˜ N (r)dr. (R1 N )(t) := t 0 Then for two dimensional compact surfaces of negative curvature one has  1 ˜ )(t) = Vol(M ) t2 + 1 (R1 N ), κ(x)dx + O( 12π 12π log(t)2 where κ(x) is the scalar curvature at the point x ∈ M . This can be inferred in the case of constant curvature hyperbolic surfaces from Selberg’s trace formula (see

204

A. STROHMAIER

[13]), but also can be shown to hold true in the case of negative variable curvature ([16]). In the case of hyperbolic surfaces one obtains

˜ )(t) = Vol(M ) t2 − 1 + O( 1 ). (R1 N 12π log(t)2 The strategy is to compute the Riesz means from a set of computed eigenvalues. That is, if {μ0 , . . . , μN } is a set of eigenvalues we compute the function ˜test (t) := #{√μj ≤ t} N and plot

˜test )(t) − Vol(M ) t2 − 1 . Ftest(t) := (R1 N 12π This is done in Fig. 12 for the Bolza surface. The red line was computed with an eigenvalue missing. One can clearly see this in the plot, and this also allows one to say roughly where the missing eigenvalue was. If an eigenvalue is missing somewhere this will result in the function not going to zero. In this way one can even detect roughly where the missed eigenvalue is located and how many eigenvalues may be missing.

1.0

0.5

5

10

15

20

-0.5

Figure 12. Ftest(t) as a function of t for the Bolza surface, the red line is the function with λ89 ≈ (9.563)2 missing

Acknowledgements The author would like to thank the organizers of the summer school for the perfect organization and the hospitality. The author is also grateful to Joseph Cook for carefully reading these notes and for providing some numerical work on the Bolza surface as well as diagrams.

MPS AND GEOMETRY

205

References [1] R. Aurich and F. Steiner, Periodic-orbit sum rules for the Hadamard-Gutzwiller model, Phys. D 39 (1989), no. 2-3, 169–193, DOI 10.1016/0167-2789(89)90003-1. MR1028714 [2] R. Aurich and F. Steiner, Energy-level statistics of the Hadamard-Gutzwiller ensemble, Phys. D 43 (1990), no. 2-3, 155–180, DOI 10.1016/0167-2789(90)90131-8. MR1067908 [3] A.H. Barnett and T. Betcke. MPSpack: A MATLAB toolbox to solve Helmholtz PDE, wave scattering, and eigenvalue problems, 2008–2012. [4] A. H. Barnett and A. Hassell, Boundary quasi-orthogonality and sharp inclusion bounds for large Dirichlet eigenvalues, SIAM J. Numer. Anal. 49 (2011), no. 3, 1046–1063, DOI 10.1137/100796637. MR2812557 [5] Pierre H. B´ erard, On the wave equation on a compact Riemannian manifold without conjugate points, Math. Z. 155 (1977), no. 3, 249–276, DOI 10.1007/BF02028444. MR0455055 [6] Timo Betcke and Lloyd N. Trefethen, Reviving the method of particular solutions, SIAM Rev. 47 (2005), no. 3, 469–491, DOI 10.1137/S0036144503437336. MR2178637 [7] David Borthwick, Sharp upper bounds on resonances for perturbations of hyperbolic space, Asymptot. Anal. 69 (2010), no. 1-2, 45–85. MR2732192 [8] Peter Buser, Geometry and spectra of compact Riemann surfaces, Modern Birkh¨ auser Classics, Birkh¨ auser Boston, Inc., Boston, MA, 2010. Reprint of the 1992 edition. MR2742784 [9] Isaac Chavel, Eigenvalues in Riemannian geometry, Pure and Applied Mathematics, vol. 115, Academic Press, Inc., Orlando, FL, 1984. Including a chapter by Burton Randol; With an appendix by Jozef Dodziuk. MR768584 [10] J. Cook PhD-thesis, in preparation. [11] L. Fox, P. Henrici, and C. Moler, Approximations and bounds for eigenvalues of elliptic operators, SIAM J. Numer. Anal. 4 (1967), 89–102, DOI 10.1137/0704008. MR0215542 [12] F. Hecht, New development in freefem++, J. Numer. Math. 20 (2012), no. 3-4, 251–265. MR3043640 [13] Dennis A. Hejhal, The Selberg trace formula for PSL(2, R). Vol. I, Lecture Notes in Mathematics, Vol. 548, Springer-Verlag, Berlin-New York, 1976. MR0439755 [14] Henryk Iwaniec, Spectral methods of automorphic forms, 2nd ed., Graduate Studies in Mathematics, vol. 53, American Mathematical Society, Providence, RI; Revista Matem´ atica Iberoamericana, Madrid, 2002. MR1942691 [15] Kamil Mroz and Alexander Strohmaier, Explicit bounds on eigenfunctions and spectral functions on manifolds hyperbolic near a point, J. Lond. Math. Soc. (2) 89 (2014), no. 3, 917–940, DOI 10.1112/jlms/jdu010. MR3217656 [16] Kamil Mroz and Alexander Strohmaier, Riesz means of the counting function of the Laplace operator on compact manifolds of non-positive curvature, J. Spectr. Theory 6 (2016), no. 3, 629–642, DOI 10.4171/JST/134. MR3551179 [17] P. Schmutz, Riemann surfaces with shortest geodesic of maximal length, Geom. Funct. Anal. 3 (1993), no. 6, 564–631, DOI 10.1007/BF01896258. MR1250756 [18] Alexander Strohmaier and Ville Uski, An algorithm for the computation of eigenvalues, spectral zeta functions and zeta-determinants on hyperbolic surfaces, Comm. Math. Phys. 317 (2013), no. 3, 827–869, DOI 10.1007/s00220-012-1557-1. MR3009726 School of Mathematics, University of Leeds, Leeds, LS2 9JT, United Kingdom E-mail address: [email protected]

Contemporary Mathematics Volume 700, 2017 http://dx.doi.org/10.1090/conm/700/14188

Scales, blow-up and quasimode constructions Daniel Grieser Abstract. In this expository article we show how the concepts of manifolds with corners, blow-ups and resolutions can be used effectively for the construction of quasimodes, i.e. of approximate eigenfunctions of the Laplacian on certain families of spaces, mostly exemplified by domains Ωh ⊂ R2 , that degenerate as h → 0. These include standard adiabatic limit families and also families that exhibit several types of scaling behavior. An introduction to manifolds with corners and resolutions, and how they relate to the ideas of (multiple) scales and matching, is included.

Contents 1. Introduction 2. A short introduction to manifolds with corners and resolutions 3. Generalities on quasimode constructions; the main steps 4. Regular perturbations 5. Adiabatic limit with constant fibre eigenvalue 6. Adiabatic limit with variable fibre eigenvalue 7. Adiabatic limit with ends 8. Summary of the quasimodes constructions References

1. Introduction This article gives an introduction to the ideas of blow-up and resolution, and how they can be used for the construction of quasimodes for the Laplacian in singular perturbation problems. Blow-up is a rigorous geometric tool for describing multiple scales, which appear in many analytic problems in pure and applied mathematics. The construction of quasimodes is a low-tech yet non-trivial problem where this tool can be used effectively. The idea of scales. One of the fundamental ideas in analysis is scale. As an illustration consider the function x , x ∈ [0, 1] (1.1) fh (x) = x+h where h is a ‘small’ positive number, see Figure 1 for h = 0.1 and h = 0.01. Observe that at x = 0 the function takes the value 0 while for ‘most’ values of x it is ‘close’ 1991 Mathematics Subject Classification. Primary 35-02; Secondary 35B25, 35P05, 58J37. c 2017 American Mathematical Society

207

208

DANIEL GRIESER

1

1

fh

x 1

resc

h→

0

h→

0

aled

f0

x 1

x=

g

1 hX

X Figure 1. Graph of fh for h = 0.1 and h = 0.01, and limits at two scales to 1. On the other hand, taking x = h we get fh (h) = 12 , and more generally if x is ‘on the order of h’ then fh (x) will be somewhere ‘definitely between 0 and 1’. This may be the way a physicist describes the function fh , even without the quotation marks; to a mathematician the quotes create a sense of uneasiness, so we search for a precise statement. We then realize that we are really talking about the family of functions (fh )h>0 and its limiting behavior as h → 0. More precisely, we first have the pointwise limit ' 0 if x = 0 (1.2) lim fh (x) = f0 (x) := h→0 1 if x > 0. On the other hand, we have the rescaled limit where we set x = hX and fix X while letting h → 0: (1.3)

lim fh (hX) = g(X) :=

h→0

X , X +1

X ≥ 0.

The function g shows how the transition from the value 0 to almost 1 happens in fh . We call this the limit of fh at the scale x ∼ h, while (1.2) is the limit at the scale x ∼ 1. We could also consider other scales, i.e. limits limh→0 fh (ha X) with a ∈ R, but they don’t give new insights in this case: if a < 1 then we get the jump function f0 while for a > 1 we just get zero. Summarizing, we see that the family (fh ) has non-trivial behavior at two scales, x ∼ 1 and x ∼ h, for h → 0. Geometric resolution analysis and matched asymptotic expansions. This rough first explanation of scales will be made more precise in Section 2. But let us now turn to real problems: Consider a differential equation whose coefficients depend on a parameter h, and have non-trivial behavior at several scales as h → 0. We then ask how the solutions behave as h → 0. Of course we expect them to exhibit

SCALES, BLOW-UP AND QUASIMODE CONSTRUCTIONS

209

several scales also.1 The same phenomenon arises in so-called singular perturbation problems, where the type of the equation changes at h = 0.2 Similarly, we could think of a partial differential equation on a domain which depends on h and has parts that scale in different ways, or which degenerates to a lower-dimensional domain as h → 0, or both. For examples see Figures 13 and 15. Such problems arise frequently in global and geometric analysis as well as in applied analysis (sometimes under the name of boundary layer problems). We will call them singular problems. There is a standard method to attack such problems, called matched asymptotic expansions (MAE) and commonly used in applied analysis since the mid-1900s (see e.g. [31]): Roughly speaking, for each scale appearing in the problem you make an ansatz for the Taylor expansion (in h) of the solution at this scale and plug it into the equation. This yields recursive sets of equations for the Taylor coefficients. The fact that the solutions at different scales must ‘fit together’ yields boundary conditions that make these equations well-posed (and often explicitly solvable). Of more recent origin is a different but closely related method, which has been used frequently in global and geometric analysis and which we call geometric resolution analysis3 (GRA): the starting point is a shift in perspective, which in the example above is to consider f : (x, h) → fh (x) as a function of two variables rather than as a family of functions of one variable. Then f has singular behavior at (x, h) = (0, 0), and the scaling considerations above can be restated as saying that this singularity can be resolved by blowing up the point (0, 0) in (x, h)-space, as will be explained in Section 2. In order to analyze the solutions of a singular differential equation we first resolve its singularities by suitably blowing up (x, h)-space; then the asymptotic behavior of solutions is obtained by solving model problems at the h = 0 boundary faces of the blown-up space. The model problems are simpler than the original problem and correspond to the recursive sets of equations of MAE. Eigenfunctions and quasimodes. The purpose of this article is to introduce the concepts needed for geometric resolution analysis and apply them to problems in spectral theory. The needed concepts are manifolds with corners, blow-up and resolution. The spectral problem is to analyze solutions λ ∈ R, u : Ω → R of the equation −Δu = λu where Δ is the Laplacian on a bounded domain Ω ⊂ R2 , and the Dirichlet boundary condition u = 0 at ∂Ω is imposed. This problem has natural generalizations to higher dimensions, manifolds and other boundary conditions, some of which will occasionally also be considered. The eigenvalues form a sequence 0 < λ1 ≤ λ2 ≤ · · · → ∞ and can usually not be calculated explicitly. But if we look at families of domains Ωh which degenerate to a line segment as h → 0 then we have a chance to analyze the asymptotic behavior of λk (h) (and associated eigenfunctions) as h → 0. Here we fix k while letting h → 0. Other regimes are also interesting, e.g. k going 1 Although it is not essential for this article, as a warm-up exercise you may analyze the behavior as h → 0 of the solution of the differential equation u + fh (x)u = 0, u(0) = 1, or (more difficult) of u + fh (x)u = 0, u(0) = 0, u (0) = 1. 2 As an example, consider the equation hu + u = 0, u(0) = 1. At h = 0 this is not even a differential equation! For h > 0 it has the solution uh (x) = e−x/h , which exhibits scaling behavior as h → 0 similar to fh . 3 As far as I know, no name has been coined for the method in the literature. This name must not be confused with the so-called geometric multi-resolution analysis, a method for the analysis of high dimensional data.

210

DANIEL GRIESER

to ∞ like h−1 , but we don’t consider them here. One expects that the leading term in the asymptotics can be calculated by solving a one-dimensional (ODE) problem. This is indeed the case also for higher order terms, but the details of how this works depend crucially on how Ωh degenerates (the ‘shape’ of Ωh ). We will analyze several interesting cases of such degenerations. A standard approach to analyzing such eigenvalue problems is to first construct so-called quasimodes, i.e. pairs (λ, u) which solve the eigenvalue equation up to a small, i.e. O(hN ), error, and then to show that the quasimodes are close to actual solutions. The construction yields the full asymptotics (i.e. up to errors O(hN ) for any N ) of quasimodes, and then of actual eigenvalues and eigenfunctions as h → 0. It is in the construction of quasimodes where GRA (or MAE) is used, and we will focus on this step in this article. The second step is quite straight-forward if the operator is scalar and the limit problem is one-dimensional, as is the case for all problems considered here. See Remark 5.5, [12], [20], [53] and point (V) below. For higher dimensional limit problems quasimodes need not be close to modes, see [4]. Why GRA? The methods of geometric resolution analysis and matched asymptotic expansions are closely related: they are really different ways to encode the same calculational base. GRA requires you to learn and get used to some new concepts, like manifolds with corners and blow-up, while MAE is very ‘down-to-earth’. Here are some points why it may be worth to invest the effort to learn about GRA. I hope they will become clear while you read this article. (I) GRA provides a rigorous framework for the powerful idea of MAE. For example, the ‘expansions at different scales’ of a putative solution u(x, h) are simply Taylor expansions at different faces of u when considered on (i.e. pulled back to) the blown-up space. (II) GRA provides conceptual clarity. In GRA the ‘singular’ aspects of a problem are dealt with in the geometric operation of blow-up. Then the analysis (solution of differential equations) is reduced to non-singular model problems, and to a version of the standard Borel lemma. In this way essential structures of a problem are clearly visible, while notationally messy (but essentially trivial) calculations involving multiple Taylor series run invisibly in the background. This also helps to identify common features of seemingly different problems. (III) GRA helps to stay sane in complex settings. Often more than two model problems appear, and remembering how they fit together (the ‘matching conditions’ of MAE) may be a torturous task. In GRA each model problem corresponds to a boundary face of the resolved space, and their relations can be read off from how these faces intersect. (IV) GRA may guide the intuition. The true art in solving singular problems is to identify the scales that can be expected to appear in the solutions. The geometric way of thinking about singularities often helps to ‘see’ how to proceed, see Section 7 for a nice example. An added complication is that sometimes solutions exhibit more scales than the data (i.e. the coefficients or the domains), as the setting in Section 6 shows. It is desirable to have systematic methods to find these. These are beyond the scope of this article however, and we refer to [51].

SCALES, BLOW-UP AND QUASIMODE CONSTRUCTIONS

211

(V) GRA can be refined to provide a systematic way to extend modern PDE methods like the pseudodifferential calculus to singular problems, and embeds them in a larger mathematical framework, see [50], [51], [14]. In this way one may also carry out the second step mentioned above, proving that quasimodes are close to actual solutions, in the framework of GRA, by analyzing the resolvent on a blown-up double space, see e.g. [43]. In this article we use simple examples to explain structures which also arise in more elaborate contexts. We consider planar domains and the scalar Laplacian, but the methods generalize without much extra work to manifolds and systems of elliptic PDEs (for example the Hodge Laplacian on differential forms). This is indicated at the end of each section. The methods can also be extended to study many other types of singular degenerations (with more work!), for example families of triangles degenerating to a line (ongoing work with R. Melrose, see also [7], [53]), domains from which a small ball is removed etc. The results presented here are not new, and in some cases more precise or more general results have been obtained by other methods, as is indicated in the subsections on generalizations. In the PDE literature blow-up methods have mostly been used in the context of microlocal analysis. Our purpose here is to illustrate their use on a more elementary level, and to introduce a systematic setup for applying them to quasimode constructions. A minor novelty seems to be the use of the quasimode and remainder spaces E(M ), R(M ) and their associated leading part maps, see Section 3 and Definitions 5.1, 7.1 and 7.3, although it is reminiscent of and motivated by the rescaled bundles used for example in [43]. Outline of the paper. In Section 2 we introduce the main objects of geometric resolution analysis (manifolds with corners, blow-up and resolution) and explain how they relate to the idea of scales. If you are mostly interested in quasimode constructions it will suffice to skim this section and only use it for reference; however, for Section 7 more of this material will be needed. In the remaining sections we show how quasimodes can be constructed using geometric resolution analysis. The examples are ordered to have increasing complexity, so that later examples use ideas introduced in previous examples plus additional ones. For easier reading the main steps of the constructions are outlined in Section 3. To set the stage, we first consider regular perturbation problems in Section 4. All further problems are eigenvalue problems on families of domains Ωh which degenerate to a line segment as h → 0. Such problems are sometimes called ‘adiabatic limit problems’. The simplest setting for these, where the cross section has constant lowest eigenvalue, is considered in Section 5. The treatment is general enough to apply to fibre bundles with Riemannian submersion metrics. Variable eigenvalues of the cross section, which occur for example when Ωh is an ellipse with half axes 1 and h, will introduce new scales, and this is analyzed in Section 6. Then in Section 7 we consider a problem where Ωh scales differently in some parts than in others. Here it will be especially apparent how the geometric way of thinking guides us to the solution. The quasimode results are formulated in Theorems 4.3, 5.4, 6.1, 7.6. In Section 8 we summarize the main points of the various quasimode constructions. Related literature. The book [51] (unfinished, available online) introduces and discusses in great generality and detail manifolds with corners and blow-ups and their use in analysis. The big picture is outlined in [48]. The focus in the

212

DANIEL GRIESER

present article is on problems depending on a parameter h, where singularities only appear as h → 0 (so-called singular perturbation problems). Closely related are problems which do not depend on a parameter but where the underlying space (or operator) is singular, and the methods of geometric resolution analysis can be and have been applied extensively in this context. A basic introduction to this is given by the author in [14], with applications to microlocal analysis, including many references to the literature. Other frameworks for manifolds with corners have been proposed, see for example [33] and references there. Blow-up methods have also been used in the context of dynamical systems, e.g. in celestial mechanics [46], for analyzing geodesics on singular spaces [13] or in multiple time scale analysis, see for example [8], [38], [57] and the book [37], which gives an excellent overview and many more references. The survey [17] discusses various types of ‘thin tube’ problems including the ones discussed here; their origin as well as various methods and results are explained. The books [41], [42] discuss many singular perturbation problems of geometric origin and their solution by a method called ‘compound asymptotic expansions’ there, which is similar to matched asymptotic expansions. More references are given at the end of each section. Acknowledgements. These notes are based on a series of lectures that I gave at the summer school ‘Geometric and Computational Spectral Theory’ at the Centre de Recherches Math´ematiques in Montreal. I am grateful to the organizers of the school for inviting me to speak and for suggesting to write lecture notes. I thank Leonard Tomczak for help with the pictures and D. Joyce, I. Shestakov, M. Dafinger and the anonymous referee for useful comments on previous versions of these notes. My biggest thanks go to Richard Melrose for introducing many of the concepts discussed here, and for many inspiring discussions. 2. A short introduction to manifolds with corners and resolutions In this section the basic concepts of geometric resolution analysis are introduced: manifolds with corners, polyhomogeneous functions, blow-up, resolutions. We emphasize ideas and introduce concepts mostly by example or picture (after all, we are talking geometry here!), hoping that the interested reader will be able to supply precise definitions and proofs herself, if desired. Many details can be found in [50] and [51]. To see where we’re heading consider the example from the introduction: x , x, h ≥ 0, (x, h) = (0, 0) . f (x, h) = x+h Recall its h → 0 limits at two scales: ' 0 (x = 0) X , , g(X) = lim f (hX, h) = f0 (x) = lim f (x, h) = h→0 h→0 X +1 1 (x > 0) see Figure 1. The ‘geometry’ (of geometric resolution analysis) resides in the spaces on which these functions are defined, i.e. their domains: dom f = R2+ , dom f0 = R+ , dom g = [0, ∞] where R+ := [0, ∞) .

SCALES, BLOW-UP AND QUASIMODE CONSTRUCTIONS

dom f

213

[dom f, (0, 0)]

h R1 R2

β

←− x

dom g

dom f0

Figure 2. Domains of f and of its rescaled limits f0 and g, and how they relate to each other. Dotted arrows mean ‘identify with’.

See Figure 2. Actually, f is not defined at (0, 0), but we ignore this for the moment. For g we have added ∞ to its domain, where we set g(∞) = limX→∞ g(X) = 1. We’ll see in a moment why this makes sense. These spaces are simple examples of manifolds with corners. We ask: Can we understand f0 and g as restrictions of f to suitable subsets of its domain? For f0 this is easy: if we identify dom f0 with the lower edge of dom f then f0 is simply the restriction of f . (Again we should exclude the point (0, 0).) How about understanding g as a restriction of f ? Can we reasonably identify dom g as a subset of dom f ? This is less obvious. Note that, for any X ≥ 0, h → f (hX, h) is the restriction of f to the ray RX := {(x, h) ∈ dom f : x = hX, h > 0} . By definition, g(X) is the limit of this restriction as h → 0, so it should be the value of f at the endpoint of RX . This remains true for X = ∞ if we set R∞ = {(x, 0) : x > 0}. Now this endpoint is (0, 0), so we have two problems: First, f is not defined there, and second, the endpoints of all rays RX (with different X) coincide. That’s why we don’t find dom g in dom f . But there is a way out, and this is the idea of blow-up: we simply add a separate endpoint for each ray RX to the picture. That is, we remove (0, 0) from dom f and replace it by a quarter circle as in Figure 2. This produces a new space, denoted [dom f, (0, 0)] and called the blow-up of (0, 0) in dom f . A precise definition is given in Section 2.3. It involves polar coordinates, and the quarter circle corresponds to r = 0. We denote the quarter circle by ff (‘front face’). Each point of the blown-up space corresponds to a point of dom f , as is indicated in Figure 2 by the dashed rays. We encode this by a map β : [dom f, (0, 0)] → dom f which maps ff to (0, 0) and is bijective between the complements of these sets. Under this correspondence, f translates into the function β ∗ f := f ◦ β on [dom f, (0, 0)].

214

DANIEL GRIESER

Essentially, we will see that β ∗ f is ‘f written in polar coordinates’. This simple construction solves all our problems: • β ∗ f is defined on all of [dom f, (0, 0)], including its full boundary. It is actually smooth, once we define what smoothness means on [dom f, (0, 0)]. • If we identify ff with [0, ∞] (the endpoint of the ray RX being identified with X ∈ [0, ∞]) then g is the restriction of β ∗ f to ff. • The pointwise limit f0 (x) = limh→0 f (x, h) is, for x > 0, still the restriction of f to the lower part of the boundary of [dom f, (0, 0)]. In addition, as we will see later, β ∗ f also encodes how f0 and g relate to each other (so-called ‘matching’). Summarizing, the multiple scales behavior of f is completely encoded by the behavior of β ∗ f near the boundary of [dom f, (0, 0)], and different scales correspond to different segments (later called boundary hypersurfaces) of the boundary. 2.1. Manifolds with corners. Even if we wanted to study problems on domains in Rn only, the natural setting for our theory is that of manifolds, for (at least) two reasons: (1) Just as finite dimensional vector spaces are like Rn without choice of a basis, manifolds are locally like Rn without choice of a (possibly non-linear) coordinate system – and foregoing such a choice leads to greater conceptual clarity. To put it more mundanely, it will be useful to use different coordinate systems (e.g. polar coordinates, projective coordinates), and it is reassuring to know that all constructions are independent of such choices. (2) Globally, a manifold represents how various local objects fit together – and one of our goals is to fit different scales together. In fact, even if the problem to be studied is topologically trivial, there may be non-trivial topology (or combinatorics) in the way that different scales relate to each other. To get an idea what a manifold with corners is, look at Figure 3. The most complicated specimen appearing in this text is on the right in Figure 16. Recall that a manifold is a space which can locally be parametrized by coordinates. For a manifold with corners some coordinates will be restricted to take only non-negative values. As before we use the notation R+ := [0, ∞) and write Rk+ = (R+ )k . Definition 2.1. A manifold with corners (mwc) of dimension n is a space M which can locally be parametrized by open subsets of the model spaces Rk+ × Rn−k , for various k ∈ {0, . . . , n}. In addition, we require that the boundary hypersurfaces be embedded, as explained below. The model space condition is meant as in the standard definition of manifolds, for which only k = 0 is allowed. So for each point p ∈ M there is k ∈ {0, . . . , n} and ˜ , with U ˜ ⊂ Rk+ × Rn−k open, a neighborhood U of p with a coordinate map U → U

SCALES, BLOW-UP AND QUASIMODE CONSTRUCTIONS

215

1 1

0

0

2

2

1 0

1

R+

2

1

2

1 0 1

2

1

R2+

Figure 3. Examples of manifolds with corners, with codimensions of points indicated

and it is required that coordinate changes are smooth.4 The smallest k which works for a fixed p is called the codimension of p. See Figures 3, 4 for some examples and non-examples of mwc. The set of points of codimension 0 is the interior int(M ) of M . The closure of a connected component of the set of points of codimension k is called a boundary hypersurface (bhs) if k = 1, and a corner of codimension k if k ≥ 2. So the examples in Figure 3 have 1, 2, 3, 2 boundary hypersurfaces. It is clear that each boundary hypersurface itself satisfies the local model condition, with n replaced by n − 1. However, as in the example on the right in Figure 4, it may happen that a boundary hypersurface ‘intersects itself’, that is, it is an immersed rather than an embedded submanifold (with corners). So according to our definition it is not a manifold with corners. The embeddedness requirement is equivalent to the existence of a boundary defining function for each bhs H, i.e. a smooth function x : M → R+ which vanishes precisely on H and whose differential at any point of H is non-zero. A boundary defining function x can be augmented to a trivialization near H, i.e. an identification of a neighborhood U of H with [0, ε) × H for some ε > 0, where x is the first component and each y ∈ H ⊂ U corresponds to (0, y). Each bhs and each corner of a mwc M is a mwc. But if M has corners then its full boundary is not a manifold with corners. Some authors, e.g. D. Joyce [32], define manifolds with corners without the embeddedness condition on boundary hypersurfaces. Also, Joyce defines the notion of boundary of a mwc differently, so that it is also a mwc. Taylor’s theorem implies the following simple fact which we need later. Lemma 2.2. Let M be a manifold with corners and S a finite set of boundary hypersurfaces of M . Let h be a total boundary defining function for S, i.e. the product of defining functions for all H ∈ S. u Then any u ∈ C ∞ (M ) which vanishes at each H ∈ S can be written as u = h˜ with u ˜ ∈ C ∞ (M ).

˜  ⊂ Rn with U ˜  ∩ (Rk × Open means relatively open, that is, there is an open subset U + ˜ ˜ ⊂ = U . For example, [0, 1) is open in R+ . A smooth function on an open subset U ˜  . A map U ˜ → Rk ×Rn−k Rk+ ×Rn−k is a function which extends to a smooth function on such a U + is smooth if each component function is smooth. The space of smooth functions on M (which are sometimes called ‘smooth up to the boundary’ for emphasis) is denoted by C ∞ (M ). 4

Rn−k )

216

DANIEL GRIESER

Figure 4. Not manifolds with corners. The cone and pyramid are understood as 3-dimensional bodies. The teardrop satisfies the local condition of a mwc, but the boundary line is not embedded. Exercise: Prove this. Show that the analogous statement would not be true for the pyramid in Figure 4.5 Remark 2.3. The corners of a mwc should not be considered as a problem, but as (part of ) a solution – of all kinds of problems involving singularities. They should not be thought of as corners in a metric sense, only in a differential sense (i.e. some coordinates are ≥ 0). For example, suppose you want to analyze the behavior of harmonic functions near the vertex of R2+ or of a cone or of the pyramid in Figure 4 (where the Laplacian is the standard Laplacian for the Euclidean metric on these spaces). The essential first step towards a solution would be to introduce polar coordinates around the vertex, and in the case of the pyramid also cylindrical coordinates around the edges. Geometrically this corresponds to the operation of blow-up, discussed below. This results in manifolds with corners. The fact that the original (metric) R2+ happens to be a mwc also is irrelevant. Remark 2.4. Manifolds with corners are an oriented analogue of manifolds with normal crossings divisors as used in real algebraic geometry. ‘Oriented’ means that the boundary hypersurfaces, which correspond to the components of the divisor, have a relative orientation, i.e. possess a transversal vector field. The use of manifolds with corners allows for greater flexibility in many analytic problems. See also Remark 2.12. 2.2. Polyhomogeneous functions. All functions we consider will be smooth in the interior of their domains. Our interest will lie in their boundary behavior – partly because we have a much better chance to analyze their boundary behavior than their interior properties. Functions smooth up to the boundary (see Footnote 4) have the following important properties:  k 1. A smooth function on R+ has a Taylor expansion f (x) ∼ ∞ k=0 ak x as x → 0, i.e. at ∂R+ . 2a. A smooth function on R2+ has Taylor expansions (2.1)

f (x, y) ∼

∞  k=0

ak (y)xk as x → 0,

f (x, y) ∼

∞ 

bl (x)y l as y → 0

l=0

at the boundary hypersurfaces x = 0 and y = 0 of R2+ , with ak , bl smooth on R+ . 5 If you understand this then you understand one of the main points about manifolds with corners!

SCALES, BLOW-UP AND QUASIMODE CONSTRUCTIONS

217

2b. (Matching) For each k, l ∈ N0 the l-th Taylor coefficient of ak at y = 0 equals (‘matches’) the k-th Taylor coefficient of bl at x = 0. This corresponds to the Taylor expansion of f at the corner (0, 0). 2c. (Borel lemma) Conversely, given ak , bl ∈ C ∞ (R+ ) satisfying these matching (or compatibility) conditions for all k, l, there is a function f ∈ C ∞ (R2+ ) satisfying (2.1), and it is unique modulo functions vanishing to infinite order at the boundary of R2+ . It turns out that requiring smoothness up to the boundary is too restrictive for many purposes. The class of polyhomogeneous functions6 is obtained by replacing the powers xm , m ∈ N0 in these expansions by terms xz logj x where z ∈ C and j ∈ N0 , and is big enough for many problems. Apart from this, polyhomogeneous functions enjoy the analogous properties as listed above. Properties 2b. and 2c. will be essential for our purpose of analyzing multiple scale solutions of PDEs. 2.2.1. Definition and examples. We will define the space of polyhomogeneous functions on a manifold with corners M . The essence of the definition can be grasped from two special cases: M = R+ × Rn where n ∈ N0 and M = R2+ . The terms permitted in an expansion are characterized by a set E ⊂ C × N0 satisfying (2.2)

{(z, j) ∈ E : Re z ≤ r} is finite for every r ∈ R .

This guarantees that the expansion (2.3) below makes sense. Definition 2.5. A polyhomogeneous function on M = R+ × Rn or M = is a smooth function u on int(M ) satisfying: (a) For M = R+ × Rn : u has an asymptotic expansion  az,j (y) xz logj x as x → 0 (2.3) u(x, y) ∼ R2+

(z,j)∈E

for each y ∈ R , for a set E as above, where each az,j ∈ C ∞ (Rn ). The set of these functions with E fixed is denoted AE (R+ × Rn ). (b) For M = R2+ : u has an asymptotic expansion (2.3) for each y > 0, where each az,j ∈ AF (R+ ), for sets E, F ⊂ C × N0 satisfying (2.2). Also, the same condition is required to hold with x, E and y, F interchanged. The set of these functions with E, F fixed is denoted AE,F (R2+ ). By definition, we understand asymptotic expansions always ‘with derivatives’, i.e. ∂x u has the asymptotic series with each term differentiated, and similarly for ∂y u and higher derivatives. In addition, certain uniformity conditions are required. n

All asymptotic expansions occuring in the problems in this article have no logarithms, so E ⊂ C × {0}.7 6 These

are called ‘nice functions’ in [14]. However, logarithms are included in the definition since they appear in the solutions of many differential equations even if they don’t appear in their coefficients. For example, the equation  x + 1 which for fixed u = fh , u(0) = 0 with fh as in (1.1) has solution uh (x) = x − h log h positive x has the expansion 7

uh (x) ∼ x + h log h − h log x + O(h2 ) as h → 0. The appearance of the log term here can be predicted without calculating integrals, using geometric resolution analysis via the push-forward theorem of Melrose [49], as is explained √ in [14] for the related example where fh (x) = x2 + h2 , see also [18].

218

DANIEL GRIESER

The ‘asymptotics with derivatives’ condition is equivalent to : ⎛ ⎞: : :  : : j ⎠: z r :(x∂x )α ∂yβ ⎝u(x, y) − a (y) x log x (2.4) z,j : ≤ Cr,α,β x : : : (z,j)∈E,Re z≤r for all r ∈ R and all α ∈ N0 , β ∈ Nn0 . Here Cr,α,β may depend on y. For M = R+ × Rn the local uniformity condition is that for any compact K ⊂ Rn the same constant can be chosen for all y ∈ K. For M = R2+ this is required for all compact K ⊂ (0, ∞), plus a local uniformity near (x, y) = (0, 0): there is N ∈ R so that estimate (2.4) holds for all y ∈ (0, 1), with ∂y replaced by y∂y and Cr,α,β by Cr,α,β y −N . We now give examples and then formulate the general definition. Examples 2.6. (1) u(x) = x1 is in AE (R+ ) for E = {(−1, 0)}. (2) If E = F = N0 ×{0} then u ∈ AE,F (R2+ ) if and only if u extends smoothly to the boundary of R2+ .8 x (3) u(x, y) = x+y is smooth on R2+ \ {(0, 0)}, but not polyhomogeneous (for any index sets) on R2+ . To see this, we expand u as x → 0 for fixed y > 0: (2.5)

u(x, y) =

x 1 x = x+y y1+

x y

=

1 1 1 x − 2 x2 + 3 x3 − + · · · y y y

We see that u has an expansion as in (2.3), but the coefficients ak,0 (y) = (−1)k y −k become more and more singular (for y → 0) as k increases, so there is no index set F for which all coefficients lie in AF (R+ ). Note that this is precisely our first example (1.1). A set E ⊂ C × N0 satisfying (2.2) and in addition (z, j) ∈ E, l ≤ j ⇒ (z, l) ∈ E is called an index set. This condition guarantees that AE (R+ × Rn ) is invariant under the operator x∂x . If, in addition, (z, j) ∈ E ⇒ (z + 1, j) ∈ E then E is called a smooth (or C ∞ ) index set. This guarantees coordinate independence, i.e. any self-diffeomorphism of R+ × Rn preserves the space AE (R+ × Rn ). The index set E in Example 2.6.1 is not smooth; the smallest smooth index set containing E is {−1, 0, 1, . . . } × {0}. We now consider general manifolds with corners. Of course we want to say a function is polyhomogeneous if it is so in any coordinate system. Since we want to allow corners of higher codimension, we give an inductive definition. An index family for M is an assignment E of a C ∞ index set E(H) to each boundary hypersurface H of M . Recall that there is a trivialization near each H, i.e. we may write points near H as pairs (x, y) where x ∈ [0, ε) and y ∈ H, for some ε > 0. Definition 2.7. Let M be a manifold with corners and E an index family for M . A polyhomogeneous function on M with index family E is a smooth function u on int(M ) which has an expansion as in (2.3) at each boundary hypersurface H, in some trivialization near H, where E = E(H) and the functions az,j are polyhomogeneous on H with the induced index family for H.9 8 Exercise:

prove this. index family E for M induces an index family EH for the mwc H as follows: Any boundary hypersurface H  of H is a component of a set H ∩ G where G is boundary hypersurface 9 The

SCALES, BLOW-UP AND QUASIMODE CONSTRUCTIONS

219

The set of these functions is denoted AE (M ). Again, if E(H) = N0 × {0} for all H then u ∈ AE (M ) if and only if u extends to a smooth function on all of M . Remark 2.8. In our terminology a ‘polyhomogeneous function on a manifold with corners M ’ needs to be defined on the interior int(M ) only. The terminology is justified since its behavior near the boundary is prescribed; C ∞ (int(M )) is the much larger space of functions without prescribed boundary behavior. More formally, AE defines a sheaf over M , not over int(M ). 2.2.2. Matching conditions and Borel lemma. A central point of polyhomogeneity is to have ‘product type’ asymptotic expansions at corners. This is most clearly seen in the case of R2+ . To ease notation we formulate this only for the case without logarithms. Lemma 2.9 (Matching conditions). Let E, F ⊂ C × {0} be index sets for R2+ . Suppose u ∈ AE,F (R2+ ), and assume u has expansions  az (y) xz as x → 0 u(x, y) ∼ (2.6)

(z,0)∈E

u(x, y) ∼



bw (x) y w

as y → 0

(w,0)∈F

where az ∈ AF (R+ ), bw ∈ AE (R+ ) for each (z, 0) ∈ E, (w, 0) ∈ F . Expand   (2.7) az (y) ∼ cz,w y w , bw (x) ∼ cz,w xz (w,0)∈F

(z,0)∈E

as y → 0 resp. x → 0. Then (2.8)

cz,w = cz,w

for all z, w.

This has a converse, which is a standard result: Lemma 2.10 (Borel lemma). Let E, F be as in the previous lemma, and assume that functions az , bw satisfying (2.7) are given. If (2.8) holds then there is u ∈ AE,F (R2+ ) satisfying (2.6). It is uniquely determined up to errors vanishing to infinite order at the boundary. This will be a central tool in our analysis since it allows us to construct approximate solution of a PDE from solutions of model problems. 2.3. Blow-up and resolution. We now introduce blow-up, which is what makes the whole manifolds with corners business interesting. Here are the most important facts about blow-up. They will be explained in this section: • Blow-up is a geometric and coordinate free way to introduce polar coordinates. • Blow-up serves to desingularize singular objects. of M uniquely determined by H  . Then we let EH (H  ) := E(G). We require az,j ∈ AEH (H) for each (z, j) ∈ E(H) and each H. If this is true in one trivialization then it is true in any other, since each E(H) is a C ∞ index set. Local uniformity is also required, analogous to the explanation after equation (2.4). This definition is inductive over the highest codimension of any point in M .

220

DANIEL GRIESER

β

← −

[R2 , 0]

R2

Figure 5. Blow-up of 0 in R2 , with a few rays (dashed) and a pair of corresponding circles (dotted) drawn; the white disk is not part of [R2 , 0]; its inner boundary circle is the front face

↓

↓

↓

↓

(a)

(b)

(c)

(d)

Figure 6. Some examples of blow-up; in each bottom picture the submanifold being blown up is drawn fat, and the blown-up space is in the top picture. The vertical arrow is the blow-down map. The third and fourth example are 3-dimensional, and only the edges are drawn. • Blow-up helps to understand scales and transitions between scales – and therefore to solve PDE problems involving different scales. We first explain the idea in the case of blow-up of 0 in R2 and then give the general definition in Subsection 2.3.2. After discussing resolutions and projective coordinates we return to our motivating example (1.1) in Example 2.18. There is also a short discussion of quasihomogeneous blow-up, which occurs naturally in Section 6. 2.3.1. The idea. We first explain the idea in the case of blowing up the point 0 in R2 , see Figure 5: Consider the set of rays (half lines) in R2 emanating from 0. They are pairwise disjoint except that they all share the common endpoint 0. The blow-up of 0 in R2 is the space constructed from R2 by removing 0 and replacing it by one separate endpoint for each ray. This space is denoted by [R2 , 0]. So we

SCALES, BLOW-UP AND QUASIMODE CONSTRUCTIONS

221

replace 0 by a circle, and each point on the circle corresponds to a direction of approach to 0. This circle is called the front face of the blow-up. Thus, blowingup 0 in R2 means taking out 0 from R2 and then choosing a new ‘compactification at 0’ of R2 \ 0, by adding the front face instead of 0. Here is a concrete mathematical model realizing this idea: As a space we take [R2 , 0] = R+ × S 1 , where S 1 = {ω ∈ R2 : |ω| = 1} is the unit circle. The front face is ff := {0} × S 1 . In Figure 5 the rays are the sets ω = const, the unbroken circle is ff and the dotted circle is {1} × S 1 . We then need to specify how points of [R2 , 0] correspond to points of R2 . This is done using the map β : [R2 , 0] → R2 ,

β(r, ω) = rω

called the blow-down map. Note that β is a diffeomorphism from (0, ∞) × S 1 to R2 \ {0}; this means that it provides an identification of [R2 , 0] \ ff with R2 \ {0}. The sets ω = const are mapped to rays, and two different such sets have different endpoints on ff. All these endpoints are mapped to 0 by β. Thus, this model and β do precisely what they were supposed to do. In addition, the model gives [R2 , 0] a differentiable structure, making it a smooth manifold with boundary and β a smooth map. Note that if we parametrize S 1 by ω = (cos ϕ, sin ϕ) then β is just the polar coordinates map (2.9)

(r, ϕ) → (x, y),

x = r cos ϕ, y = r sin ϕ .

Recall that ‘polar coordinates on R2 ’ are not coordinates at the origin. So [R2 , 0] is the space on which polar coordinates are actual coordinates – also at r = 0. Exercise 2.11. Show that points of ff correspond to directions at 0 not only of rays, but of any regular curve. That is: Let γ : [0, 1) → R2 be a smooth curve with γ(0) = 0, γ(0) ˙ = 0 and γ(t) = 0 for t = 0. Show that there is a unique smooth ˙ . curve γ˜ : [0, 1) → [R2 , 0] lifting γ, i.e. satisfying β ◦ γ˜ = γ, and that γ˜ (0) = γ(0) γ(0) ˙ 2.3.2. Definition and examples. The general operation of blow-up associates to any manifold X and submanifold Y ⊂ X a manifold with boundary, denoted [X, Y ], and a surjective smooth map β : [X, Y ] → X. We say that [X, Y ] is obtained from blowing up Y in X and call β the blow-down map.10 X, Y may also be manifolds with corners, then a local product assumption (see below) must be placed on Y , and [X, Y ] is a manifold with corners. The preimage β −1 (Y ) is called the front face ff of the blow-up. It is a boundary hypersurface of [X, Y ], and β maps diffeomorphically [X, Y ] \ ff → X \ Y . See Figure 6 for some examples. To define blow-up we use local models as in the previous subsection, but you should always keep the original idea of adding endpoints of rays in mind. We start with blow-up of an interior point, then generalize this in two ways: blow-up of a point on the boundary, and blow-up of a subspace (by taking products). Finally both generalizations are combined to yield the most general case.

10 For this to be defined Y must have codimension at least one. We will always assume that the codimension is at least two, the other case being less interesting.

222

DANIEL GRIESER

Definition of blow-up for the local models. Recall that a model space is a space of the form Rn−k × Rk+ (or Rk+ × Rn−k ). We consider these first. (1) Blow-up of11 0 in Rn : Define [Rn , 0] := R+ × S n−1 ,

β(r, ω) = rω

= {ω ∈ R : |ω| = 1} is the unit sphere, n ≥ 1. where S Note that [Rn , 0] is a manifold with boundary. (2) Blow-up of 0 in the upper half plane R × R+ : Define n−1

n

1 [R × R+ , 0] := R+ × S+ , β(r, ω) = rω 1 where S+ = S 1 ∩ (R × R+ ) is the upper half circle. See Figure 6(a). This is simply the upper half of case (1) with n = 2. This generalizes in an obvious way to the blow-up of zero in any model space:

[Rn−k × Rk+ , 0] := R+ × Skn−1 , β(r, ω) = rω where Skn−1 := S n−1 ∩ (Rn−k × Rk+ ). See Figure 6(b) for n = k = 2 and Figure 6(c) for n = k = 3. Note that [Rn−k × Rk+ , 0] has corners if k ≥ 1. (3) Blow-up of the x-axis R × {0} in R3 : Define [R3 , R × {0}] := R × [R2 , 0], β(x, y, z) = (x, β0 (y, z)) with β0 : [R2 , 0] → R2 from case (1). So the line Y = R × {0} is blown up to a cylinder, the front face of this blow-up. Any point p ∈ Y is blown up to a circle β −1 (p). Points on the front face correspond to a pair consisting of a point p ∈ Y and a direction of approach to p, modulo directions tangential to Y . This generalizes in an obvious way to the blow-up of Rn−m × {0} in n R : [Rn , Rn−m × {0}] = Rn−m × [Rm , 0] (write Rn = Rn−m × Rm and take out the common factor Rn−m ). (4) Blow-up of R+ × {0} in R3+ . Combining cases (2) and (3) we define [R3+ , R+ × {0}] = R+ × [R2+ , 0] see Figure 6(d). The main point here is the product structure. In general, for model spaces X, W, Z, (2.10)

for X = W × Z, Y = W × {0} define

[X, Y ] = W × [Z, 0]

with [Z, 0] defined in (2). In the example, X = R3+ , W = R+ , Z = R2+ . Definition of blow-up for manifolds (possibly with corners). It can be shown (see [47], [51]) that these constructions are invariant in the following sense: for model spaces X, Y as in case (4), any self-diffeomorphism of X fixing Y pointwise lifts to a unique self-diffeomorphism of [X, Y ].12 Now if X is a manifold and Y ⊂ X simplify notation we often write 0 instead of {0}. Also 0 denotes the origin in any Rk . the case of [R2 , 0] this can be rephrased as follows: let x, y be standard cartesian coordinates and r, ϕ corresponding polar coordinates. Let x , y  be some other coordinate system defined near 0 (possibly non-linearly related to x, y), with x = y  = 0 corresponding to the point 0. Define polar coordinates in terms of x , y  , i.e. x = r  cos ϕ , y  = r  sin ϕ . Then (r, ϕ) → (r  , ϕ ) is a smooth coordinate change on [R2 , 0]. 11 To 12 In

SCALES, BLOW-UP AND QUASIMODE CONSTRUCTIONS

223

a submanifold, then Y ⊂ X is locally Rn−m × {0} ⊂ Rn , in suitable coordinates. Therefore, the blow-up [X, Y ] is well-defined as a manifold, along with the blowdown map β : [X, Y ] → X.13 If X is a manifold with corners then a subset Y ⊂ X is called a p-submanifold if it is everywhere locally like the models (2.10) (p is for product). Therefore, the blow-up [X, Y ] is defined for p-submanifolds Y ⊂ X. For example, the fat subsets in the bottom line of Figure 6 are p-submanifolds, as are the dashed rays in the top line. However, the dashed rays in (b), (c) and (d) in the bottom line are not p-submanifolds. Put differently, a subset Y ⊂ X is a p-submanifold if near every q ∈ Y there are local coordinates centered at q so that Y and every face of X containing q is a coordinate subspace, i.e. a linear subspace spanned by some coordinate axes, locally. The preimage β −1 (Y ) ⊂ [X, Y ] is a boundary hypersurface of [X, Y ], called the front face of the blow-up. The other boundary hypersurfaces of [X, Y ] are in 1-1 correspondence with those of X. Remark 2.12. This notion of blow-up, sometimes called oriented blow-up, is closely related to (unoriented) blow-up as defined in real algebraic geometry, where one ‘glues in’ a real projective space instead of a sphere. Unoriented blow-up can be obtained from oriented blow-up by identifying pairs of antipodal points of this sphere. This results in an interior hypersurface (usually called exceptional divisor) rather than a new boundary hypersurface as front face. Compare Remark 2.4. Unoriented blow-up has the virtue of being definable purely algebraically, so it extends to other ground fields, e.g. to complex manifolds. See [24], where also a characterization of blow-up by a universal property is given (Proposition 7.14). 2.3.3. Multiple blow-ups. Due to the geometric nature of the blow-up operation, it can be iterated. So if X is a manifold with corners and Y a p-submanifold, we can first form the blow-up β1 : [X, Y ] → X. Next, if Z is a p-submanifold of [X, Y ] then we can form the blow-up β2 : [[X, Y ], Z] → [X, Y ]. The total blow-down map is then the composition β = β1 ◦ β2 : [[X, Y ], Z] → X . See Figure 7 for a simple example. Of course one may iterate any finite number of times. 2.3.4. Resolutions via blow-up. The main use of blow-ups is that they can be used to resolve singular objects, for example functions and sets.

It is in this sense that blow-up is a coordinate free way of introducing polar coordinates: the result does not depend on the (cartesian) coordinates chosen initially. This is important, for example, for knowing that we may choose coordinates at our convenience. For example, when doing an iterated blow-up we may choose projective coordinates after the first blow-up, or polar coordinates, and will get the same mathematics in the end. 13 The original idea that points on the front face correspond to directions at 0 can be used directly as an invariant definition: Let M be a manifold and p ∈ M . The set of directions at p is Sp M := (Tp M \ {0})/R>0 where Tp M is the tangent space and R>0 acts by scalar multiplication. Then [M, p] = (M \ {p}) ∪ Sp M , with β the identity on M \ {p} and mapping Sp M to p. One still needs local coordinates to define the differentiable structure on [M, p].

224

DANIEL GRIESER

β1

←−

β2

←−

Z

0 R2+

[R2+ , 0]

[[R2+ , 0], Z]

Figure 7. A double blow-up Definition 2.13 (Resolving functions). Let β : X  → X be a (possibly iterated) blow-down map of manifolds with corners and f : int(X) → C a function. We say that f is resolved by β if β ∗ f is a polyhomogeneous function on X  . Here β ∗ f := f ◦ β is the pull-back. Recall that β ∗ f need only be defined on the interior of X  , compare Remark x on R2+ \ {0} is 2.8. In Example 2.18 we will see that the function f (x, y) = x+y resolved by blowing up zero. For subsets we need a slight generalization of p-submanifolds. A d-submanifold of a manifold with corners X is a subset Y ⊂ X which is everywhere locally modelled on (2.11)

X = W × Z × Rl ,

Y = W × {0} × Rl+

for some l ≥ 0 and model spaces W, Z (d means decomposable). This is a psubmanifold iff l = 0, see (2.10). For example, R2+ ⊂ R2 is a d-submanifold which is not a p-submanifold. Definition 2.14 (Resolving subsets). Let β : X  → X be a (possibly iterated) blow-down map of manifolds with corners and S ⊂ X a subset. We say that S is resolved by β if β ∗ S is a d-submanifold of X  . Here the lift14 β ∗ S under a blow-down map [X, Y ] → X is defined as β ∗ S = β −1 (S \ Y ) if S ⊂ Y,

β ∗ S = β −1 (S) if S ⊂ Y.

For an iterated blow-down map β = β1 ◦ · · · ◦ βk we define β ∗ S = βk∗ . . . β1∗ S. For example, the solid cone S ⊂ R3 (left picture in Figure 4) is resolved by blowing up 0 in R3 . Here β ∗ S ⊂ [R3 , 0] is a manifold with corners, the local model at the corner is (2.11) with W = R × R+ , Z = {0} and l = 1. The boundary of the cone is also resolved by β, its lift is even a p-submanifold. Note that in general the lift β ∗ S is almost the preimage, but not quite. In the cone example, the preimage β −1 S would be the union of β ∗ S and the front face of the blow-up, which is a 2-sphere. We consider β ∗ S since it contains the only interesting information about S. See Figure 8(d) for another example (dashed lines) and Figure 16 for an example of a resolution by a multiple blow-up. Both of them will be used later. Of course we can combine Definitions 2.14 and 2.13: If S ⊂ X then a function f on S ∩int(X) is resolved by β : X  → X if S is resolved and β ∗ f is polyhomogeneous on β ∗ S. 14 The

lift is also called the strict transform in the algebraic geometry literature.

SCALES, BLOW-UP AND QUASIMODE CONSTRUCTIONS

225

lf z

x y

y

ff

y z

y x x

z x z

x y z

rf ↓

↓

y

↓ z

x

x y

(b)

z

x y

(c)

(d)

Figure 8. Projective coordinates for examples (b), (c), (d) of Figure 6. Dashed lines in (d) indicate a singular subset (below) and its resolution (above). Note that in these definitions we consider polyhomogeneous functions and dsubmanifolds as ‘regular’ and more general functions resp. subsets as ‘singular’. Regular objects in this sense remain regular after blow-up, as is easy to see using projective coordinates, introduced below.15 Remark 2.15. By a deep famous theorem of Hironaka every algebraic variety S ⊂ CP n can be resolved by a sequence of blow-ups (in the algebraic geometric sense, see Remark 2.12). Similar statements hold for algebraic (or even semi- or subalgebraic) subsets of Rn , see [30] and [27] for a more entertaining and low-tech survey. Remark 2.16. There is a generalization of blow-up which is sometimes useful when resolving several scales simultaneously, see [32], [36]. 2.3.5. Projective coordinates. Projective coordinates simplify calculations with blow-ups and also provide the link of blow-ups to the discussion of scales. We first discuss this for the space [R2+ , 0], see Figure 8(b). Recall that points of [R2+ , 0] correspond to pairs consisting of a ray (in R2+ , emanating from 0) and a point on that ray. Now, with x, y standard coordinates on R2+ , y points on a ray ↔ values of x rays ↔ values of , x except if the ray is the y-axis (which would correspond to xy = ∞). Here xy ≥ 0 and x ≥ 0, and x = 0 is the endpoint of the ray. 15 For a d-submanifold S ⊂ X to lift to a d-submanifold under blow-up of Y ⊂ X we must require that S and Y intersect cleanly (which might be called ‘normal crossings’ by algebraic geometers), i.e. near every intersection point there are coordinates in which X, S and Y are given by model spaces.

226

DANIEL GRIESER

This means that xy and x provide a coordinate system for [R2+ , 0] \ lf, where lf (‘left face’) is the lift of the y-axis:16 y (2.12) (x, ) : [R2+ , 0] \ lf → R2+ x We need to check that this is a smooth coordinate system. This means: 1. The function xy , which is defined and smooth on [R2+ , 0] \ (lf ∪ ff), extends smoothly to [R2+ , 0] \ lf. 2. The map (2.12) is a diffeomorphism. Both statements refer to the differentiable structure on [R2+ , 0], which was defined 1 1 by writing [R2+ , 0] = R+ × S++ where S++ is the quarter circle. If we use the angle π 1 then we need to check that the map (r, ϕ) → (x, xy ) coordinate ϕ ∈ [0, 2 ] on S++ extends smoothly from r > 0, ϕ < π2 to r ≥ 0, ϕ < π2 and is a diffeomorphism R+ × [0, π2 ) → R+ × R+ . This can be seen from the explicit formulas x = r cos ϕ, = 2 y = tan ϕ, and for the inverse map r = x 1 + xy , ϕ = arctan xy . x By symmetry, we have another smooth coordinate system given by xy and y on the set [R2+ , 0] \ rf, where rf (‘right face’) is the lift of the x-axis. Note that in the coordinate system x, xy the boundary defining function of the front face is x, and in the coordinate system y, xy it is y. Projective coordinates can be used to check that a function is resolved under a blow-up: Lemma 2.17. A function f on R2>0 is resolved by the blow up of 0 if and only if f is polyhomogeneous as a function of xy , y and as a function of x, xy . This is clear since polyhomogeneity (or smoothness) of a function on a manifold means polyhomogeneity (or smoothmess) in each coordinate system of an atlas. x on R2+ \ 0 again. We Example 2.18. We consider the function f (x, y) = x+y saw in Example 2.6(3) that f is not polyhomogeneous at 0. However, X x in coordinates X = , y : β ∗ f = y X +1 1 y in coordinates x, Y = : β ∗ f = x 1+Y and both of these functions are smooth for (X, y) ∈ R2+ resp. (x, Y ) ∈ R2+ , the respective ranges of these coordinates. So f is resolved by β, and β ∗ f is even smooth on [R2+ , 0]. x X . Here β ∗ f2 = X+1+Xy and As another example, consider f2 (x, y) = x+y+xy 1 ∗ β f2 = 1+Y +xY in the two coordinate systems, so f2 is also resolved by β. Note that these agree with β ∗ f at y = 0 and x = 0 respectively, which means β ∗ f2 = β ∗ f at the front face. This is clear a priori since xy vanishes to second order at x = y = 0.

Remark 2.19 (Relation of projective coordinates to scaled limit). Suppose a function f on R2>0 is resolved by β : [R2+ , 0] → R2+ , and assume β ∗ f is even smooth. To emphasize the relation to the discussion of scales, we denote coordinates by x, h and write fh (x) = f (x, h). 16 It

would be formally better to write

β∗ y β∗ x

and β ∗ x instead of

y x

and x, but this quickly

becomes cumbersome. Note that β ∗ x vanishes on lf ∪ ff and β ∗ y vanishes on ff ∪ rf.

SCALES, BLOW-UP AND QUASIMODE CONSTRUCTIONS

227

(1) The rescaled limit g(X) = limh→0 fh (hX) is simply the restriction of β ∗ f to the front face ff, when parametrizing ff by the projective coordinate X. To see this, note that in the projective coordinate system X, h the map β is given by β(X, h) = (hX, h) (this is the meaning of writing X = hx ), so (β ∗ f )(X, h) = f (hX, h), and h = 0 is the front face. (2) That f is resolved by β contains additional information beyond existence of this scaled limit: information on derivatives as well as information on the behavior of g(X) as X → ∞. Note that X = ∞ corresponds to the ‘lower’ corner in [R2+ , 0]. More precisely, g is smooth at ∞ in the sense that η → g( η1 ) is smooth at η = 0. Here η is the coordinate hx in the second projective coordinate system. For more general blow-ups it is useful to have: Quick practical guide to finding projective coordinate systems: Point blow-up of 0 in Rk+ ×Rn−k : Near the (lift of the) x-axis y projective coordinates are x and xj , where yj are the variables other than x. These are coordinates except on the (lift of the) set {x = 0}. Similarly for any other axis. Blow-up of coordinate subspace Y ∈ Rk+ × Rn−k : Apply the previous to variables x, yj vanishing on Y . Other variables remain unchanged. It may be useful to think of x as ‘dominant’ variable on the coordinate patch: for y any compact subset of the patch there is a constant C so that |yj | ≤ Cx. So xj is bounded there. Note: dominant variable = boundary defining function of front face For the examples in Figure 6(a),(c),(d) we get the projective coordinate systems, see also Figure 8(c),(d) (where only one system is indicated): (a) [R × R+ , 0]: near the interior of the front face: y, xy ; in a neighborhood of the lift of the x-axis: x, xy .17 (c) [R3+ , 0]: outside the left boundary hypersurface: x, xy , xz ; outside the back boundary hypersurface: y, xy , yz ; outside the bottom boundary hypersurface: z, xz , yz . (d) [R3+ , R+ × {0}]: outside the back boundary hypersurface: x, y, yz ; outside the bottom boundary hypersurface: x, z, yz .  x2 + xy + y 3 on R2+ is Exercise 2.20. Show that the function f (x, y) = resolved by the double blow-up in Figure 7, but not by the simple blow-up of 0 ∈ R2+ .18 latter are really two coordinate patches, one for x ≥ 0 (near right corner) and one for y instead so the x ≤ 0 (near left corner). Near the left corner it is more customary to use |x|, |x| 17 The

dominant variable is positive. √ 18 Solution: In coordinates x, Y = y the function β ∗ f = x 1 + Y + xY 3 is polyhomogeneous 1 x  x since it is smooth. In coordinates X = y , y the function β1∗ f = y X 2 + X + y is polyhomogeneous outside X = y = 0, but not at this point. Therefore we blow up X = y = 0, which √ is the point Z in Figure 7. Let β = β1 ◦ β2 . In coordinates y the function β ∗ f = X 3/2 η X + 1 + η is polyhomogeneous. In coordinates ξ = X ,y X, η = X y  the function β ∗ f = y 3/2 ξ 2 y + ξ + 1 is polyhomogeneous. So f is resolved by β.

228

DANIEL GRIESER

2.3.6. Quasihomogeneous blow-up. In many problems scalings other than x ∼ y √ appear, for example x ∼ y in the function f (x, y) = x21+y . These can be understood either by multiple blow-ups, as in Exercise 2.20, or by the use of quasihomogeneous blow-up. This occurs, for example, in Section 6, and also for the heat kernel (where y is time), see e.g. [50], [15], [45]. For simplicity we only consider the quasihomogeneous blow-up of 0 in R2+ , with √ x scaling like y. We denote it by [R2+ , 0]q . This is sometimes called parabolic blowup. The idea is analogous to regular blow-up, except that the rays in R2+ through 0 are replaced by ‘parabolas’, by which we mean the sets {y = Cx2 } including the cases C = 0, i.e. the x-axis, and C = ∞, i.e. the y-axis. Then the blown-up space is constructed by removing 0 and replacing it by one separate endpoint for each parabola. These endpoints can be thought of as forming a quarter circle again, so the blown-up space looks just like, and in fact will be diffeomorphic to, [R2+ , 0]. However, the blow-down map β will be different.  Here is a local model realizing this idea: Let r(x, y) = x2 + y and Sq1 = {(ω, η) ∈ R2+ : r(ω, η) = 1}. Then we let [R2+ , 0]q = R+ × Sq1 with blow-down map β(r, (ω, η)) = (rω, r 2 η). This is constructed so that β maps each half line {(ω, η) = const} to a parabola, so that indeed endpoints of parabolas correspond to points of ff := {0} × Sq1 . Also, β maps ff ⊂ [R2+ , 0]q to 0 ∈ R2+ and is a diffeomorphism between the complements of these sets.19 Projective coordinates are as shown in Figure 9. The coordinates near A seem quite natural: x smoothly parametrizes the points on each parabola {y = Cx2 } (except C = ∞), and the parabolas are parametrized by the value of C = xy2 , so pairs ( xy2 , x) parametrize pairs (parabola, point on this parabola). On the other hand, the coordinates near B require explanation. One way to understand them is to check in the model that these are indeed coordinates (compare the explanation after (2.12); do it!). Without reference to the model the exponents that occur can be understood from three principles: √ (a) The coordinate ‘along the front face’ should reflect the scaling x ∼ y. (b) β should be smooth, so both x and y must be expressible as monomials in the coordinates,20 near A and near B (c) The smooth structure on [R2+ , 0]q should be the minimal one satisfying (a) and (b), i.e. the exponents should be maximal possible. So for the system near A, (b) implies that in the coordinate along ff the exponent 1 for some m ∈ N, and then (c) implies m = 1. Hence the coordinate of y must be m y must be x2 by (a). The exponent of x in the other coordinate must be 1 by (b) and (c). Similarly, near B in the coordinate along ff we need x in first power by 19 Maybe

you ask: why this model, not another one? In fact, the precise choice or r and Sq1 are irrelevant – any choice of positive smooth function r which is 1-homogeneous when giving x the weight 1 and y the weight 2, and any section transversal to all parabolas which stays away from the origin will do, with the same definition of β. Choosing Sq1 = r −1 (1) has the nice feature that use of the letter r is consistent in that r(β(R, (ω, η))) = R. 20 This means that we require β to be a b-map, a condition stronger than smoothness, see [51].

SCALES, BLOW-UP AND QUASIMODE CONSTRUCTIONS

y

229

lf √

β

← − x

y

B

x √ y

ff

A

y x2

x

rf

Figure 9. Projective coordinate systems for quasihomogeneous blow-up [R2+ , 0]q √ (b) and (c), and (a) gives √xy . Then (b) and (c) leave no choice but to have y as the other coordinate.21 Projective coordinates can be used as in Lemma 2.17 to check whether quasihomogeneous blow-up resolves a function. For more details, including the question of coordinate invariance, see [9] and [51]; see also [19]. A more general blow-up procedure is introduced in [36], see also [32], [35]. This is closely related to blow-up in toric geometry, see [5]. 2.4. Summary on blow-up and scales; further examples. We first sumx : f is smooth on R2+ \ 0 but has marize our discussion of the function f (x, h) = x+h no continuous extension to 0. The behavior of f near 0 can be described by saying that the scaling limit limh→0 f (hX, h) = g(X) exists for all X. This can be restated in terms of the blow-up of 0 in R2+ with blow-down map β : [R2+ , 0] → R2+ and front face ff = β −1 (0): the function β ∗ f , defined on [R2+ , 0] \ ff, extends continuously to ff, and g is the restriction of this extension to ff when ff is parametrized by X. Here X = hx is part of the projective coordinate system X, h. In fact, we saw that the extension of β ∗ f is not only continuous but even smooth on [R2+ , 0]. That is, f is resolved by β in the sense of Definition 2.13. The fact that g is not constant leads to the discontinuity of f at 0. The example suggests that the vague idea of scaling behavior is captured by the notion of resolution, which is defined rigorously in Definition 2.13. We note a few details of this definition: (1) The resolved function β ∗ f is required to be polyhomogeneous, which means in particular: • the asymptotics holds with all derivatives • full asymptotics is required, not just leading order asymptotics To include derivatives is natural since we want to deal with differential equations. To require full asymptotics is then natural since for example smoothness at a boundary point means having a full asymptotic series (the Taylor series). Only the combination of both conditions yields a unified theory.22 different way to understand the coordinates xy2 , √xy along ff is to note that xy2 is a defining function of rf in its interior x > 0, and √xy is a defining function of lf in its interior y > 0. 21 A

This reflects the fact that only the point 0 ∈ R2+ is affected by the blow-up, that is, that β is a diffeomorphism between the complements of β −1 (0) and {0}. In particular, quasihomogeneous √ blow-up is not the same as first replacing the variable y by y and then doing a standard blow-up. 22 Of course one could define finite order (in number of derivatives or number of asymptotic terms) theories, and this may be useful for some problems. However, many problems do admit

230

DANIEL GRIESER

(2) On the other hand, requiring β ∗ f to be smooth would be too restrictive (compare Footnote 7). What really matters is the product structure near corners as explained in Section 2.2.2. (3) Of course any function can be ‘over-resolved’, for example if f is smooth on R2+ then we may still look at β ∗ f which is still smooth. This would correspond to ‘looking at f at scale x ∼ h’.23 We give some more examples to illustrate these points. Examples 2.21. In these examples we denote coordinates on R2+ by x, h to emphasize the relation to scaling. β is always the blow-down map for the blow-up of 0 in R2+ . (1) f (x, h) = x + h is smooth on R2+ . In scaled coordinates f (hX, h) = h(X + 1) √ is the expansion of β ∗ f at the front face. (2) f (x, h) = x + h is not polyhomogeneous on R2+ as can be seen from the Taylor expansion as h → 0 for fixed x > 0: > ∞   √ √ 1 h  12 x+h= x 1+ = x 2 −k hk x k k=0

compare (2.5). However, note that fh = f (·, h) converges uniformly to f0 on R+ . But already√ fh does not converge uniformly to f0 . The same is true for f2 (x, h) = x2 + h2 even though f0 is smooth. Both f and f2 are resolved by blowing up 0 in R2+ . These examples show that non-trivial scaling behavior may only be visible in the √ derivatives. (3) f (x, h) = x2 + xh + h3 is resolved by the double blow-up in Figure 7, see Exercise 2.20. The two front faces correspond to the scales x ∼ h and x ∼ h2 . Any problem involving f needs to take into account both of these scales. To end this section we consider an example in three dimensions where a set is resolved by two blow-ups. This will be used in Section 7. Consider the family of plane domains Ωh ⊂ R2 , h > 0, shown in Figure 10: The 1 × h rectangle [0, 1) × (0, h) with a fixed triangle (e.g. a right-angled isosceles triangle), scaled to have base h, attached at one end. Again we want to describe the behavior of Ωh as h → 0. As in the first example, different features emerge at different scales: (1) We can consider B := limh→0 Ωh . This is just an interval.24 Many features of Ωh are lost in the limit: the thickness h, the triangular shape at the end. infinite order asymptotics – once the scales are correctly identified. Requiring less than the best possible sometimes obfuscates the view towards the structure of a problem. 23 So really we should not say that a function ‘exhibits the scale x ∼ h’, since every function does. More appropriate may be ‘f requires scale x ∼ h’, or ‘The scale x ∼ h is relevant for f ’. In any case, ‘f is resolved by β’ is a well-defined statement giving an upper bound on the ‘badness’ of f . 24 The precise meaning of the limit is irrelevant for this motivational discussion. You may think of Hausdorff limits.

SCALES, BLOW-UP AND QUASIMODE CONSTRUCTIONS

231

y h 0

Ωh

1

x

Y

Y

y x B

x A

X S

Figure 10. A family of domains Ωh and three rescaled limits as h→0 (2) More information is retained by noting that y scales like h, hence considering (2.13)

Ah := {(x, Y ) : (x, hY ) ∈ Ωh },

the domain obtained from stretching by the factor h−1 in the y-direction. Then A := limh→0 Ah is the square (0, 1) × (0, 1). This still forgets the triangular shape at the end. (3) At the left end, both x and y scale like h. So we consider (2.14)

Sh := {(X, Y ) : (hX, hY ) ∈ Ωh } = h−1 Ωh . Then S := limh→0 Sh is a half infinite strip of width one with a triangle attached at the left end. This limit remembers the triangle, but not that Ωh has essentially length 1 in the x-direction.

For the asymptotic analysis of the eigenvalue problem on Ωh in Section 7 it will be essential to understand A and?S as parts of one bigger space, which arises Ωh × {h} ⊂ R3 . This resolution is shown as resolution of the closure of Ω = h>0

in Figure 16 and explained there. Note that A and S are boundary hypersurfaces. The limit interval B occurs as the base of a natural fibration of the face A. 3. Generalities on quasimode constructions; the main steps In this section we give an outline of the main steps of the quasimode constructions that will be carried out in the following sections. For each h > 0 let Ωh be a bounded domain in R2 , and let Ph = −Δ be the Laplacian on Ωh , acting on functions that vanish at the boundary ∂Ωh . We assume that ∂Ωh is piecewise smooth.25 25 More generally one can consider families of compact manifolds with (or without) boundary and differential operators on them which are elliptic and self-adjoint with respect to given measures and for given boundary conditions. The methods are designed to work naturally in this context. Non-smooth boundary may require extra work.

232

DANIEL GRIESER

A quasimode for the family (Ωh )h>0 is a family (λh , uh )h>0 where λh ∈ R and uh is in the domain of Ph (in particular, uh = 0 at ∂Ωh ), so that (Ph − λh )uh = O(h∞ )

(3.1) ∞

as h → 0.

N

Here O(h ) means O(h ) for each N . We are ambitious in that we require these estimates to hold uniformly, also for all derivatives with respect to x ∈ Ωh and with respect to h. We reformulate this as follows: Consider the total space @ Ωh × {h} ⊂ R2 × R+ . Ω= h>0

We assume that Ωh depends continuously on h in the sense that Ω is open. A family of functions uh on Ωh corresponds to a single function u on Ω defined by u(x, h) = uh (x). The operators Ph define a single operator P on Ω via (P u)(·, h) = Ph (uh ). The operator P differentiates only in the Ωh directions, not in h. Then a quasimode is a pair of functions λ : R>0 → R, u : Ω → R satisfying the boundary conditions and (P − λ)u = O(h∞ ) as h → 0. How can we find quasimodes? Since the only issue is the behavior as h → 0, one expects that finding λ and u reduces to solving PDE problems ‘at h = 0’, along with an iterative construction: first solve with O(h) as right hand side, then improve the solution so the error is O(h2 ) etc. This is straightforward in the case of a regular perturbation, i.e. if the family (Ωh )h>0 has a limit Ω0 at h = 0, and the resulting family is smooth for h ≥ 0. This essentially means that the closure Ω of the total space Ω is a manifold with corners, see Section 4.1 for details. In particular, Ω0 is still a bounded domain in R2 . Then the problem at h = 0 is the model problem (P0 − λ0 )v = g

on Ω0 ,

v = 0 at ∂Ω0

where P0 = −Δ on Ω0 . Solving the model problem is the only analytic input in the quasimode construction. As we recall in Section 4 the iterative step reduces to solving this equation, plus some very simple algebra. However, our main focus will be on singular perturbations, where a limit Ω0 exists but Ωh does not depend smoothly on h at h = 0, so Ω has a singularity at h = 0. For example, if Ω0 is an interval or a curve, then this singularity looks approximately like an edge, see Figures 12, 14 and 16. We will consider several concrete such families. Their common feature is that this singularity can be resolved by (possibly several) blow-ups, yielding a manifold with corners M and a smooth map β : M → Ω. As explained in Section 2 this corresponds to a certain scaling behavior in the family (Ωh )h>0 as h → 0. The boundary hypersurfaces of M at h = 0, whose union is ∂0 M := β −1 (Ω ∩ {h = 0}), will now take the role of Ω0 , i.e. they will carry the model problems whose solution is used for constructing quasimodes. Since in the singular case several model problems are involved, the algebra needed for the quasimode construction is more complicated than in the regular

SCALES, BLOW-UP AND QUASIMODE CONSTRUCTIONS

233

case. However, this can be streamlined, and unified, by cleverly defining function spaces E(M ) and R(M ) which will contain putative quasimodes u and remainders f = (P − λ)u, respectively, along with suitable notions of leading part (at h = 0). The leading parts will lie in spaces E(∂0 M ) and R(∂0 M ), and are, essentially, functions on ∂0 M . All model problems together define the model operator (P − λ)0 : E(∂0 M ) → R(∂0 M ). Denoting for the moment by LP the leading part map, the needed algebra will be summarized in a Leading part and model operator lemma, which states26 a) the exactness of the sequences (3.2)

0 → hE(M )

→ E(M )

LP

→ 0

LP

→ 0;

−−→ E(∂0 M )

0 → hR(M ) → R(M ) −−→ R(∂0 M )

The main points here are exactness at R(M ) and at E(∂0 M ), explicitly: f ∈ R(M ), LP(f ) = 0 ⇒ f ∈ hR(M ), and any v ∈ E(∂0 M ) is the leading part of some u ∈ E(M ); b) the commutativity of the diagram (3.3)

E(M )

LP

P −λ

 R(M )

LP

/ E(∂0 M ) 

(P −λ)0

/ R(∂0 M )

That is, (P − λ)0 encodes the leading behavior of P − λ at h = 0. Summarizing, the main steps of the quasimode constructions are: (1) Resolve the geometry, find the relevant scales (2) Find the correct spaces for eigenfunctions and remainders (3) Find the correct ‘leading part’ definition for eigenfunctions and remainders. Identify model operators, prove Leading part and model operator lemma. (4) Study model operators (solvability of homogeneous/non-homogeneous PDE problems) (5) Carry out the construction: Initial step, inductive step The examples are progressively more complex, so that some features will occur only in later examples. Of course the process of finding the correct spaces etc. may be non-linear, as usual. The ‘meat’ is in step 4. After this, step 5 is easy. Steps 1-3 are the conceptual work needed to reduce the construction of quasimodes to the study of model operators. The results are formulated in Theorems 4.3, 5.4, 6.1, 7.6. They all have the same structure: given data for λ and u at h = 0 there is a unique quasimode having this data. For λ the data is the first or first two asymptotic terms, for u the data 26 This is analogous to the algebra needed for the parametrix construction in the classical pseudodifferential calculus, as explained in [14]: LP corresponds to the symbol map, the model operator is the constant coefficient operator obtained by freezing coefficients at any point. Invertibility of the model operator (which amounts to ellipticity) allows construction of a parametrix, which is the analogue of the construction of a quasimode.

234

DANIEL GRIESER

is the restriction to the boundary hypersurfaces of M at h = 0. Both cannot be freely chosen but correspond to a boundary eigenvalue problem. There are many other types of singular perturbations which can be treated by the same scheme. For example, Ω0 could be a domain with a corner and Ωh be obtained from Ω0 by rounding the corner at scale h. Or Ωh could be obtained from a domain Ω0 by removing a disk of radius h. Remark 3.1 (Are the blow-ups needed?). Our constructions yield precise asymptotic information about u as a function of h, x, y. Different boundary hypersurfaces of M at h = 0 correspond to different asymptotic regimes in the family Ωh . This is nice, but is it really needed if we are only interested in λ, say? The leading asymptotic term for λ and u as h → 0 is often easier to come by and does not usually require considering different regimes. But in order to obtain higher order terms of λ, it is necessary to obtain this detailed information about u along the way. As we will see, all regimes of the asymptotics of u will ‘influence’ the asymptotics of λ, often starting at different orders of the expansion. Another mechanism is that justifying a formal expansion up to a certain order usually requires knowing the expansion to a higher order (as is explained in [20], for example). 4. Regular perturbations To set the stage we first consider the case of a regular perturbation. Here basic features of any quasimode construction are introduced: the reduction to an initial and an inductive step, the identification of a model operator, and the use of the solvability properties of the model operator for carrying out the initial and inductive steps. 4.1. Setup. Let Ωh , h ≥ 0 be a family of bounded domains in R2 with smooth boundary.27 We say that this family is a regular perturbation of Ω0 if one of the following equivalent conditions is satisfied: (A) There are diffeomorphisms Φh : Ω0 → Ωh so that Φh is smooth in x ∈ Ω0 and h ≥ 0, and Φ0 = IdΩ0 . (B) The closure of the total space @ Ωh × {h} ⊂ R2 × R+ M =Ω= h≥0

is a manifold with corners, with boundary hypersurfaces @ ∂Ωh . X := Ω0 , ∂D M := h≥0 28

See Figure 11. Note that the two boundary hypersurfaces play different roles: At ∂D M , the ‘Dirichlet boundary’, we impose Dirichlet boundary conditions. The 27 Everything works just as well in Rn or in a smooth Riemannian manifold. Also the smoothness of the boundary can be relaxed, for example the Ωh could be domains with corners, then the requirement (B) below is that M be a d-submanifold of Rn × R+ , as defined before Definition 2.14. 28 To prove the equivalence of (A) and (B) note that Ω × R is a manifold with corners and 0 + that the Φh define a trivialization (diffeomorphism) Φ : Ω0 × R+ → Ω, (x, h) → (Φh (x), h), and conversely a trivialization defines Φh . (B) could also be reformulated as: M is a p-submanifold of R2 × R+

SCALES, BLOW-UP AND QUASIMODE CONSTRUCTIONS

235

h

∂D M X

Figure 11. The total space M for a regular perturbation quasimode construction proceeds at X = {h = 0}. To unify notation, we denote the boundary of X by ∂D X. In the geometric spirit of this article, and to prepare for later generalization, we use condition (B). For explicit calculations the maps Φh in (A) are useful, as we indicate in Subsection 4.3. As explained in Section 3 the quasimode construction problem is to find u and λ satisfying (P − λ)u = O(h∞ ), where u is required to satisfy Dirichlet boundary conditions. Here P = −Δ on each Ωh . 4.2. Solution. The idea is this: Rather than solve (P −λ)u = O(h∞ ) directly, we proceed inductively with respect to the order of vanishing of the right hand side: Initial step: Find λ, u satisfying (P − λ)u = O(h). Inductive step: Given λ, u satisfying (P − λ)u = O(hk ) where k ≥ 1, ˜ u ˜ u = O(hk+1 ). find λ, ˜ satisfying (P − λ)˜ Before carrying this out, we prepare the stage. We structure the exposition of the details so that it parallels the later generalizations. 4.2.1. Function spaces, leading part and model operator. For a regular perturbation we expect λ and u to be smooth up to h = 0. Therefore we introduce the function spaces ∞ (M ) = {u ∈ C ∞ (M ) : u = 0 at ∂D M } CD ∞ CD (X) = {v ∈ C ∞ (X) : v = 0 at ∂D X}

and hk C ∞ (M ) = {hk f : f ∈ C ∞ (M )},

h∞ C ∞ (M ) =

#

hk C ∞ (M ).

k∈N ∞

∞

So h C (M ) is the space of smooth functions on M vanishing to infinite order at the boundary h = 0. For simplicity we always consider real-valued functions. ∞ (M ) for which the remainders f = (P − λ)u lie in hk C ∞ (M ) We seek u ∈ CD for k = 1, 2, 3, . . . . Our final goal is: ∞ (M ) so that (P − λ)u ∈ h∞ C ∞ (M ) . Find λ ∈ C ∞ (R+ ), u ∈ CD ∞ The leading part of u ∈ CD (M ) and of f ∈ C ∞ (M ) is defined to be the restriction to h = 0:

uX := u|X ,

fX := f|X .

The following lemma is obvious. In (a) use Taylor’s theorem.

236

DANIEL GRIESER

Leading part and model operator lemma (regular perturbation). a) If f ∈ C ∞ (M ) then f ∈ h C ∞ (M ) if and only if fX = 0. b) For λ ∈ C ∞ (R+ ) we have ∞ P − λ : CD (M ) → C ∞ (M )

and [(P − λ)u]X = (P0 − λ0 )uX where P0 = −Δ is the Laplacian on Ω0 and λ0 = λ(0). We call ∞ (X) → C ∞ (X) P 0 − λ0 : C D

the model operator of P − λ, since it models its action at h = 0. Thus, the leading part of (P − λ)u is obtained by applying the model operator to the leading part of u. Remark 4.1. In the uniform notation of Section 3, see (3.2), (3.3), we have ∞ ∞ ∂0 M = X and E(M ) = CD (M ), E(∂0 M ) = CD (X), R(M ) = C ∞ (M ), R(∂0 M ) = ∞ C (X) and LP(u) = uX , LP(f ) = fX , (P − λ)0 = P0 − λ0 . 4.2.2. Analytic input for model operator. The core analytic input in the construction of quasimodes is the following fact about P0 . Lemma 4.2. Let λ0 ∈ R and g ∈ C ∞ (X). Then there is a unique γ ∈ Ker(P0 − λ0 ) so that the equation (P0 − λ0 )v = g + γ

(4.1)

∞ has a solution v ∈ CD (X). Also, γ = 0 if and only if g ⊥ Ker(P0 − λ0 ). The solution v is unique up to adding an element of Ker(P0 − λ0 ).

Note that the lemma is true for any elliptic, self-adjoint elliptic operator on a compact manifold with boundary. Proof. By standard elliptic theory, self-adjointness of P0 in L2 (X) and elliptic regularity imply the orthogonal decomposition C ∞ (X) = Ran(P0 − λ0 ) ⊕ Ker(P0 −  λ0 ). This implies the lemma. 4.2.3. Inductive construction of quasimodes. Initial step We want to solve (P − λ)u ∈ hC ∞ (M ).

(4.2)

By the leading part and model operator lemma this is equivalent to [(P − λ)u]X = 0 and then to (P0 − λ0 )uX = 0. Therefore we choose λ0 = an eigenvalue of P0 u0 = a corresponding eigenfunction

SCALES, BLOW-UP AND QUASIMODE CONSTRUCTIONS

237

then any u having uX = u0 will solve (4.2). For simplicity we make the29 (4.3)

Assumption: the eigenspace Ker(P0 − λ0 ) is one-dimensional.

Inductive step Inductive step lemma (regular perturbation). Let λ0 , u0 be chosen ∞ (M ) as in the initial step, and let k ≥ 1. Suppose λ ∈ C ∞ (R+ ), u ∈ CD satisfy (P − λ)u ∈ hk C ∞ (M ) ∞ and λ(0) = λ0 , uX = u0 . Then there are μ ∈ R, v ∈ CD (M ) so that

˜ u ∈ hk+1 C ∞ (M ) (P − λ)˜ ˜ = λ + hk μ, u for λ ˜ = u + hk v. The number μ is unique, and vX is unique up to adding constant multiples of u0 . More precisely, μ and vX (modulo Ru0 ) are uniquely determined by λ0 , u0 and the leading part of h−k (P − λ)u. ˜ = λ + hk μ, u Proof. Writing (P − λ)u = hk f and λ ˜ = u + hk v we have ˜ u = hk [f − μu + (P − λ)v − hk μv] (P − λ)˜ This is in hk+1 C ∞ (M ) if and only if the term in brackets is in hC ∞ (M ), which by the leading part and model operator lemma (and by k ≥ 1) is equivalent to fX − μuX + (P0 − λ0 )vX = 0, i.e. (using uX = u0 ) to (4.4)

(P0 − λ0 )vX = −fX + μu0 . This equation can be solved for μ, vX by applying Lemma 4.2 to g = −fX , since Ker(P0 − λ0 ) = {μu0 : μ ∈ R} by (4.3). Having vX we extend it to a smooth function v on M . Lemma 4.2 also gives the uniqueness of μ and  the uniqueness of vX modulo multiples of u0 .

The initial and inductive steps give eigenvalues and quasimodes to any order hN , and this is good enough for all purposes. It is still nice to go to the limit and also consider uniqueness. We get the final result: Theorem 4.3 (quasimodes for regular perturbation). Assume the setup of a regular perturbation as described in Section 4.1. Given a simple eigenvalue λ0 and ∞ associated eigenfunction u0 of P0 , there are λ ∈ C ∞ (R+ ), u ∈ CD (M ) satisfying (P − λ)u ∈ h∞ C ∞ (M ) and λ(0) = λ0 ,

u X = u0 .

Furthermore, λ and u are unique in Taylor series at h = 0, up to replacing u by a(h)u where a is smooth and a(0) = 1. Clearly, u cannot be unique beyond what is stated. 29 The method can be adjusted to the case dim Ker(P − λ ) > 1. The main difference is that 0 0 generically, not every eigenfunction u0 of P0 will arise as a limit of quasimodes uh with h > 0.

238

DANIEL GRIESER

Proof. Let u(k) , λ(k) be as obtained in the initial step (if k = 0) or the inductive step (if k ≥ 1), respectively. Then u(k+1) = u(k) + O(hk ), λ(k+1) = λ(k) + O(hk ) for all k by construction, so by asymptotic summation (Borel Lemma, cf. Lemma 2.10) we obtain λ, u as desired. To prove uniqueness, we show inductively that for λ, λ and u, u having the same leading terms, the assumptions (P −λ)u ∈ hk C ∞ (M ), (P −λ )u ∈ hk C ∞ (M ) ∞ (M ) for a smooth function imply that λ − λ = O(hk ) and u − a(k) (h)u ∈ hk CD a(k) , a(k) (0) = 1. For k = 1 there is nothing to prove. Suppose the claim is true for k, and let (P − λ)u ∈ hk+1 C ∞ (M ), (P − λ )u ∈ hk+1 C ∞ (M ). By the inductive hypoth∞ (M ). Since the leading esis, we have λ − λ = O(hk ) and u − a(k) (h)u ∈ hk CD −k −k   terms of h (P − λ)u and h (P − λ )u both vanish, the uniqueness statement in the inductive step lemma implies that λ − λ = O(hk+1 ) and u − a(k) (h)u − ∞ (M ) for some c ∈ R. Then a(k+1) (h) = a(k) (h) + chk satisfies chk u0 ∈ hk+1 CD ∞ (M ). Now define a from the a(k) by asymptotic summau − a(k+1) (h)u ∈ hk+1 CD tion.  4.3. Explicit formulas. The proof of Theorem 4.3 is constructive: it gives a method for finding u(x, h) and λ(h) to any order in h, under the assumption that the model problem (4.1) can be solved. We present two standard alternative ways of doing the calculation. We use the maps Φh : X → Ωh , see (A) in Section 4.1, where X = Ω0 . In fact, only the restriction of Φh to ∂D X is needed, as will be clear from the first method presented below. ˙ where the dot denotes the 4.3.1. Boundary perturbation. We will compute λ, first derivative in h at h = 0. This is the first order perturbation term since λ(h) = λ(0) + hλ˙ + O(h2 ). Differentiating the equation (P − λ)u ∈ h2 C ∞ (M ) in h at h = 0 we obtain ˙ 0 + (P0 − λ0 )u˙ = 0 . (4.5) (P˙ − λ)u In our case P˙ = 0. The boundary condition is u(x, h) = 0 for all x ∈ ∂D Ωh and all h, so u(Φh (y), h) = 0 for y ∈ ∂D X. Differentiating in h yields the boundary condition for u: ˙ V u0 + u˙ = 0 on ∂D X where V u0 is the derivative of u0 in the direction of the vector field V = (∂h Φh )|h=0 . Now take the L2 (X) scalar product of (4.5) with u0 . We write the second summand using Green’s formula as  ˙ u0 = (−∂n u˙ · u0 + u˙ · ∂n u0 ) dS + u, ˙ (P0 − λ0 )u0

(P0 − λ0 )u, ∂D X

where ∂n denotes the outward normal derivative. Using u0|∂D X = 0, (P0 −λ0 )u0 = 0 we obtain  ˙λ = − 1 V u0 · ∂n u0 dS u0 2 ∂D X where u0  is the L2 (X)-norm of u0 . Commonly one chooses Φh so that V = a∂n for a function a on ∂D X. This means that the boundary is perturbed in the vertical direction at velocity a. For L2 -normalized u0 this yields Hadamard’s formula (see [23]) λ˙ = − ∂D X a(∂n u0 )2 dS. Higher order terms are computed in a similar way.

SCALES, BLOW-UP AND QUASIMODE CONSTRUCTIONS

239

Note that we did not need to solve the model problem. Its solution is only needed to compute u˙ or higher derivatives of λ and u. 4.3.2. Taylor series ansatz. Here is a different method where in a first step all operators are transferred to the h-independent space X. Using the maps Φh : X → Ωh pull back the operator Ph to X: Ph = Φ∗h Ph . Now Ph is a smooth family of elliptic operators on X, so we can write Ph ∼ P0 + hP1 + . . . Here P0 is the Laplacian on X since Φ0 is the identity. We also make the ansatz u ∼ u0 + hu1 + . . . ,

λ ∼ λ0 + hλ1 + . . .

∞ (X), multiply out the left side of where all ui ∈ CD

(P0 + hP1 + · · · − λ0 − hλ1 − . . . )(u0 + hu1 + . . . ) ∼ 0, order by powers of h and equate each coefficient to zero. The h0 term gives the initial equation (P0 − λ0 )u0 = 0

(4.6)

and the h term, k ≥ 1, gives the recursive set of equations k

(4.7)

(P0 − λ0 )uk = −(P1 − λ1 )uk−1 − · · · − (Pk − λk )u0 =: −fk + λk u0

where fk is determined by u0 , . . . , uk−1 and λ0 , . . . , λk−1 . This is the decomposition of Lemma 4.2 for g = fk , so it can be solved for λk , uk . We can solve (4.7) explicitly as follows: Taking the scalar product with u0 and using (P0 − λ0 )uk , u0 = uk , (P0 − λ0 )u0 = 0 we get (4.8)

λk =

fk , u0

, u0 2

for example P1 u0 , u0

, u0 2 Here u1 , u2 etc. are computed as λ1 =

λ2 =

(P1 − λ1 )u1 + P2 u0 , u0

u0 2

uk = (P0 − λ0 )−1 (−fk + λk u0 ) where (P0 − λ0 )−1 is a generalized inverse of P0 − λ0 , i.e. a left inverse defined on Ran(P0 − λ0 ). The choice (4.8) of λk guarantees that −fk + λk u0 ∈ Ran(P0 − λ0 ). Remark 4.4. This method seems simpler and more effective than the one presented in Section 4.2. However, in the context of singular perturbations, where several model problems occur, it will pay off to have a geometric view and not to have to write down asymptotic expansions. The relation between these two methods becomes clearer if we formulate the  present one in terms of the operator P  on the space Ω = X × R+ . The product   structure of Ω allows us to extend functions on X to functions on Ω in a canonical way (namely, constant in h). This yields the explicit formulas. In comparison, for Ω there is no such canonical extension.

240

DANIEL GRIESER

4.4. Generalizations. Theorem 4.3 generalizes to any smooth family of uniformly elliptic operators Ph with elliptic boundary conditions on a compact manifold with boundary, supposing P0 is self-adjoint. Note that Ph for h > 0 need not be self-adjoint. If Ph has complex coefficients then u and λ will be complex valued, and if all Ph are self-adjoint then λ can be chosen real-valued. The method in Subsection 4.3.2 can be formulated abstractly for any family of operators Ph on a Hilbert space which has a regular Taylor expansion in h as h → 0. Using contour integration one may find the asymptotics of eigenfunctions and eigenvalues, not just quasimodes, directly and show that they vary smoothly in the parameter h under the simplicity assumption (4.3). See [34]. 5. Adiabatic limit with constant fibre eigenvalue The adiabatic limit30 is a basic type of singular perturbation which will be part of all settings considered later. Its simplest instance is the Laplacian on the family of domains Ωh = (0, 1) × (0, h) ⊂ R2 .

(5.1)

Since the domain of the variable y is (0, h) it is natural to use the variable Y = (0, 1) instead. Then (5.2)

y h

∈

Δ = ∂x2 + ∂y2 = h−2 ∂Y2 + ∂x2 .

Although it is not strictly needed for understanding the calculations below, we explain how this is related to blow-up, ? in order to prepare for later generalizations: The closure of the total space Ω = h>0 Ωh × {h} ⊂ R2 × R+ has a singularity (an edge) at h = 0.31 This singularity can be resolved by blowing up the x-axis L = {y = h = 0} in R2 × R+ . If β : [R2 × R+ , L] → R2 × R+ is the blow-down map then the lift M = β∗Ω is contained in the domain of the projective coordinates system x, Y = hy , h, compare Figure 8(d). In these coordinates the set M is given by x ∈ [0, 1], Y ∈ [0, 1], h ∈ R+ . See Figure 12. Note that the operators Δ turn into the ‘singular’ family of operators (5.2) on M . This example, and the generalization needed in Section 6, motivates considering the following setting. See Section 5.4 for more examples where this setup occurs. 5.1. Setup. Suppose B, F are compact manifolds, possibly with boundary. For the purpose of this article you may simply take B, F to be closed intervals (but see Subsection 5.5 for a generalization needed later). We consider a family of differential operators depending on h > 0 (5.3)

P (h) ∼ h−2 PF + P0 + hP1 + . . .

on A = B × F . We assume (5.4) 30 The

PF is a self-adjoint elliptic operator on F

word adiabatic originally refers to physical systems that change slowly. In their quantum mechanical description structures similar to the ones described here occur, where x corresponds to time and h−1 to the time scale of unit changes of the system. This motivated the use of the word adiabatic limit in global analysis in this context. 31 The precise meaning of this is that Ω is not a d-submanifold of R2 × R , as defined before + Definition 2.14. This is what distinguishes it from a regular perturbation. Note that Ω happens to be a submanifold with corners of R3 , but this is irrelevant here.

SCALES, BLOW-UP AND QUASIMODE CONSTRUCTIONS

y

h

241

h β

← − h

x

x

x A

y Y Ωh

M

Ω

Figure 12. Domain Ωh , total space Ω and resolution M of Ω for adiabatic limit. Compare Figure 8(d). On the right only the part of the blown-up space [R2 × R+ , {y = h = 0}] where the ‘top’ projective coordinates h, Y = hy are defined is shown. Dotted lines are fibres of the natural fibration of the front face A. where boundary conditions are imposed if F has boundary.32 For example, if F = [0, 1] then we could take PF = −∂Y2 with Dirichlet boundary conditions. P0 , P1 , . . . are differential operators on A. A condition on P0 will be imposed below, see equations (5.11), (5.12). One should think of A as the union of?the fibres (preimages of points) of the {x} × F , see also Remark 5.6 below. projection π : A = B × F → B, i.e. A = x∈B

We call F the fibre and B the base. The analysis below generalizes to the case of fibre bundles A → B, see Section 5.5. The letter A is used for ‘adiabatic limit’. We will denote coordinates on B by x and on F by Y . This may seem strange but serves to unify notation over the whole article, since this notation is natural in the following sections. 5.2. What to expect: the product case. To get an idea what happens, we consider the case of a product operator, i.e. P (h) = h−2 PF + PB where PB , PF are second order elliptic operators on B and F , self-adjoint with given boundary conditions. An example is (5.1), (5.2) where B = F = [0, 1] and PB = −∂x2 , PF = −∂Y2 . More generally, PB , PF could be the Laplacians on compact Riemannian manifolds (B, gB ), (F, gF ). Then P would be the Laplacian on A with respect to the metric h2 gF ⊕ gB in which the lengths in F -direction are scaled down by the factor h. By separation of variables P (h) has the eigenvalues λk,l = h−2 λF,k +λB,l where λF,k , λB,l are the eigenvalues of PF , PB respectively, with eigenfunctions33 φk ⊗ ψl . Although we have solved the problem, we now rederive the result using formal expansions, in order to distill from it essential features that will appear in the general case. We make the ansatz u = u0 + hu1 + . . . , λ = h−2 λ−2 + . . . 32 Formally it would be more correct to write Id ⊗ P instead of P in (5.3), but here and B F F in the sequel we will use the simplified notation. 33 For functions φ : B → R and ψ : F → R we write φ ⊗ ψ : B × F → R, (x, Y ) → φ(x)ψ(Y ).

242

DANIEL GRIESER

and plug in (5.5) (h−2 PF + PB − h−2 λ−2 − h−1 λ−1 − λ0 − . . . )(u0 + hu1 + h2 u2 + . . . ) = 0. The h−2 term gives (PF − λ−2 )u0 = 0

(5.6)

so λ−2 must be an eigenvalue of PF . Suppose it is simple and let ψ be a normalized eigenfunction. It follows that u0 (x, Y ) = φ(x)ψ(Y ) for some yet unknown function φ. How can we find φ? The h−1 term gives (PF − λ−2 )u1 = λ−1 u0 . Taking the scalar product with u0 and using self-adjointness of PF we get λ−1 = 0. The h0 term then gives (PF − λ−2 )u2 = −(PB − λ0 )u0 . By Lemma 4.2, applied to PF for fixed x ∈ B, this has a solution u2 if and only if (5.7)

(PB − λ0 )u0 (x, ·) ⊥ ψ

in L2 (F ) for each x ∈ B.

Now the left side is [(PB − λ0 )φ(x)] ψ, so we get (PB − λ0 )φ = 0 on B. Thus, λ0 is an eigenvalue of PB with eigenfunction φ. This solves the problem since φ ⊗ ψ is clearly an eigenfunction of P (h) with eigenvalue h−2 λ−2 + λ0 . From these considerations, we see basic features of the adiabatic problem: • λ−2 is an eigenvalue of the fibre operator PF . • λ0 is an eigenvalue of the base operator PB . • The leading term of the eigenfunction, u0 , is the tensor product of the eigenfunctions on fibre and base. It is determined from the two ‘levels’, h−2 and h0 of (5.5). For a general operator (5.3) we cannot separate variables since P0 (and the higher Pi ) may involve Y -derivatives (or Y -dependent coefficients). However, the ‘adiabatic’ structure of P (h) still allows separation of variables to leading order: The h−2 term of (5.5) still yields u0 = φ ⊗ ψ and the h−1 term yields λ−1 = 0. The h0 term now yields condition (5.7) with PB replaced by P0 . This shows that φ must be an eigenfunction of the operator U → (Π ◦ P0 )(U ⊗ ψ) where Πu = u, ψ F is the L (F ) scalar product with ψ. This motivates the definition of the horizontal operator PB below. 2

5.3. Solution. The solution of the formal expansion equation (5.5) is complicated by the fact that a single ui is only determined using several hk . It is desirable to avoid this, in order to easily progress to more complex problems afterwards. Thus, we need a procedure where consideration of a fixed hk gives full information on the corresponding next term in the u expansion. This can be achieved by redefining the function space containing the remainders f = (P − λ)u in the iteration, as well as their notion of leading part. As before, we consider a family (uh )h≥0 of functions on B × F as one function on the total space M = B × F × R+

SCALES, BLOW-UP AND QUASIMODE CONSTRUCTIONS

243

and consider a differential operator P acting on functions on M and having an expansion as in (5.3). Let A := B × F × {0} be the boundary at h = 0 of M . 5.3.1. A priori step: Fixing a vertical mode. The horizontal operator. A priori we fix λ−2 = a simple eigenvalue of PF (5.8) ψ = an L2 (F )-normalized corresponding eigenfunction. We will seek (quasi-)eigenvalues of P of the form h−2 λ−2 + C ∞ (R+ ). Every f ∈ C ∞ (F ) may be decomposed into a ψ component and a component perpendicular to ψ: f = f, ψ F ψ + f ⊥ ,

(5.9)

f ⊥ ⊥F ψ

where , F is the L2 (F ) scalar product. The same formula defines a fibrewise decomposition of f in C ∞ (A) or in hk C ∞ (M ), k ∈ Z. The coefficient of ψ defines projections Π : C ∞ (A) → C ∞ (B) f → f, ψ F Π : hk C ∞ (M ) → hk C ∞ (B × R+ ) By self-adjointness of PF Π ◦ (PF − λ−2 ) = 0

(5.10)

on the domain of PF . Motivated by the consideration at the end of the previous section we define the horizontal operator34 (5.11)

∞ PB : CD (B) → C ∞ (B),

U → ΠP0 (U ⊗ ψ).

We can now formulate the assumption on P0 : (5.12)

PB is a self-adjoint elliptic differential operator on B

where self-adjointness is with respect to some fixed density on B and given boundary conditions. This notation is consistent with the use of PB in the product case. 5.3.2. Function spaces, leading part and model operator. We will seek quasi∞ (M ), the space of smooth functions on M modes u in the solution space CD ∞ (M ) is defined satisfying the boundary conditions. The leading part of u ∈ CD to be ∞ (A). uA := u|h=0 ∈ CD The following definition captures the essential properties of the remainders f = (P − λ)u arising in the iteration. Definition 5.1. The remainder space for the adiabatic limit is R(M ) := {f ∈ h−2 C ∞ (M ) : Πf is smooth at h = 0} = {f = h−2 f−2 + h−1 f−1 + · · · : Πf−2 = Πf−1 = 0}. The leading part of f ∈ R(M ), f = h−2 f−2 + h−1 f−1 + . . . is35   f−2 fAB := ∈ C ∞ (A)Π⊥ ⊕ C ∞ (B) Πf0 34 P B is also called the effective Hamiltonian, e.g. in [56]. 35 The notation f AB is meant to indicate that the leading

functions on A and on B.

part has components which are

244

DANIEL GRIESER

where C ∞ (A)Π⊥ := {v ∈ C ∞ (A) : Πv = 0} . For functions in the solution space we clearly have: ∞ ∞ Let u ∈ CD (M ). Then u ∈ hCD (M ) ⇐⇒ uA = 0 .

The definition of the leading part of f ∈ R(M ) is designed to make the corresponding fact for f true: Leading part and model operator lemma (adiabatic limit). a) If f ∈ R(M ) then f ∈ hR(M ) if and only if fAB = 0. b) For λ ∈ h

−2

λ−2 + C ∞ (R+ ) we have ∞ (M ) → R(M ) P − λ : CD

(5.13) and

[(P − λ)u]AB

(5.14)

  (PF − λ−2 )uA = Π(P0 − λ0 )uA

where λ0 is the constant term of λ.   PF − λ−2 is called the model operator for The operator (P − λ)A := Π(P0 − λ0 ) P − λ at A. Proof. a) Let f = h−2 f−2 + h−1 f−1 + f0 + . . . with Πf−2 = Πf−1 = 0. Suppose fAB = 0, so f−2 = 0 and Πf0 = 0. Then f = h−1 f−1 + f0 + O(h) with Πf−1 = Πf0 = 0, so f ∈ hR(M ). The converse is obvious. ∞ (M ) then (P − λ)u = h−2 (PF − λ−2 )u + (P0 − λ0 )u + O(h) is b) If u ∈ CD in R(M ) by (5.10), and then the definition of leading part implies (5.14).  Remark 5.2. In the uniform notation of Section 3, see (3.2), (3.3), we have ∞ ∞ ∂0 M = A and E(M ) = CD (M ), E(∂0 M ) = CD (A), R(M ) is defined in Definition ∞ ∞ 5.1, R(∂0 M ) = C (A)Π⊥ ⊕ C (B), and LP(u) = uA , LP(f ) = fAB , (P − λ)0 = (P − λ)A . 5.3.3. Analytic input for model operator. For the iterative construction of quasimodes we need the solution properties of the model operator, analogous to Lemma 4.2. The main additional input is the triangular structure of the model operator, equation (5.16) below. By definition ∞ (A) → C ∞ (A)Π⊥ ⊕ C ∞ (B) (P − λ)A : CD ∞ (A) into In the proof below it will be important to decompose functions v ∈ CD ⊥ their fibrewise Π and Π components. More precisely, the decomposition (5.9) defines an isomorphism (5.15) C ∞ (A) ∼ = C ∞ (A)Π⊥ ⊕ C ∞ (B) , v → (v ⊥ , Πv) D

so that v = v ⊥ + (Πv) ⊗ ψ.

D

D

SCALES, BLOW-UP AND QUASIMODE CONSTRUCTIONS

245

Lemma 5.3. Let P be an operator on M having an expansion as in (5.3), (5.4), ∞ (F ) satisfy (5.8), (5.12). and assume P and λ−2 ∈ R, ψ ∈ CD ∞ Then for each g ∈ C (A)Π⊥ ⊕ C ∞ (B) and λ0 ∈ R there is a unique γ ∈ ∞ (B) so that the equation Ker(PB − λ0 ) ⊂ CD   0 (P − λ)A v = g + γ ∞ has a solution v ∈ CD (A). This solution is unique up to adding w ⊗ ψ where w ∈ Ker(PB − λ0 ). ∞ (M ) as in (5.15). Then (PF − λ−2 )v = (PF − Proof. Decompose v ∈ CD ⊥ λ−2 )v and Π(P0 − λ0 )v = ΠP0 v ⊥ + (PB − λ0 )Πv since Πv ⊥ = 0 and by definition of PB . Therefore, we may write (P − λ)A as a 2 × 2 matrix: ⎞ ⎛ ∞ C ∞ (A)Π⊥ 0 (A)Π⊥ CD PF − λ−2 ⎠ ⎝ ⊕ ⊕ → : (5.16) (P − λ)A = ∞ CD C ∞ (B) P B − λ0 (B) ΠP0    ⊥  ⊥ 0 v g we write v = In order to solve (P − λ)A v = g + and g = and get γ vΠ gΠ the system

(PF − λ−2 )v ⊥ = g ⊥ ΠP0 v ⊥ + (PB − λ0 )vΠ = gΠ + γ The first equation has a unique solution v ⊥ by Lemma 4.2 applied to PF . Then by Lemma 4.2 applied to PB , there is a unique γ ∈ Ker(PB − λ0 ) so that the second  equation has a solution vΠ , and vΠ is unique modulo Ker(PB − λ0 ). 5.3.4. Inductive construction of quasimodes. We now set up the iteration. Initial step: We want to solve (5.17)

(P − λ)u ∈ hR(M ). By the leading part and model operator lemma this is equivalent to [(P − λ)u]AB = 0 and then to (PF − λ−2 )uA = 0,

Π(P0 − λ0 )uA = 0.

By (5.8) the first equation implies uA = φ ⊗ ψ for some function φ on B, and then the second equation is equivalent to (PB − λ0 )φ = 0 by (5.11), so if we choose λ0 = an eigenvalue of PB φ = a corresponding eigenfunction of PB then any u having uA = φ ⊗ ψ satisfies (5.17). Again, we make the (5.18)

Assumption: the eigenvalue λ0 of PB is simple From now on, we fix the following data: λ−2 , λ0 ∈ R,

∞ u0 := φ ⊗ ψ ∈ CD (A).

246

DANIEL GRIESER

Inductive step: Inductive step lemma (adiabatic limit). Let λ−2 , λ0 and u0 be as ∞ (M ) satisfy above, and let k ≥ 1. Suppose λ ∈ h−2 C ∞ (R+ ), u ∈ CD (P − λ)u ∈ hk R(M ) ∞ (M ) and λ = h−2 λ−2 +λ0 +O(h), uA = u0 . Then there are μ ∈ R, v ∈ CD so that ˜ u ∈ hk+1 R(M ) (P − λ)˜ k ˜ = λ + h μ, u ˜ = u + hk v. The number μ is unique, and vA is unique for λ up to adding constant multiples of u0 .

˜ = λ + hk μ, u Proof. Writing (P − λ)u = hk f and λ ˜ = u + hk v we have ˜ u = hk [f − μu + (P − λ)v − hk μv] (P − λ)˜ This is in hk+1 R(M ) if and only if the term in brackets is in hR(M ), which by the initial step and model operator lemma is equivalent to [f − μu + (P − λ)v]AB = 0 and then to   0 (P − λ)A vA = −fAB + μφ where we used (h2 u)h=0 = 0 and ΠuA = Πu0 = φ. Now Lemma 5.3 gives the result.  We obtain the following theorem. Theorem 5.4 (quasimodes for adiabatic limit). Suppose the operator P in (5.3) satisfies (5.4) and (5.12), where PB is defined in (5.11). Given simple eigenvalues λ−2 , λ0 of PF , PB with eigenfunctions ψ, φ respectively, there are λ ∈ h−2 C ∞ (R+ ), ∞ (M ) satisfying u ∈ CD (P − λ)u ∈ h∞ C ∞ (M ) and λ = h−2 λ−2 + λ0 + O(h), uA = φ ⊗ ψ . Furthermore, λ and u are unique in Taylor series at h = 0, up to replacing u by a(h)u where a is smooth and a(0) = 1. Proof. This follows from the initial and inductive step as in the proof of Theorem 4.3.  Remark 5.5 (Quasimodes vs. modes). This construction works for any simple eigenvalues λ−2 , λ0 of PF , PB respectively. However, when we ask whether a quasimode (λ, u) is close (for small h) to an actual eigenvalue/eigenfunction pair we need to be careful: while λ will still be close to a true eigenvalue, u may not be close to an eigenfunction unless λ−2 is the smallest eigenvalue of PF (‘first vertical mode’). This is in contrast to the case of a regular perturbation where this problem does not arise. The reason is that closeness of u to an eigenfunction can only be proved (and in general is only true) if we have some a priori knowledge of a spectral gap, i.e. separation of eigenvalues. Such a separation is guaranteed for small h only for the smallest λ−2 . For example, in the case of intervals B = F = [0, π] we have

SCALES, BLOW-UP AND QUASIMODE CONSTRUCTIONS

247

eigenvalues λl,m = h−2 l2 +m2 , k, l ∈ N. Then for each m there are hi → 0 and mi ∈ N so that λ2,m = λ1,mi for each i. Then besides u2,m also au2,m + bu1,m , a, b ∈ R are eigenfunctions for these eigenvalues, and in fact under small perturbations (i.e. if P1 = 0) only the latter type may ‘survive’. If one fixes k and considers the kth eigenvalue λk (h) of Ωh then, for sufficiently small h, it will automatically correspond to the first vertical mode. This is clear for the rectangle but follows in general from the arguments that show that such a quasimode is close to an eigenfunction. Remark 5.6 (Why fibres?). Why is it natural to think of the subsets Fx := {x} × F of A = B × F as ‘fibres’ (and not the sets B × {Y }, for example)? The reason is that these sets are inherently distinguished by the operator P : if u is a smooth function on M = A × R+ then P u is generally of order h−2 . But it is bounded as h → 0 if and only if u and ∂h u are constant on each set Fx . Put invariantly, P determines the fibres Fx to second order at the boundary h = 0. In the geometric setup of the problem, which is sketched in Figure 12, the fibres arise naturally as fibres (i.e. preimages of points) of the blow-down map β restricted to the front face. Exercise 5.7. Find a formula for the first non-trivial perturbation term λ1 . 5.4. Examples. We already looked at the trivial example of a rectangle. A non-trivial example will be given in Section 6. Tubes around curves provide another interesting example: Let γ : I → R2 be a smooth simple curve in the plane parametrized by arc length, where I ⊂ R is a compact interval. The tube of width h > 0 around γ is Th = {γ(x) + hY n(x) : x ∈ I, Y ∈ [− 12 , 12 ]} where n(x) is a unit normal at γ(x). For h small the given parametrization is a diffeomorphism, and in coordinates x, Y the euclidean metric on Th is a2 dx2 +h2 dY 2 where a(x, Y ) = 1 − hY κ(x) with κ the curvature of γ, so the Laplacian is Δ = a−1 ∂x a−1 ∂x +h−2 a−1 ∂Y a∂Y , which is selfadjoint for the measure adxdY . This does not have the desired form. However, the operator P = −a1/2 Δa−1/2 is unitarily equivalent to −Δ and self-adjoint in L2 (I × [− 12 , 12 ], dxdY ), and short calculation gives 1 P = −h−2 ∂Y2 − ∂x2 − κ2 + O(h). 4 Theorem 5.4 now yields quasimodes where λ−2 = π 2 k2 and λ0 is a Dirichlet eigenvalue of the operator −∂x2 − 14 κ2 on I. See [17] and [11] for details. In all previous examples (and also in the example of Section 6) the operators PF and P0 commute. Here is a simple example where this is not the case. Take B = F = [0, 1], PF = −∂Y2 and P0 = −∂x2 + b(x, Y ) for some smooth function b. Then 1 PB = −∂x2 + c(x) where c(x) = b(x, Y )ψ(Y ), ψ(Y ) F = 12 0 b(x, Y ) sin2 πY dY if λ−2 = π 2 is the lowest eigenvalue of PF . Here P0 commutes with PF iff b = b(x), and then c = b.

248

DANIEL GRIESER

y

x

Figure 13. Thin domain Ωh of variable thickness 5.5. Generalizations. Fibre bundles. The product B × F can be replaced by a fibre bundle π : A → B with base B and fibres Fx = π −1 (x). We assume P is given as in (5.3), where PF differentiates only in the fibre directions. That is, for each x ∈ B there is an operator PFx on the fibre Fx . We assume that PFx has the same eigenvalue λ−2 for each x ∈ B, with one-dimensional eigenspace Kx . Under this assumption there are no essential changes, mostly notational ones: The Kx form a line bundle K over B. Sections of K → B may be identified with functions on A which restricted to Fx are in Kx , for each x, so C ∞ (B, K) ⊂ C ∞ (A). The line bundle K → B may not have a global non-vanishing section (replacing ψ). We deal with this by replacing functions on B by sections of K → B. The projections C ∞ (Fx ) → Kx fit together to a map Π : C ∞ (M ) → C ∞ (B × R+ , K) and then

∞ PB = ΠP0 i : CD (B, K) → C ∞ (B, K) ∞ ∞ where i : CD (B, K) → CD (A) is the inclusion. We replace C ∞ (B) by C ∞ (B, K) ∞ ∞ (A), an eigensection of PB , everywhere. Then and φ ⊗ ψ by u0 ∈ CD (B, K) ⊂ CD the construction of formal eigenvalues and eigenfunctions works as before. The adiabatic limit for fibre bundles has been considered frequently in the global analysis literature, see for example [43], [6]. Multiplicities. The construction can be generalized to the case where λ−2 and λ0 are multiple eigenvalues. In the case of fibre bundles it is important that the multiplicity of λ−2 is independent of the base point, otherwise new analytic phenomena arise. Noncompact base. The base (or fibre) need not be compact as long as PB (resp. PF ) has compact resolvent (hence discrete spectrum) and the higher order (in h) terms of P behave well at infinity. For example, the case B = R with PB = −∂x2 + V (x) where V (x) → ∞ as |x| → ∞ arises in Section 6.

6. Adiabatic limit with variable fibre eigenvalue In this section we consider thin domains of variable thickness, see Figure 13. We will see that the nonconstancy of the thickness makes a big difference to the behavior of eigenfunctions and hence to the construction of quasimodes. However, using a suitable rescaling, reflected in the second blow-up in Figure 14, we can reduce the problem to the case considered in the previous section.

SCALES, BLOW-UP AND QUASIMODE CONSTRUCTIONS

249

We consider a family of domains Ωh ⊂ R2 defined as follows. Let I ⊂ R be a bounded open interval and a− , a+ : I → R be functions satisfying a− (x) < a+ (x) for all x ∈ I. Let Ωh = {(x, y) ∈ R2 : ha− (x) < y < ha+ (x), x ∈ I}

(6.1)

for h > 0. We assume that the height function a := a+ − a− has a unique, nondegenerate maximum, which we may assume to be at 0 ∈ I. More precisely (6.2)

for each ε > 0 there is a δ > 0 so that |x| > ε ⇒ a(x) < a(0) − δ, and a is smooth near 0 and a (0) < 0

The conditions in the second line sharpen the first condition near 0. See Section 6.3 for generalizations. As before, we want to construct quasimodes (λh , uh ) for the Laplacian on Ωh with Dirichlet boundary conditions, as h → 0. Our construction will apply to ‘low’ eigenvalues, see Remark 6.2 below. As in the previous section we rescale the y-variable to lie in a fixed interval, independent of x: Let (6.3)

Y =

y − ha− (x) ∈ (0, 1). ha(x)

The change of variables (x, y) → (x, Y ) transforms the vector fields ∂x , ∂y to36 ∂x  ∂x + b(x, Y )∂Y , ∂y 

b=

a a ∂Y =− − −Y ∂x a a

∂Y ∂Y = h−1 a−1 ∂Y ∂y

Therefore

Δ = h−2 a−2 ∂Y2 + (∂x + b∂Y )2 This is reminiscent of the adiabatic limit considered in Section 5, but the fibre operator a−2 ∂Y2 has first eigenvalue π 2 a(x)−2 depending on x, so the analysis developed there is not directly applicable. We deal with this by expanding around x = 0 and rescaling the x-variable. 6.1. Heuristics: Finding the relevant scale. The assumption a (0) < 0 implies that the Taylor series of a−2 around 0 is (6.4)

a−2 (x) ∼ c0 + c2 x2 + . . . ,

c0 > 0, c2 > 0

so (6.5)

Δ = c0 h−2 ∂Y2 + c2 h−2 x2 ∂Y2 + · · · + (∂x + b∂Y )2

near x = 0. Which behavior do we expect for the eigenfunctions with small eigenvalues, say the first? Such an eigenfunction u will minimize the Rayleigh-quotient R(u) =

−Δu, u

u2

36 This is common but terrible notation. For calculational purposes it helps to write (x , Y )  ∂ ∂ ∂ for the new coordinates, related to (x, y) via x = x and (6.3). Then ∂x = ∂x + ∂Y = ∂x ∂x ∂x ∂Y ∂ ∂ ∂ , Y )  by x to simplify notation. + b(x and similarly for . In the end replace x ∂x ∂Y ∂y

Put differently,  means push-forward under the map F (x, y) = (x, Y (x, y)).

250

DANIEL GRIESER

among functions satisfying Dirichlet boundary conditions. Let us see how the different terms in (6.5) contribute to R(u): • The h−2 ∂Y2 term contributes at least c0 π 2 h−2 , since −∂Y2 ψ, ψ [0,1] ≥ π 2 ψ2[0,1] for any ψ : [0, 1] → R having boundary values zero.37 • The h−2 x2 ∂Y2 term contributes a positive summand which is O(h−2 ), but can be much smaller if the eigenfunction is large only for x near zero. Specifically, if u concentrates near x = 0 on a scale of L, i.e. x u(x, Y ) ≈ φ( )ψ(Y ) L for a function φ on R that is rapidly decaying at infinity then this term will be of order h−2 L2 ˜ x ) for φ(ξ) ˜ since x2 φ( Lx ) = L2 φ( = ξ 2 φ(ξ) and φ˜ is bounded38 . If L → 0 L for h → 0 then this is much smaller than h−2 . • On the other hand, the ∂x2 term will be of order L−2 if u concentrates on a scale of L near x = 0. • The other terms are smaller. We can now determine the scale L (as function of h) for which the sum of the h−2 x2 ∂Y2 and ∂x2 terms is smallest: For fixed h the sum h−2 L2 + L−2 is smallest when h−2 L2 = L−2 (since the product of h−2 L2 and L−2 is constant), i.e. L = h1/2 . The expectation of concentration justifies using the Taylor expansions around x = 0. The heuristic considerations of this section are justified by the construction of quasimodes in the next section. 6.2. Solution by reduction to the adiabatic limit with constant fibre. The scaling considerations suggest to introduce the variable x (6.6) ξ = 1/2 h in (6.5). Expanding also b(x, Y ) in Taylor series around x = 0 and substituting x = ξh1/2 we obtain Δ∼h

(6.7)

−2

c0 ∂Y2

+h

−1

∞  2

2 2 ∂ ξ + ξ c 2 ∂Y + hj/2 Pj j=−1

where Pj are second order differential operators in ξ, Y whose coefficients are polynomial in ξ (of degree at most j + 4) and linear in Y . The right hand side of (6.7) is a formal series of differential operators which are defined for Y ∈ (0, 1) and ξ ∈ R. Now we may apply the constructions of Section 5, with F = [0, 1] and B = R. More precisely, −Δ = h−1 P where, with t = h1/2 , P ∼ t−2 PF + P0 + tP1 + . . . 37 This 38 It

is just the fact that the smallest eigenvalue of the Dirichlet Laplacian on [0, 1] is π 2 . x is useful to think of this as follows: ‘φ( L ) contributes only for x ≈ L, and then x2 ≈ L2 ’.

SCALES, BLOW-UP AND QUASIMODE CONSTRUCTIONS

h

251

h ←

←

∂D M

A x y

x

Z

Z

Y Ω

M0

M

Figure 14. Total space for adiabatic limit with variable thickness and its resolution with PF = −c0 ∂Y2 and P0 = −∂ξ2 − ξ 2 c2 ∂Y2 . These operators act on bounded functions satisfying Dirichlet boundary conditions at Y = 0 and Y = 1. Using the first eigenvalue, λ−2 = c0 π 2 , of PF we get the horizontal operator (see (5.11)) √ PB = −∂ξ2 + ω 2 ξ 2 , ω = c0 c2 π. This is the well-known quantum harmonic oscillator, with eigenvalues μm = ω(2m+ 1), m = 0, 1, 2, . . . , and eigenfunctions √ 2 1 (6.8) ψm (ξ) = Hm ( ωξ)e− 2 ωξ where Hm is the mth Hermite polynomial. The exponential decay of ψm as |ξ| → ∞ justifies a posteriori the scaling limit considerations above. It means that quasimodes concentrate on a strip around x = 0 whose width is of order h1/2 . By Theorem 5.4 in Section 5 the operator P has quasimodes sin πY ψm (ξ)+O(t). To get quasimodes for −Δ on Ωh we simply substitute the coordinates Y, ξ as in (6.3), (6.6). In addition, we should introduce a cutoff near the ends of the interval I so that Dirichlet boundary conditions are satisfied there. We state the result in terms of resolutions. Introducing the ? singular coordinates Y and ξ corresponds to a resolution of the total space Ω = h>0 Ωh × {h} by two blow-ups as shown in Figure 14: Ω ←− [Ω, {y = h = 0}] =: M0 ←− [M0 , {x = h = 0}]q =: M . The blow-up of Ω in the x axis corresponds to introducing Y , as in Section 5, and results in the space M0 . The quasihomogeneous blow-up (see Subsection 2.3.6) of M0 in the Y -axis corresponds to introducing ξ = √xh . Compare Figure 9 (with y √ replaced by h): ξ and h = t, the variables used for the operator P , are precisely the ‘top’ projective coordinates defined away from the right face. Denote the total blow-down map by β : M → Ω. Each of the two blow-ups creates a boundary hypersurface of M at h = 0: the first blow-up creates Z, the second blow-up creates A (for?‘adiabatic’). In addition, M has the Dirichlet boundary ∂D M which is the lift of h>0 (∂Ωh ) × {h} ⊂ Ω. The essence of these blow-ups is that we can construct quasimodes as smooth functions on M .39 Their expansion at A is the one obtained using the analysis of 39 Of course this means that we construct quasimodes on Ω so that their pull-backs to M extend smoothly to the boundary of M .

252

DANIEL GRIESER

P . Since the quasimodes of P are exponentially decaying as ξ → ±∞, we may just take the zero expansion at Z (hence the letter Z). Summarizing, we obtain the following theorem. We denote √ ∞ (R+ ) = {μ : R+ → R : μ(h) = μ ˜( h) for some μ ˜ ∈ C ∞ (R+ )} . C1/2 Theorem 6.1 (quasimodes for adiabatic limit with variable fibre eigenvalue). Consider the family of domains Ωh defined in (6.1) and satisfying (6.2). Define ∞ ∞ (R+ ), um ∈ CD (M ) M as above. Then for each m ∈ N there are λm ∈ h−2 C1/2 satisfying (−Δ − λm )um ∈ h∞ C ∞ (M ) and λm ∼ c0 π 2 h−2 +

√

c0 c2 π(2m + 1)h−1 + O(h−1/2 )

um = sin πY ψm (ξ) at A,

um = 0 at Z

where c0 = a(0)−2 , c2 = −a (0)a(0)−1 . In addition, um vanishes to infinite order at Z. In the original coordinates on Ωh the conditions on um translate to  −N 2 y − ha− (x) x (6.9) um (h, x, y) = sin π ) ψm ( 1/2 ) + O(h1/2 1 + xh ha(x) h for all N . There is also a uniqueness statement similar to the one in Theorem 5.4. Proof. Choose a function um on M satisfying the following conditions: The expansion of um at the face A is given by the expansion for the quasimodes of P discussed above. The expansion of um at the face Z is identically zero; and um is zero at the Dirichlet boundary of M . Since ψm is exponentially decaying and all Pj have coefficients which are polynomial in ξ, all terms in the expansion at A are exponentially decaying as ξ → ∞. Since ξ = ∞ corresponds to the corner A ∩ Z, ∞ (M ) the matching conditions of the Borel Lemma 2.10 are satisfied, so um ∈ CD exists having the given expansions. Since both expansions satisfy the eigenvalue equation to infinite order, so does um . The extra decay factor in the error term 2 of um in (6.9) corresponds to the infinite order vanishing at Z, since xh defines Z near A ∩ Z, see Figure 9.  Remark 6.2. The scaling considerations depended on the assumption that u concentrates near x = 0 as h → 0, and this was justified a posteriori by Theorem 6.1. On the other hand, it can also be shown a priori using Agmon estimates that eigenfunctions for eigenvalues λk (h), where k is fixed as h → 0, behave in this way (and this can be used to prove closeness of quasimodes to eigenfunctions, see [1], [7] for example). Quasimodes can also be constructed for higher vertical modes, i.e. taking λ−2 = l2 c0 π 2 for any l ∈ N. However, the same caveat as in Remark 5.5 applies. Exercise 6.3. Compute the next term in the expansion of λm , i.e. the coefficient of h−1/2 .

SCALES, BLOW-UP AND QUASIMODE CONSTRUCTIONS

253

Y y 1 h 1

ΩL

x

1

x

Ωh

Ωh

Figure 15. Example of domains ΩL and Ωh , and rescaling after first blow-up

6.3. Generalizations. Degenerate maximum. A very similar procedure works if a has a finitely degenerate maximum, i.e. if the condition a (0) < 0 in (6.2) is replaced by (6.10)

a(j) (0) = 0 for j < 2p,

a(2p) (0) < 0

for some p ∈ N. The order is even by smoothness. The expansion (6.4) is replaced by a−2 (x) ∼ c0 + c2p x2p + . . . with c2p > 0, and then the correct scaling is found 1 1 from the equation h−2 L2p = L−2 , so L = h p+1 . So we set ξ = xt where t = h p+1 , then −Δ = t−2 P where P = t−2p (−c0 ∂Y2 ) + (−∂ξ2 − c2p ξ 2p ∂Y2 ) + tP1 + . . . The adiabatic limit analysis works just as well with t−2p as with t−2 in the leading term (do it!), and the eigenfunctions of the operator −∂ξ2 + ω 2 ξ 2p are still rapidly decaying at infinity, so we obtain 2 −2

λm ∼ c 0 π h

+

∞ 

j

djm h p+1

j=−2

and a similar statement for um . This problem with weaker regularity assumptions (and also allowing half-integer p in (6.10)) was analyzed in [12], by a different method. Several maxima. If the height function a has several isolated maxima then each one will contribute quasimodes. For instance, consider the case of two maxima at x = x1 and x = x2 , with ai = a(xi ), and let λk (h) be the kth eigenvalue of Ωh for fixed k ∈ N. If a1 > a2 then the leading term of the quasi-eigenvalue constructed at a1 is smaller than the one at a2 , and therefore the aymptotics of λk (h) as h → 0 is determined from the Taylor series of a around x1 , and the eigenfunction concentrates near x1 alone. On the other hand, if a1 = a2 then both maxima will generally contribute, and it is interesting to analyze their interaction (so-called tunnelling). A special case of this was analyzed in [53], and a detailed study of tunnelling for Schr¨odinger operators with potentials was carried out in [29] and [28]. Other approaches. A different, more operator-theoretic approach to the problem considered here (and more general ones, e.g. higher dimensions) is taken in [40], [22], [39], see also the book [56].

254

DANIEL GRIESER

7. Adiabatic limit with ends We consider the following problem, see Figure 15 left and center: Let ΩL ⊂ R2 be a bounded domain contained in the left half plane x < 0, having {0} × [0, 1] as part of its boundary. For h > 0 consider the domain (7.1)

Ωh = hΩL ∪ Rh ⊂ R2 ,

Rh = [0, 1) × (0, h)

i.e. a 1 × h rectangle with the ‘end’ ΩL , scaled down by the factor h, attached at its left boundary. To simplify notation we assume that ΩL is such that the boundary of Ωh is smooth, except for the right angles at the right end; however, this is irrelevant for the method. We denote coordinates on Ωh by x, y. We will construct quasimodes (λh , uh ) for the Laplacian Δh = ∂x2 + ∂y2 on Ωh , with Dirichlet boundary conditions, as h → 0. The central difficulty, and new aspect compared to the adiabatic limit, is the fact that there are two different scalings in the problem: • in the rectangular part of Ωh only the y-direction scales like h, • in the left end both x- and y-directions scale like h. This leads to different ways in which these two parts of Ωh influence eigenvalues and eigenfunctions. This is a simple case of a much more general setup arising in contexts such as surgery in global analysis and ‘fat graph’ analysis, see Section 7.4. The essential structures, however, already appear in this simple case. An explicit analysis using matched asymptotic expansions was carried out in [20]. We will rederive the quasimode expansions in a more conceptual way using the idea of resolutions. 7.1. Resolution. First, we construct a space on which we may hope the eigenfunctions (and quasimodes) to be smooth. We start with the total space on which these are functions, which is @ Ωh × {h} ⊂ R3 Ω= h>0

see the left picture in Figure 16. Really we want to consider the closure Ω since we are interested in the behavior of quasimodes as h → 0, compare Remark 2.8. This set is not a manifold with corners, let alone a d-submanifold of R2 × R+ (compare Footnote 31). At y = h = 0 the set Ω has an adiabatic limit type singularity as in the case of Section 5. In addition, it has a conical singularity (with singular base) at the point x = y = h = 0. So we blow up these two submanifolds of R3 and find the lift (see Definition 2.14) of Ω: The blow up of {y = h = 0} results in the space M0 in the center of Figure 16. Projective coordinates are x, Y = hy and h, globally on M0 since |y| ≤ Ch on Ω.40 The bottom face of M0 is h = 0, and the preimage of the point x = y = h = 0 is the bold face line x = h = 0 in M0 . So we blow up this line and define M to be the lift of M0 .41  make sense of the picture for M0 it may help to note that M0 is the closure of h>0 Ωh × y  {h} where Ωh = {(x, Y ) : (x, y) ∈ Ωh , Y = h } is depicted on the right in Figure 15. 41 You may wonder if we would have obtained a different space if had first blown up the point x = y = h = 0 and then the (lift of the) line y = h = 0. It can easily be checked that this results in the same space M – more precisely that the identity on the interiors of this space 40 To

SCALES, BLOW-UP AND QUASIMODE CONSTRUCTIONS

h

Ω

h

M0

←

255

M

←

∂D M S

x

y

x

A

Y πa B Figure 16. Total space and its resolution for adiabatic limit with ends, with fibration of the adiabatic face A; solid lines are codimension 2 corners of M , dashed or dotted lines are not 1 Y S

A

0 X 0

x ∞ 0

1

Figure 17. ‘Flattened’ picture of h = 0 boundary of resolved total space M , with coordinates for each face As always we will use x, y, h to denote the pull-backs of the coordinate functions x, y, h on Ω to M . Projective coordinate systems for the second blow-up give coordinates h, X = hx , Y on M \ A and hx , x, Y in a neighborhood of A. The space M has two types of boundary hypersurfaces: • The ‘Dirichlet boundary’ ∂D M , which corresponds to the boundary of Ωh . This is the union of the two ‘vertical’ faces in the right picture of Figure 16: < ; @ −1 ∂D M = β ∂Ωh × {h} h>0

where β : M → Ω is the total blow-down map. • The boundary at h = 0, ∂0 M = S ∪ A where A and S are the front faces of the two blow-ups, which meet in the corner S ∩ A.42 Our interest lies in the behavior of quasimodes at A and S. All functions will be smooth at the Dirichlet boundary. and of M extends to the boundary as a diffeomorphism. This also follows from the fact that {x = y = h = 0} ⊂ {y = h = 0} and a general theorem about commuting blow-ups, see [51]. 42 A is for adiabatic and S is for surgery, see Section 7.4 for an explanation.

256

DANIEL GRIESER

The faces A and S are rescaled limits of Ωh , see the discussion at the end of Section 2.4. The adiabatic face A is naturally a rectangle y A ≡ [0, 1] × [0, 1] with coordinates x and Y = . h It is the limit as h → 0 of {(x, hy ) : (x, y) ∈ Rh } – this is precisely what the blow-up means, in terms of projective coordinates. The Laplacian in these coordinates is Δ = h−2 ∂Y2 + ∂x2 . Thus, we have an adiabatic problem, with base B = [0, 1]x and fibre F = [0, 1]Y and PF = −∂Y2 , PB = −∂x2 . The corresponding projection is (7.2)

πA : UA → UB ,

(x, Y, h) → (x, h)

where UB = B × [0, ε) for some ε > 0 and UA is a neighborhood of A. There is a difference to the setup in Section 5 in that h is not a defining function for A. This leads to various issues below. The interior of the surgery face S can be identified with the plane domain Ω∞ obtained by taking h−1 Ωh and letting h → 0 (again, by definition of the blowup): y x (7.3) int(S) ≡ Ω∞ := ΩL ∪ ([0, ∞) × (0, 1)) with coordinates X = and Y = , h h and the Laplacian is 2 Δ = h−2 (∂X + ∂Y2 ). The corner S ∩ A is the interval [0, 1] and corresponds to x = 0 in A and to X = ∞ in S. Coordinates near the corner are x, defining S locally, and hx = X −1 , defining A locally and even globally. Note that the face A carries naturally a non-trivial fibration, compare Remark 5.6, but the face S does not: locally near any point of S no direction is distinguished. 7.2. Solution. The construction of quasimodes builds on the construction for the adiabatic limit in Section 5. The presence of the extra scale, i.e. the left end of Ωh , leads to a number of new features. To emphasize the relation with previous sections and motivated by the considerations above we will use the notation (7.4)

P = −Δ = −∂x2 − ∂y2 2 PF = −∂Y2 , PB = −∂x2 , PS = −∂X − ∂Y2

7.2.1. A priori step: Fixing the vertical mode. Since an adiabatic limit is involved, we fix a priori λ−2 = a simple eigenvalue of PF on [0, 1], with Dirichlet boundary conditions ψ = an L2 -normalized corresponding eigenfunction. Here we take the lowest fibre eigenvalue43 λ−2 = π 2 ,

ψ(Y ) =

√ 2 sin πY

We will seek (quasi-)eigenvalues of P of the form λ(h) ∈ h−2 λ−2 + C ∞ (R+ ). 43 One could also consider higher fibre modes, but this would change the analysis at S, see also Remark 5.5.

SCALES, BLOW-UP AND QUASIMODE CONSTRUCTIONS

257

7.2.2. Function spaces, leading parts and model operators. We want to define spaces E(M ) and R(M ) which will contain the eigenfunctions/quasimodes and remainders in the construction, respectively. Our resolution was chosen so that eigenfunctions have a chance of being smooth on M , so E(M ) ⊂ C ∞ (M ). Since hx is a defining function for A, functions u ∈ C ∞ (M ) have an expansion at A ∞  j  h u∼ u ˜j (x, Y ), u ˜j ∈ C ∞ (A). x j=0 In the sequel it will be convenient44 to write this as ∞  (7.5) u∼ hj uj (x, Y ) j=0 −j

where uj = x u ˜j . Note that uj may be not smooth at x = 0, i.e. at S ∩ A, even though u ∈ C ∞ (M ). We posit that quasimodes satisfy the stronger condition that uj be smooth on A (including S ∩ A) and define (7.6)

C ∞,tr (M ) = {u ∈ C ∞ (M ) : uj ∈ C ∞ (A) in the expansion (7.5)}

See Remark 7.5 below for an explanation why we expect quasimodes to satisfy this condition. This can be reformulated as a ‘triangular’ condition on the indices in the expansion at the corner S ∩ A:  j ∞  h ajl (Y ) xl near S ∩ A If u ∈ C ∞ (M ), u ∼ x (7.7) j,l=0 then u ∈ C ∞,tr (M ) ⇐⇒ (ajl = 0 ⇒ l ≥ j) . In addition, quasimodes should vanish at the Dirichlet boundary ∂D M . As before, we indicate this by the index D in the function spaces. For functions on the ∞ (A) is the space of faces A, S, S ∩ A we use a similar notation. For example, CD smooth functions on A vanishing on the Dirichlet boundary of A, which consists of the three sides x = 1, Y = 0, Y = 1. Definition 7.1. The space of quasimodes for the adiabatic limit with ends is defined as ∞,tr (M ) E(M ) = CD i.e. smooth functions on M satisfying Dirichlet boundary conditions and the triangular condition explained above. The leading parts of u ∈ E(M ) are defined as uS := u|S , uA := u|A . What are the restrictions of elements of E(M ) to ∂0 M = S ∪ A? Define (7.8) ∞,tr ∞ ∞ CD (S) := {us ∈ CD (S) : us = a(Y ) + O(X −∞ ) as X → ∞, ∈ CD (S ∩ A)} (7.9) ∞,tr ∞ E(∂0 M ) := {(us , ua ) : us ∈ CD (S), ua ∈ CD (A), us = ua at S ∩ A} Here we use the coordinate X on S. Recall that X −1 defines the face S ∩ A of S. 44 In

order to have P u ∼

 j

hj P uj . But note that h is not a defining function of A.

258

DANIEL GRIESER

Lemma 7.2 (leading parts of quasimodes, adiabatic limit with ends). If u ∈ E(M ) then (uS , uA ) ∈ E(∂0 M ). Conversely, given (us , ua ) ∈ E(∂0 M ) there is u ∈ E(M ) satisfying (uS , uA ) = (us , ua ), and u is unique modulo hE(M ). This could be formulated as existence of a short exact sequence: 0 → hE(M ) → E(M ) → E(∂0 M ) → 0

(7.10)

where the left map is inclusion and the right map is restriction. Proof. It is clear that the restrictions of u ∈ E(M ) to S, A are smooth and agree at S ∩ A. Write the expansion of u at the corner as in (7.7). The l = 0 terms give the expansion of uS at S ∩ A, i.e. as hx → 0. The only such term is j = 0, so ∞ uS = a00 (Y ) + O( hx ). From hx = X −1 we get (uS , uA ) ∈ E(∂0 M ). Given (us , ua ) ∈ E(∂0 M ) one constructs u ∈ E(M ) having this boundary data using the Borel Lemma 2.10, as follows. We write η = X −1 for the function defining A and suppress the Y -coordinate. Write ua (x) ∼ l≥0 a0l xl , x → 0. We choose u having complete expansions u(x, η) ∼ us (η) +

∞ 

a0l xl

as x → 0,

i.e. at S

as η → 0,

i.e. at A

l=1

u(x, η) ∼ ua (x)

∞ (with error O(η ∞ ) in the second case). Such a u ∈ CD (M ) exists by the Borel 0l Lemma – the matching conditions at x = η = 0 are satisfied since ∂a ∂η = 0 for all l. ∞,tr (M ). Also, the expansions satisfy the triangular condition in (7.7), hence u ∈ CD Finally, we need to show that if u ∈ E(M ), (uS , uA ) = 0 then u ∈ hE(M ). u for Now h = hx x is a total boundary defining function for {S, A}, so u = h˜ ∞ (M ) by Lemma 2.2. In the expansion (7.5) all uj are smooth and some u ˜ ∈ CD  ∞,tr ˜ = j≥1 hj−1 uj is in CD (M ).  u0 = uA = 0, so u

The definition of the remainder space combines the triangular condition with the remainder space for the adiabatic limit. First, the choice of λ−2 defines a projection type map related to the projection π : UA → UB , see (7.2), Π : C ∞ (M ) → C ∞ (UB ),

f → f|UA , ψ F .

Then (7.11)

Π ◦ (PF − λ−2 ) = 0

where this is defined. Definition 7.3. The remainder space for the adiabatic limit with ends is defined as R(M ) = {f ∈ h−2 C ∞,tr (M ) : Πf is smooth at B}. where B := B × {0} ⊂ UB . The leading parts of f ∈ R(M ) are   f−2,A 2 f−2,S := (h f )|S , fAB := Πf0,A where f−2,A = (h2 f )|A , Πf0,A = (Πf )|B .

SCALES, BLOW-UP AND QUASIMODE CONSTRUCTIONS

259

Thus, a function f ∈ h−2 C ∞ (M ) is in R(M ) iff it has an expansion f ∼ h f−2,A + h−1 f−1,A + f0,A + . . . at A with fj,A ∈ C ∞ (A) analogous to (7.5), and Πf−2,A = Πf−1,A = 0. This defines f−2,A and f0,A in the definition of fAB . Note that f ∈ R(M ) implies that −2

f−2,S ∈ C ∞,tr (S),

(7.12)

Πf−2,S = 0 at S ∩ A.

The first statement follows as in the proof of Lemma 7.2 and the second from Π(h2 f )A = 0 and (h2 f )A = (h2 f )S at S ∩ A. As usual, the remainder space and leading part definitions are justified by the following properties. Recall the definition of the operators P, PF , PB , PS in (7.4). Leading part and model operator lemma (adiabatic limit with ends). a) If f ∈ R(M ) then f ∈ hR(M ) if and only if f−2,S = 0, fAB = 0. b) For λ ∈ h−2 λ−2 + C ∞ (R+ ) we have P − λ : E(M ) → R(M ) and (7.13)

[(P − λ)u]−2,S = (P − λ)S uS

(7.14)

[(P − λ)u]AB = (P − λ)A uA

where (P − λ)S = PS − λ−2   PF − λ−2 where (P − λ)A = Π(PB − λ0 )

where λ0 is the constant term of λ. The operators (P − λ)S , (P − λ)A are called the model operators of P − λ at S and at A. There is also a short exact sequence like (7.10) for R(M ), but we need only what is stated as a). Proof. a) “⇒” is obvious. “⇐”: If f ∈ R(M ) then h2 f ∈ C ∞,tr (M ), and f−2,S = 0, fAB = 0 imply (h2 f )S = 0, (h2 f )A = 0, so Lemma 7.2 gives h2 f ∈ hC ∞,tr (M ), so f = hf˜ with f˜ ∈ h−2 C ∞,tr (M ). Furthermore, fAB = 0 implies f−2,A = 0 and Πf0,A = 0, so Πf˜ is smooth at B, hence f˜ ∈ R(M ). b) If u ∈ E(M ) then (P − λ)u = h−2 (PS − λ−2 + O(h2 ))u near S and (P − λ)u = h−2 (PF − λ−2 )u + (PB − λ0 )u + O(h) near A. This is clearly in h−2 C ∞,tr (M ), and even in R(M ) by (7.11). The definition of leading parts directly implies (7.13), (7.14).  7.2.3. Analytic input for model operators. At the face A, i.e. for the operators PF = −∂Y2 and PB = −∂x2 on F = [0, 1]Y resp. B = [0, 1]x , with Dirichlet boundary conditions, we have the standard elliptic solvability result, Lemma 4.2. The solvability properties of the model operator (P − λ)S are of a different nature, essentially since this operator has essential spectrum. Recall from (7.3) that the interior of S can be identified with the unbounded domain Ω∞ ⊂ R2 , see Figure 17. This set is the union of a compact set and an

260

DANIEL GRIESER

infinite strip, hence an example of a space with infinite cylindrical ends, and we can use the standard theory for such spaces. We assume: (7.15)

Non-resonance assumption: The resolvent z → (PS − z)−1 of the Laplacian PS on Ω∞ has no pole at z = λ−2 = π 2 .

It is well-known that this condition is equivalent to the non-existence of bounded solutions of (PS − λ−2 )v = 0, and also to the unique solvability of (PS − λ−2 )v = f for compactly supported f , with bounded v, see [50, Proposition 6.28]. Also, this condition is satisfied for convex sets Ω∞ , see [20, Lemma 7], and holds for generic ΩL . Lemma 7.4. Assume Ω∞ ⊂ R2 satisfies the non-resonance assumption (7.15). If fs ∈ C ∞,tr (S), Πfs = 0 at S ∩ A then the equation (7.16)

(PS − λ−2 )vs = fs

∞,tr has a unique bounded solution vs , and vs ∈ CD (S).

Proof. Uniqueness holds since (PS − λ−2 )v = 0 has no bounded solution. For existence, we first reduce to the case of compactly supported fs , then use the non-resonance assumption to get a bounded solution vs and then show that ∞,tr vs ∈ CD (S). The first and third step can √ be done by developing fs and vs for each 2 fixed X > 0 in eigenfunctions ψk (Y ) = 2 sin kπY ∞ of the ‘vertical’ operator −∂Y on [0, 1] with Dirichlet conditions: fs (X, Y ) = k=1 fk (X)ψk (Y ). Then (7.16) is d2 2 2 equivalent, in X > 0, to the ODEs (− dX 2 + μk )vk = fk where μk = (k − 1)π , and these can be analyzed explicitly. For example, if k > 1 and fk (X) = 0 for large X then any bounded solution vk must be exponentially decaying. For details see [20, Lemma 6 and Lemma 9 (with p = 0)].  Remark 7.5. This lemma explains why we expect the ‘triangular’ condition on the Taylor series of quasimodes: for compactly supported fs the solution vs lies in ∞,tr CD (S). This leads to the definition of E(∂0 M ). Then E(M ) must be defined so that the sequence (7.10) is exact. 7.2.4. Inductive construction of quasimodes. Initial step We want to solve (P − λ)u ∈ hR(M ),

u ∈ E(M ).

By the leading part and model operator lemma this means (7.17)

(PS − λ−2 )uS = 0

(7.18)

(PF − λ−2 )uA = 0

(7.19)

(PB − λ0 )ΠuA = 0 where we used that Π commutes with PB = −∂x2 . Also, (uS , uA ) ∈ E(∂0 M ), defined in (7.9). First, Lemma 7.4 implies uS = 0. Therefore uA = 0 at S ∩ A, hence uA satisfies Dirichlet boundary conditions at all four sides of the square A. Thus we have an adiabatic problem as treated in Section 5, so λ0 must be a Dirichlet eigenvalue of −∂x2 on B = [0, 1], i.e. √ λ0 = π 2 m2 , uA = u0 := φ ⊗ ψ, φ(x) = 2 sin πmx

SCALES, BLOW-UP AND QUASIMODE CONSTRUCTIONS

261

for some m ∈ N. Since B is one-dimensional, λ0 is a simple eigenvalue as required in (5.18). From now on we fix m, φ, λ−2 = π 2 , λ0 , and u0 = φ ⊗ ψ. Inductive step Inductive step lemma (adiabatic limit with ends). Let λ−2 , λ0 and u0 be chosen as above in the initial step, and let k ≥ 1. Suppose λ ∈ h−2 C ∞ (R+ ), u ∈ E(M ) satisfy (P − λ)u ∈ hk R(M ) and λ = h−2 λ−2 + λ0 + O(h) and uS = 0, uA = u0 . Then there are μ ∈ R, v ∈ E(M ) so that ˜ u ∈ hk+1 R(M ) (P − λ)˜ ˜ = λ + hk μ, u for λ ˜ = u + hk v. The number μ and the restriction vS are unique, and vA is unique up to adding constant multiples of u0 . Proof. Writing (P − λ)u = hk f , f ∈ R(M ) we have ˜ u = hk [f − μu + (P − λ)v − hk μv] (P − λ)˜ This is in hk+1 R(M ) if and only if the term in brackets is in hR(M ), which by the leading part and model operator lemma is equivalent to (7.20)

(PS − λ−2 )vS = −f−2,S

(7.21)

(PF − λ−2 )vA = −f−2,A

(7.22)

(PB − λ0 )ΠvA = −Πf0,A + μφ where we have used that (h2 u)S = 0, (h2 u)A = 0 and ΠuA = Πu0 = φ. We first solve at S: We have f ∈ R(M ), so by (7.12) we can apply ∞,tr Lemma 7.4 with fs = −f−2,S and obtain vS ∈ CD (S) solving (7.20). This determines in particular v|S∩A , i.e. the boundary value of vA at S ∩ A. Now at A we need to solve an adiabatic problem, but with an inhomogeneous boundary condition at S ∩ A. To this end we extend v|S∩A ∞ to v  ∈ CD (A). Writing vA = v  + v  we then need to find v  satisfying homogeneous boundary conditions also at S ∩ A, and solving (7.21), (7.22) with f−2,A modified to f−2,A + (PF − λ−2 )v  and f0,A modified to f0,A + (PB − λ0 )v  . This is an adiabatic problem, so Lemma 5.3 guarantees the existence of a solution v  . Note that since Π commutes with PB there is no offdiagonal term in (5.16) (where P0 = PB in current notation). The uniqueness follows directly from (7.20)-(7.22): The difference between two solutions v would satisfy the same equations with f = 0, so would have to vanish at S and therefore solve the adiabatic problem at A with homogeneous boundary condition at S ∩A, for which we have already  shown that μ is unique and vA is unique up to multiples of u0 .

262

DANIEL GRIESER

Now by the same arguments as for Theorem 4.3 we obtain from the initial and inductive steps: Theorem 7.6 (quasimodes for adiabatic limit with ends). Consider the family of domains Ωh defined in (7.1). Suppose the non-resonance assumption (7.15) is ∞,tr satisfied. Then for each m ∈ N there are λm ∈ h−2 C ∞ (R+ ), um ∈ CD (M ) satisfying (P − λm )um ∈ h∞ C ∞ (M ) and λm = h−2 π 2 + m2 π 2 + O(h) uA = 2 sin mπx sin πY , uS = 0 There is also a uniqueness statement similar to the one in Theorem 5.4. Remark 7.7 (Quasimodes vs. modes). It is shown in [20] that for convex Ωh all eigenfunctions are captured by this construction. That is, for each m ∈ N there is h0 > 0 so that for h < h0 the mth eigenvalue of Ωh is simple, and both eigenvalue and (suitably normalized) eigenfunction are approximated by λm , um with error O(h∞ ). However, if Ωh is not convex then there may be an additional finite number of eigenvalues not captured by this construction. Essentially, these arise from L2 -eigenvalues of the Laplacian on Ω∞ below the essential spectrum. See [16] and references there for a detailed discussion. 7.3. Explicit formulas. The inductive step yields a method for finding any number of terms in the expansions of λm and um as h → 0 in terms of solutions of the model problems. In [20] the next two terms for λm are computed: λm = h−2 π 2 + m2 π 2 (1 + ah)−2 + O(h3 ) where a > 0 is determined by the scattering theory of −Δ on Ω∞ at the infimum of the essential spectrum, which equals π 2 . More precisely, there is, up to scalar multiples, a unique polynomially bounded solution v of (Δ + π 2 )v = 0 on Ω∞ , and it has the form v(X, Y ) = (X + a) sin πY + O(e−X ) as X → ∞. This fixes a. Another description is a = 12 γ  (0) where γ(s) is the scattering phase at frequency π 2 + s2 . 7.4. Generalizations. The structure of Ωh may be described as ‘thin cylinder with end attached’. A natural general setup for this structure is obtained by replacing the Y -interval [0, 1] by a compact Riemannian manifold N , of dimension n − 1, and the end ΩL by another compact Riemannian manifold, of dimension n, which has an isometric copy of N as part of its boundary. One may also add another Riemannian manifold as right end. This is studied in global analysis (where it is sometimes called ‘analytic surgery’) as a tool to study the glueing behavior of spectral invariants, see [25], [26], [44] for example. Other degenerations which have been studied by similar methods include conic degeneration [21], [54], [55] and degeneration to a (fibred) cusp [2], [3]. Another generalization is to have several thin cylinders meeting in prescribed ways, so that in the limit h = 0 one obtains a graph-like structure instead of an interval. This is called a ‘fat graph’. For example, consider a finite graph embedded in Rn with straight edges, and let Ωh be the set of points of Rn having distance at

SCALES, BLOW-UP AND QUASIMODE CONSTRUCTIONS

263

most h from this graph. This was studied in detail in [10], [16] and [52], see also [17] for a discussion and many more references. The methods in these papers actually yield a stronger result: λm (h) is given by a power series in h which converges for small h, plus an exponentially small error term. 8. Summary of the quasimodes constructions We summarize the essential points of the quasimode constructions, continuing the outline given in Section 3. In the case of a regular perturbation we introduced the iterative setup that allowed us to reduce the quasimode construction to the solution of a model problem (Lemma 4.2). It involves spaces of quasimodes and remainders and notions of leading part. In this case these are simply smooth functions and their restriction to h = 0. For the adiabatic limit problem with constant fibre eigenvalue this needs to be refined: the different scaling in fibre and base directions requires a new definition of remainder space and leading part of remainders (Definition 5.1). The model operator combines fibre and horizontal operators, and its triangular structure with respect to the decomposition of functions in fibrewise λ−2 modes and other modes, Equation (5.16), enables us to solve the model problem. The adiabatic limit problem with variable fibre eigenvalue can be reduced to the previous case by expanding the fibre eigenvalue (as function on the base) around its maximum and by rescaling the base variable. This rescaling balances the leading non-constant term in the expansion of the eigenvalue with the leading term of the base operator. The rescaling is encoded geometrically by a blow-up of the total space. The adiabatic limit problem with ends carries the new feature of having two regions with different scaling behavior. Geometrically this corresponds to two boundary hypersurfaces, A and S, at h = 0 in the resolved total space. The model problem at A is the same as for the adiabatic limit with constant fibre eigenvalue. The model problem at S is a scattering problem, i.e. a spectral problem on a noncompact domain. The properties of the solutions of the scattering problem lead to the triangular condition on the Taylor series at the corner S ∩ A in the spaces of quasimodes and remainders. Once this setup is installed the construction proceeds in a straight-forward way as in the other cases. References [1] Shmuel Agmon, Bounds on exponential decay of eigenfunctions of Schr¨ odinger operators, Schr¨ odinger operators (Como, 1984), Lecture Notes in Math., vol. 1159, Springer, Berlin, 1985, pp. 1–38, DOI 10.1007/BFb0080331. MR824986 [2] Pierre Albin, Fr´ed´ eric Rochon, and David Sher. Analytic torsion and R-torsion of Witt representations on manifolds with cusps. Preprint math.DG/1411.1105 at arxiv.org, 2014. [3] Pierre Albin, Fr´ed´ eric Rochon, and David Sher. Resolvent, heat kernel, and torsion under degeneration to fibered cusps. Preprint math.DG/1410.8406 at arxiv.org, 2014. zen. 6 (1972), no. 2, [4] V. I. Arnold, Modes and quasimodes (Russian), Funkcional. Anal. i Priloˇ 12–20. MR0297274 [5] David A. Cox, John B. Little, and Henry K. Schenck, Toric varieties, Graduate Studies in Mathematics, vol. 124, American Mathematical Society, Providence, RI, 2011. MR2810322 [6] Xianzhe Dai and Richard B. Melrose, Adiabatic limit, heat kernel and analytic torsion, Metric and differential geometry, Progr. Math., vol. 297, Birkh¨ auser/Springer, Basel, 2012, pp. 233– 298, DOI 10.1007/978-3-0348-0257-4 9. MR3220445

264

DANIEL GRIESER

[7] Monique Dauge and Nicolas Raymond, Plane waveguides with corners in the small angle limit, J. Math. Phys. 53 (2012), no. 12, 123529, 34, DOI 10.1063/1.4769993. MR3405919 [8] Freddy Dumortier and Robert Roussarie, Canard cycles and center manifolds, Mem. Amer. Math. Soc. 121 (1996), no. 577, x+100, DOI 10.1090/memo/0577. With an appendix by Cheng Zhi Li. MR1327208 [9] C. L. Epstein, R. B. Melrose, and G. A. Mendoza, Resolvent of the Laplacian on strictly pseudoconvex domains, Acta Math. 167 (1991), no. 1-2, 1–106, DOI 10.1007/BF02392446. MR1111745 [10] Pavel Exner and Olaf Post, Convergence of spectra of graph-like thin manifolds, J. Geom. Phys. 54 (2005), no. 1, 77–115, DOI 10.1016/j.geomphys.2004.08.003. MR2135966 [11] P. Freitas and D. Krejˇ ciˇr´ık, Location of the nodal set for thin curved tubes, Indiana Univ. Math. J. 57 (2008), no. 1, 343–375, DOI 10.1512/iumj.2008.57.3170. MR2400260 [12] Leonid Friedlander and Michael Solomyak, On the spectrum of the Dirichlet Laplacian in a narrow strip, Israel J. Math. 170 (2009), 337–354, DOI 10.1007/s11856-009-0032-y. MR2506330 [13] Vincent Grandjean and Daniel Grieser. The exponential map at a cuspidal singularity. J. R. Angew. Math., 2015. DOI 10.1515/crelle-2015-0020. arXiv:1205.4554v2. [14] Daniel Grieser, Basics of the b-calculus, Approaches to singular analysis (Berlin, 1999), Oper. Theory Adv. Appl., vol. 125, Birkh¨ auser, Basel, 2001, pp. 30–84. MR1827170 [15] Daniel Grieser. Notes on heat kernel asymptotics. Unpublished notes, www.unimath.gwdg.de/iwitt/SpecGeo2014/heat.pdf, 2004. [16] Daniel Grieser, Spectra of graph neighborhoods and scattering, Proc. Lond. Math. Soc. (3) 97 (2008), no. 3, 718–752, DOI 10.1112/plms/pdn020. MR2448245 [17] Daniel Grieser, Thin tubes in mathematical physics, global analysis and spectral geometry, Analysis on graphs and its applications, Proc. Sympos. Pure Math., vol. 77, Amer. Math. Soc., Providence, RI, 2008, pp. 565–593, DOI 10.1090/pspum/077/2459891. MR2459891 [18] Daniel Grieser and Michael J. Gruber, Singular asymptotics lemma and push-forward theorem, Approaches to singular analysis (Berlin, 1999), Oper. Theory Adv. Appl., vol. 125, Birkh¨ auser, Basel, 2001, pp. 117–130. MR1827172 [19] Daniel Grieser and Eug´ enie Hunsicker, Pseudodifferential operator calculus for generalized Q-rank 1 locally symmetric spaces. I, J. Funct. Anal. 257 (2009), no. 12, 3748–3801, DOI 10.1016/j.jfa.2009.09.016. MR2557724 [20] Daniel Grieser and David Jerison, Asymptotics of eigenfunctions on plane domains, Pacific J. Math. 240 (2009), no. 1, 109–133, DOI 10.2140/pjm.2009.240.109. MR2485476 [21] Colin Guillarmou and David A. Sher, Low energy resolvent for the Hodge Laplacian: applications to Riesz transform, Sobolev estimates, and analytic torsion, Int. Math. Res. Not. IMRN 15 (2015), 6136–6210, DOI 10.1093/imrn/rnu119. MR3384474 [22] Stefan Haag, Jonas Lampart, and Stefan Teufel, Generalised quantum waveguides, Ann. Henri Poincar´ e 16 (2015), no. 11, 2535–2568, DOI 10.1007/s00023-014-0374-9. MR3411741 [23] Jacques Hadamard. M´ emoire sur le probl` eme d’analyse relatif ` a l’´ equilibre des plaques ´ elastiques encastr´ ees. M´ em. Sav. ´ etrang. (2) 33, Nr. 4, 128 p, 1908. [24] Robin Hartshorne, Algebraic geometry, Springer-Verlag, New York-Heidelberg, 1977. Graduate Texts in Mathematics, No. 52. MR0463157 [25] Andrew Hassell, Analytic surgery and analytic torsion, Comm. Anal. Geom. 6 (1998), no. 2, 255–289, DOI 10.4310/CAG.1998.v6.n2.a2. MR1651417 [26] Andrew Hassell, Rafe Mazzeo, and Richard B. Melrose, Analytic surgery and the accumulation of eigenvalues, Comm. Anal. Geom. 3 (1995), no. 1-2, 115–222, DOI 10.4310/CAG.1995.v3.n1.a4. MR1362650 [27] Herwig Hauser, The Hironaka theorem on resolution of singularities (or: A proof we always wanted to understand), Bull. Amer. Math. Soc. (N.S.) 40 (2003), no. 3, 323–403, DOI 10.1090/S0273-0979-03-00982-0. MR1978567 [28] B. Helffer and J. Sj¨ ostrand, Multiple wells in the semiclassical limit. I, Comm. Partial Differential Equations 9 (1984), no. 4, 337–408, DOI 10.1080/03605308408820335. MR740094 [29] Bernard Helffer, Semi-classical analysis for the Schr¨ odinger operator and applications, Lecture Notes in Mathematics, vol. 1336, Springer-Verlag, Berlin, 1988. MR960278 [30] Heisuke Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero. I, II, Ann. of Math. (2) 79 (1964), 109–203; ibid. (2) 79 (1964), 205–326. MR0199184

SCALES, BLOW-UP AND QUASIMODE CONSTRUCTIONS

265

[31] Mark H. Holmes, Introduction to perturbation methods, 2nd ed., Texts in Applied Mathematics, vol. 20, Springer, New York, 2013. MR2987304 [32] Dominic Joyce, A generalization of manifolds with corners, Adv. Math. 299 (2016), 760–862, DOI 10.1016/j.aim.2016.06.004. MR3519481 [33] Dominic Joyce, A generalization of manifolds with corners, Adv. Math. 299 (2016), 760–862, DOI 10.1016/j.aim.2016.06.004. MR3519481 [34] Tosio Kato, Perturbation theory for linear operators, Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer-Verlag New York, Inc., New York, 1966. MR0203473 [35] Chris Kottke. Blow-up in manifolds with generalized corners. arXiv:1509.03874, 2015. [36] Chris Kottke and Richard B. Melrose, Generalized blow-up of corners and fiber products, Trans. Amer. Math. Soc. 367 (2015), no. 1, 651–705, DOI 10.1090/S0002-9947-2014-06222-3. MR3271273 [37] Christian Kuehn, Multiple time scale dynamics, Applied Mathematical Sciences, vol. 191, Springer, Cham, 2015. MR3309627 [38] Christian Kuehn and Peter Szmolyan, Multiscale geometry of the Olsen model and non-classical relaxation oscillations, J. Nonlinear Sci. 25 (2015), no. 3, 583–629, DOI 10.1007/s00332-015-9235-z. MR3338451 [39] Jonas Lampart and Stefan Teufel, The adiabatic limit of Schr¨ odinger operators on fibre bundles, Math. Ann. 367 (2017), no. 3-4, 1647–1683, DOI 10.1007/s00208-016-1421-2. MR3623234 [40] Jonas Lampart, Stefan Teufel, and Jakob Wachsmuth, Effective Hamiltonians for thin Dirichlet tubes with varying cross-section, Mathematical results in quantum physics, World Sci. Publ., Hackensack, NJ, 2011, pp. 183–189, DOI 10.1142/9789814350365 0018. MR2885171 [41] Vladimir Mazya, Serguei Nazarov, and Boris Plamenevskij, Asymptotic theory of elliptic boundary value problems in singularly perturbed domains. Vol. I, Operator Theory: Advances and Applications, vol. 111, Birkh¨ auser Verlag, Basel, 2000. Translated from the German by Georg Heinig and Christian Posthoff. MR1779977 [42] Vladimir Mazya, Serguei Nazarov, and Boris Plamenevskij, Asymptotic theory of elliptic boundary value problems in singularly perturbed domains. Vol. II, Operator Theory: Advances and Applications, vol. 112, Birkh¨ auser Verlag, Basel, 2000. Translated from the German by Plamenevskij. MR1779978 [43] Rafe R. Mazzeo and Richard B. Melrose, The adiabatic limit, Hodge cohomology and Leray’s spectral sequence for a fibration, J. Differential Geom. 31 (1990), no. 1, 185–213. MR1030670 [44] R. R. Mazzeo and R. B. Melrose, Analytic surgery and the eta invariant, Geom. Funct. Anal. 5 (1995), no. 1, 14–75, DOI 10.1007/BF01928215. MR1312019 [45] Rafe Mazzeo and Julie Rowlett, A heat trace anomaly on polygons, Math. Proc. Cambridge Philos. Soc. 159 (2015), no. 2, 303–319, DOI 10.1017/S0305004115000365. MR3395373 [46] Richard McGehee, Singularities in classical celestial mechanics, Proceedings of the International Congress of Mathematicians (Helsinki, 1978), Acad. Sci. Fennica, Helsinki, 1980, pp. 827–834. MR562695 [47] R. B. Melrose. Real blow up. Notes for lectures at MSRI. http://wwwmath.mit.edu/∼rbm/InSisp/InSiSp.html, 2008. [48] Richard B. Melrose, Pseudodifferential operators, corners and singular limits, Proceedings of the International Congress of Mathematicians, Vol. I, II (Kyoto, 1990), Math. Soc. Japan, Tokyo, 1991, pp. 217–234. MR1159214 [49] Richard B. Melrose, Calculus of conormal distributions on manifolds with corners, Internat. Math. Res. Notices 3 (1992), 51–61, DOI 10.1155/S1073792892000060. MR1154213 [50] Richard B. Melrose, The Atiyah-Patodi-Singer index theorem, Research Notes in Mathematics, vol. 4, A K Peters, Ltd., Wellesley, MA, 1993. MR1348401 [51] Richard B. Melrose. Differential analysis on manifolds with corners. Book in preparation. http://www-math.mit.edu/∼rbm/book.html, 1996. [52] S. Molchanov and B. Vainberg, Scattering solutions in networks of thin fibers: small diameter asymptotics, Comm. Math. Phys. 273 (2007), no. 2, 533–559, DOI 10.1007/s00220-007-02208. MR2318317 [53] Thomas Ourmi` eres-Bonafos, Dirichlet eigenvalues of asymptotically flat triangles, Asymptot. Anal. 92 (2015), no. 3-4, 279–312. MR3371117 [54] Julie Rowlett, Spectral geometry and asymptotically conic convergence, Comm. Anal. Geom. 16 (2008), no. 4, 735–798, DOI 10.4310/CAG.2008.v16.n4.a2. MR2471369

266

DANIEL GRIESER

[55] David A. Sher, Conic degeneration and the determinant of the Laplacian, J. Anal. Math. 126 (2015), 175–226, DOI 10.1007/s11854-015-0015-3. MR3358031 [56] Stefan Teufel, Adiabatic perturbation theory in quantum dynamics, Lecture Notes in Mathematics, vol. 1821, Springer-Verlag, Berlin, 2003. MR2158392 [57] S. van Gils, M. Krupa, and P. Szmolyan, Asymptotic expansions using blow-up, Z. Angew. Math. Phys. 56 (2005), no. 3, 369–397, DOI 10.1007/s00033-004-1021-y. MR2142930 ¨r Mathematik, Carl von Ossietzky Universita ¨t Oldenburg, 26111 OldInstitut fu enburg E-mail address: [email protected]

Contemporary Mathematics Volume 700, 2017 http://dx.doi.org/10.1090/conm/700/14189

Scattering for the geodesic flow on surfaces with boundary Colin Guillarmou Abstract. These are lecture notes based on a mini-course given in the 2015 summer school Th´ eorie spectrale g´ eom´ etrique et computationnelle in CRM, Montr´ eal.

1. Introduction In these notes, we discuss some geometric inverse problems in 2-dimension that have been studied since the eighties, and we review some results on these questions. The problem we consider consists in recovering a Riemannian metric on a surface with boundary from measurements at the boundary: the lengths of geodesics relating boundary points and their tangent directions at the boundary. This is a non-linear problem called the lens rigidity problem, which has a gauge invariance (pull-backs by diffeomorphisms fixing the boundary). The associated linear problem consists in the analysis of the kernel of the geodesic X-ray transform, a curved version of the Radon transform. The tools to study the X-ray transform are of analytic nature, more precisely a combination of analysis of transport equations with some energy identity. Microlocal methods have also been very powerful in that study, but we won’t review this aspect in these notes. The techniques presented in these notes are quite elementary and give a short introduction to that area of research. 2. Geometric background Let (M, g) be a smooth oriented compact Riemannian surface with boundary ∂M and let M ◦ be its interior. In local coordinates x = (x1 , x2 ) the metric will be written 2  gij (x)dxi dxj , g= i,j=1

where (gij (x))ij are symmetric positive definite matrices smoothly depending on x. We will write ∇ the Levi-Civita connection of g on M . The tangent bundle of M is denoted T M and the projection on the base is written π0 : T M → M. The second fundamental form II is the symmetric tensor II : T ∂M × T ∂M → R,

II(u, w) := −g(∇u ν, w),

2010 Mathematics Subject Classification. Primary 35R30. c 2017 American Mathematical Society

267

268

COLIN GUILLARMOU

where ν is the interior pointing unit normal vector to ∂M . We say that ∂M is strictly convex for g if II is positive definite and we will assume along this course that this property holds. Exercise: Show that a neighborhood of ∂M in M is isometric to [0, ]r × ∂M with metric dr 2 + hr where hr is a smooth 1-parameter family of metrics on ∂M . In these geodesic normal coordinates, we have ν = ∂r |r=0 and II = − 12 ∂r hr |r=0 . 2.1. Geodesic flow. A geodesic on M is a C 2 curve on M such that ∇x˙ x˙ = 0 where x(t) ˙ ∈ Tx(t) M is the tangent vector to the curve, i.e x(t) ˙ := ∂t x(t). In local 2 k coordinates x = (x1 , x2 ), we can write ∇∂xi ∂xj = k=1 Γij ∂xk for some smooth functions Γkij called Christoffel symbols. The geodesic equation in these coordinates is 2  x ¨j (t) = − Γjk (x(t))x˙ k (t)x˙ (t), j = 1, 2. k, =1

By standard arguments of ordinary differential equations – Cauchy-Lipschitz–, this second order equation has a solution x(t) in some interval t ∈ [0, ) if we fix an initial condition (x(0), x(0)) ˙ = (x0 , v0 ) ∈ T M ◦ , and the solution can be extended until x(t) reaches ∂M . The geodesics are minimizers of the energy and of the length functionals: if p, q are two points in M and if γ : [0, 1] → M is a C 2 curve such that γ(0) = p and γ(1) = q, then the energy Ep,q (γ) and the length Lp,q (γ) are defined by  1  1 2 |γ(t)| ˙ Lp,q (γ) = |γ(t)| ˙ Ep,q (γ) = g(γ(t)) dt. g(γ(t)) dt, 0

0

Then the minimum of Ep,q (γ) and Lp,q (γ) among curves as above are obtained by a geodesic. In fact, one can show using variational methods and a compactness argument that in each homotopy class of curves with endpoints p, q, there is a mimimizer for Ep,q and Lp,q which is a geodesic. The miminizer is in general not unique. Definition 2.1. The geodesic flow at time t ∈ R is the map ϕt defined by ϕt : U (t) → T M,

ϕt (x, v) := (x(t), x(t)), ˙

if x(t) is the geodesic with initial condition (x(0), x(0)) ˙ = (x, v), where U (t) ⊂ T M is the set of points (x, v) ∈ T M such that the geodesic x(s) with initial condition (x, v) exists in M for all s ∈ [0, t]. The exponential map at a point x ∈ M is the map expx : Ux ⊂ Tx M → M,

expx (v) = π0 (ϕ1 (x, v)),

where Ux is the set of vector v ∈ Tx M so that ϕt (x, v) ∈ M for all t ∈ [0, 1). The map expx is a local diffeomorphism near v = 0 at each x ∈ M ◦ . Notice that for x(t) a geodesic on M , ˙ x(t))) ˙ = 2gx(t) (∇x(t) x(t), ˙ x(t)) ˙ =0 ∂t (gx(t) (x(t), ˙ and therefore ϕt acts on the unit tangent bundle SM := {(x, v) ∈ T M ; gx (v, v) = 1}. The vector field generating the flow ϕt is a smooth vector field on SM defined by Xf (x, v) = ∂t f (ϕt (x, v))|t=0 .

SCATTERING FOR THE GEODESIC FLOW ON SURFACES WITH BOUNDARY

269

The manifold SM is compact and has boundary ∂SM = π0−1 (∂M ). This boundary splits into the disjoint parts ∂− SM := {(x, v) ∈ SM ; x ∈ ∂M, gx (v, ν) > 0}, ∂+ SM := {(x, v) ∈ SM ; x ∈ ∂M, gx (v, ν) < 0}, ∂0 SM := {(x, v) ∈ SM ; x ∈ ∂M, gx (v, ν) = 0}. We call ∂− SM the incoming boundary, ∂+ SM the outgoing boundary and ∂0 SM the glancing boundary. 2.2. Hamiltonian approach. The cotangent bundle T ∗ M is a symplectic manifold, with symplectic form ω = dα where α ∈ C ∞ (T ∗ M, T ∗ (T ∗ M )) is the Liouville 1-form defined by α(x,ξ) (W ) = ξ(dπ0 (x, ξ).W ) if π0 : SM → M is the projection on the base. In local coordinates x = (x1 , x2 ), ξ = ξ1 dx1 + ξ2 dx2 , we have 2 2   α= ξi dxi , ω = dξi ∧ dxi . i=1

i=1

∗

The function p : T M → R defined by p(x, ξ) = 12 gx−1 (ξ, ξ) where g −1 is the metric induced on T ∗ M by g has a Hamiltonian vector field Xp defined by ω(Xp , ·) = dp and Xp is tangent to S ∗ M := p−1 (1/2). Exercise: Show that the duality isomorphism SM → S ∗ M given by the metric g conjugates ϕt = etX to etXp . 2.3. Geometry of SM . There are particular sets of coordinates x = (x1 , x2 ) near each points x0 ∈ M , called isothermal coordinates, such that in these coordinates, the metric has the form g = e2ρ (dx21 + dx22 ) in these coordinates, for some smooth function ρ(x) near x0 . The metric is conformal to the Euclidean metric in these coordinates, which will be a useful fact for what follows. These coordinates can be obtained by solving an elliptic equation, and more precisely a Beltrami equation (see [Ta, Chapter 5.10]). These coordinates induce a diffeomorphism Ω × R/2πZ → π0−1 (Ω) ⊂ SM,

(x, θ) → (x, v = e−ρ(x) (cos(θ)∂x1 + sin(θ)∂x2 )),

where Ω is the neighborhood of x0 where the isothermal coordinates are valid. In these coordinates, the vector field X becomes   (2.1) X = e−ρ cos(θ)∂x1 + sin(θ)∂x2 + (− sin(θ)∂x1 ρ + cos(θ)∂x2 ρ)∂θ . We start by analyzing the flow of X near the boundary, under the assumption that ∂M is strictly convex. Lemma 2.2. Geodesics in M ◦ intersect ∂M transversally, i.e. a geodesic coming from M ◦ and touching ∂M at a point x can not be tangent to ∂M Proof. There are isothermal coordinates x = (x1 , x2 ) near each x0 ∈ ∂M such that a neighborhood of x0 in M correspond to a neighborhood of 0 in the half-plane {x2 ≥ 0} and the metric is of the form e2ρ (dx21 + dx22 ), and we can assume that x0 is mapped to x = 0 by this chart. One has ∂x2 ρ|x2 =0 < 0 if ∂M is strictly convex. If x(t) is a geodesic for t ≤ t0 with x2 (t) > 0 for t < t0 and x2 (t0 ) = 0, then if ˙ 0 ) = e−ρ(0) (cos(θ0 )∂x ρ(0)). x˙ 2 (t0 ) = 0, we get by (2.1) that θ(t0 ) = 0 (or π) and θ(t 2

270

COLIN GUILLARMOU

˙ 0 ) < 0 and Let us consider the case θ(t0 ) = 0 (the case θ(t0 ) = π is similar): then θ(t −ρ ρ(0) ˙ ¨2 (t0 ) = e θ(t0 ) < 0. θ decreases as t → t0 , and since x˙ 2 (t) = e sin θ(t), we get x A Taylor expansion gives ¨2 (t0 ) + O((t − t0 )3 ), x2 (t) = 12 (t − t0 )2 x which is negative near t0 , leading to a contradiction.



Define Θt the rotation of angle +t in the fibers of SM ; in the coordinates above Θt is just (x, θ) → (x, θ + t). This smooth 1-parameter family of diffeomorphisms of SM induces a smooth vector field V defined by V f (x, v) = ∂t f (Θt (x, v))|t=0 ,

∀f ∈ C ∞ (SM ).

In the coordinates (x, θ), V = ∂θ . Next we define another vector field X⊥ := [X, V ], which in the coordinates (x, θ), is given by   X⊥ = −e−ρ − sin(θ)∂x1 + cos(θ)∂x2 − (cos(θ)∂x1 ρ + sin(θ)∂x2 ρ)∂θ . It it an elementary computation to check that the three vector fields (X, X⊥ , V ) form a global basis of T (SM ) (we recover that SM is trivialisable) and satisfy the commutation relations (2.2)

[X⊥ , V ] = −X,

[X, X⊥ ] = −κ(x)V,

where κ(x) is the Gaussian curvature of g at x. In isothermal coordinates, a computation yields κ(x) = e−2ρ(x) Δx ρ(x) where Δx = −(∂x21 + ∂x22 ). We define the Sasaki metric of g as the metric G on SM so that (X, X⊥ , V ) is an orthonormal basis, and its volume form dvG is also equal to the Liouville measure dμL obtained from the symplectic form ω = dα by setting dμL = |α ∧ dα| when we use the identification S ∗ M → SM . In isothermal coordinates, one has dμL (x, θ) = e2ρ(x) dxdθ. If W = aX + bX⊥ + cV , we have G(W, W ) = a2 + b2 + c2 = g(dπ0 (W ), dπ0 (W )) + c2 . The Sasaki metric is usually defined using the splitting of the vertical bundle and horizontal bundle (see [Pa]), but it coincides in our case with the definition above. Here (X, X⊥ ) span the horizontal bundle while V span the vertical bundle of the fibration π0 : SM → M . Exercise: Check, using Cartan formula, that the following Lie derivatives vanish: ∀Z ∈ {X, X⊥ , V }, LZ dμL = 0. ∗ = −X⊥ on C0∞ (SM ) As a consequence, we have X ∗ = −X, V ∗ = −V and X⊥ 2 ∞ with respect to the L (SM, dμL ) product, where C0 (SM ) is the set of smooth functions on SM vanishing at the boundary ∂(SM ) of SM . On SM ⊂ T M , we can consider functions which are restrictions to SM of homogeneous polynomials of order m ∈ N0 in the fibers of T M , i.e. symmetric ∗ tensors defined as sections of ⊗m S T M . There is a natural map for each m ∈ N0 (2.3)

∗ ∗ ∞ πm : C ∞ (M, ⊗m S T M ) → C (SM ),

∗ πm f (x, v) := f (x)(⊗m v).

SCATTERING FOR THE GEODESIC FLOW ON SURFACES WITH BOUNDARY

271

2.4. Conjugate points. Geodesic flows can have 1-parameter families of geodesics with the same endpoints x− , x+ : this is related to the existence of conjugate points. We say that x± ∈ M are conjugate points if there exist v± ∈ Sx± M and t0 so that ϕt0 (x− , v− ) = (x+ , v+ ), and if dϕt0 (x− , v− ).V ∈ RV where V is the vertical vector field. Equivalently, we say that x± are conjugate if there is an orthogonal Jacobi field J = J(t) along the geodesic (x(t))t∈[0,t0 ] vanishing at x− and x+ . Recall that an orthogonal Jacobi field is a vector field along x(t), orthogonal to x(t) ˙ = v(t). If we write v ⊥ (t) = Rπ/2 (v(t)) the unit orthogonal vector obtained by a rotation of angle π/2 of v(t), such a field would be of the form J(t) = a(t)v ⊥ (t), and for it to be a Jacobi field the function a(t) has to satisfy: a(0) = a(t0 ) = 0,

a ¨(t) + κ(x(t))a(t) = 0.

Exercise: Show that when the Gauss curvature κ is non-positive, there are no conjugate points. By Gauss-Bonnet theorem, for a geodesic triangle with interior angles α1 , α2 , α3 , one has  3  κ dvolg + π = αi . M

i=1

If two geodesics have the same endpoints x− , x+ , we obtain a triangle with angles π, α2 , α3 , and this forces by Gauss-Bonnet to have some positive curvature somewhere. In fact, it can be proved that absence of conjugate points implies that between two points x− , x+ , there is a unique geodesic – this is done using the index form, see [Mi, Sections 14 & 15]. 3. Scattering map, length function and X-ray transform 3.1. Lens rigidity problem. We start by making the non-trapping assumption on the geodesic flow. That is, for each (x, v) ∈ SM ◦ there is a unique + (x, v) ≥ 0 and − (x, v) ≤ 0 so that ϕ ± (x,v) (x, v) ∈ ∂SM , which means that each geodesic of SM has finite length. Exercise: Prove, using the strict convexity of ∂M and the implicit function theorem, that ± are smooth in SM ◦ and that they extend smoothly to SM \∂0 SM . Show also that ± extend continuously to SM in a way that ± |∂± SM ∪∂0 SM = 0. We will still call ± these continous extensions. Definition 3.1. The function + is called the length function and g := + |∂SM is called the boundary length function. The map Sg : ∂− SM → ∂+ SM,

Sg (x, v) = ϕ + (x,v) (x, v)

is called the scattering map of g. We introduced these objects to formulate some questions in the realm of inverse problems. They are quantities that can be measured from the boundary. The boundary length function contains the set of Riemannian distances between boundary points, but it does contain a priori more information in the case where there are several geodesics between boundary points. The scattering map tells where geodesics leave SM .

272

COLIN GUILLARMOU

Definition 3.2. The X-ray transform on functions on SM is defined as the operator  + (x,v) I : C ∞ (SM ) → C ∞ (∂− SM ), If (x, v) = f (ϕt (x, v))dt. 0

The X-ray on symmetric tensors of order m is the operator ∗ ∗ ∞ : C ∞ (M, ⊗m Im := Iπm S T M ) → C (∂− SM ).

The two main inverse problems related to these objects are: Problem 1: Determine the kernel of Im and invert Im on Ran(Im |(ker Im )⊥ ). Problem 2: Do (Sg , g ) determine the metric g up to Gauge invariance ? The natural Gauge invariance in Problem 2 is the diffeomorphism action: let Diff ∂M (M ) be the group of diffeomorphism of M which are equal to the identity on ∂M , then one has for each ψ ∈ Diff(M ) Sψ ∗ g = Sg ,

ψ∗ g = g .

Problem 2 is called the lens rigidity problem and (Sg , g ) is the lens data. There is a link between these inverse problems. Indeed, one has Lemma 3.3. Let gs = e2ρs g0 be some smooth 1-parameter family of nontrapping metrics with strictly convex boundary and no conjugate points, where ρs is a smooth family of smooth functions on M such that ρ0 = 0. If (gs , Sgs ) = (g0 , Sg0 ) for each s ∈ (− , ), then I0 (ρ0 ) = 0 if I0 is the X-ray transform on functions for g0 and ρ0 := ∂s ρs |s=0 . Proof. Fix (x, v) ∈ ∂− SM and (x , v  ) = Sg0 (x, v) = Sgs (x, v). We differentiate

 gs (x, v) =

g0 (x,v)

eρs (γs (t,x,v)) |γ˙ s (t, x, v)|g0 (γs (t,x,v)) dt

0

with respect to s, where γs (t, x, v) is the unique geodesic for gs relating x and x . Using the fact that γ0 (t, x, v) is the unique geodesic for g0 relating x and x , it minimizes the length functional among curves with endpoints x, x and thus it is direct to see that  g0 (x,v) ρ0 (γ0 (t, x, v))dt + ∂s (Lgx0− ,x+ (γs ))|s=0 = I0 (ρ0 )(x, v) 0= 0

where

0 (γ) Lgp,q

denotes the length of a curve γ joining p, q for the metric g0 .



This corresponds to analyzing deformation lens rigidity within a conformal class. More generally we have (with essentially the same proof): Lemma 3.4. Let gs be some smooth 1-parameter family of non-trapping metrics with strictly convex boundary and no conjugate points. If (gs , Sgs ) = (g0 , Sg0 ) for each s ∈ (− , ), then I2 (g0 ) = 0 if I2 is the X-ray transform on symmetric 2-tensors for g0 and g0 := ∂s gs |s=0 ∈ C ∞ (M ; ⊗2S T ∗ M ). These two lemmas are folklore and the ideas appear already in GuilleminKazhdan [GuKa] and the Siberian school (Mukhometov, Anikonov, Sharafutdinov, etc).

SCATTERING FOR THE GEODESIC FLOW ON SURFACES WITH BOUNDARY

273

Remark that if ψs : M → M is a smooth family of diffeomorphisms which are equal to Id on ∂M , we have an associated vector field Z(x) = ∂s ψs (x)|s=0 with Z|∂M = 0 and, according to Lemma 3.4, LZ g0 = ∂s (ψs∗ g0 )|s=0 satisfies I2 (LZ g0 ) = 0 if I2 is the X-ray transform for g0 on symmetric 2-tensors. The natural question about the kernel of X-ray transform is then: under which conditions on g0 do we have ker I0 = 0 ? and ker I2 = {LZ g0 ; Z ∈ C ∞ (M, T M ), Z|∂M = 0} ? 3.2. Boundary rigidity problem. When the metric g has strictly convex boundary, is non-trapping and satisfies that between each pair of points x, x ∈ M there is a unique geodesic, we say that g is simple. In this case, knowing the lens data is equivalent to knowing the restriction dg |∂M ×∂M of the Riemannian distance dg : M × M → R+ . The lens rigidity problem is called boundary rigidity problem in that setting. 4. Resolvents and boundary value problems for transport equations For a general metric g on SM with strictly convex boundary, we define the incoming tail Γ− and outgoing tails as the sets Γ− := {(x, v) ∈ SM ; + (x, v) = +∞},

Γ+ := {(x, v) ∈ SM ; − (x, v) = −∞}.

These sets correspond to the set of points which are on geodesics that are trapped inside SM ◦ in forward (for Γ− ) and backward (for Γ+ ) time. They are closed sets in SM ◦ . The trapped set is defined by K := Γ+ ∩ Γ− , it corresponds to trajectories contained entirely in the interior SM ◦ . It is a closed set in SM , invariant by the geodesic flow, and by the strict convexity of the boundary ∂M , we actually have K ⊂ SM ◦ , since each point (x, v) ∈ ∂∓ SM is not in Γ± and each point (x, v) ∈ ∂0 SM has + (x, v) = − (x, v) = 0. 4.1. Resolvents in physical half-planes. Assume that Γ± = ∅, i.e the metric is non-trapping. There are two natural boundary value problems for the transport equations associated to X. For f ∈ C ∞ (SM ), find u± in some fixed functional space solving (in the distribution sense)  −Xu± = f u± |∂± SM = 0 One is an incoming Dirichlet type boundary condition and the other one is an ougoing Dirichlet type boundary condition. Moreover, it is easy to check that  + (x,v)  0 f (ϕt (x, v))dt, u− (x, v) = − f (ϕt (x, v))dt u+ (x, v) = 0

− (x,v)

are solutions, and in fact they are the only continuous solutions: indeed the difference of two solutions would be constant along flow lines and, under the nontrapping condition each point in SM is on a flow line with two endpoints in ∂+ SM and ∂− SM . We see in particular that f ∈ C ∞ (SM ) =⇒ u± ∈ C ∞ (SM \ ∂0 SM ). Without assuming the non-trapping condition, we can proceed using the resolvent of X.

274

COLIN GUILLARMOU

Lemma 4.1. For Re(λ) > 0, there exist two operators R± (λ) : C ∞ (SM ) → C (SM ) satisfying for all f ∈ C ∞ (SM ) (in the distribution sense)  (−X ± λ)R± (λ)f = f, R± (λ)f |∂± SM = 0. 0

They are given by the expressions



+ (x,v)

R+ (λ)f (x, v) = 0

(4.1)



e−λt f (ϕt (x, v))dt,

0

R− (λ)f (x, v) = −

eλt f (ϕt (x, v))dt.

− (x,v)

Proof. Lebesgue theorem shows directly that R± (λ)f are continuous if f is continuous. The fact that they solve the desired boundary value problem for Re(λ) large enough follows from the fact that X(f (ϕt (x, v))) = ∂t (f (ϕt (x, v)) and the estimate |d(f ◦ ϕt )|G ≤ |df |G |dϕt |G , |dϕt |G ≤ CeCt for some C > 0 depending on X. Then, we get that for each f  ∈ Cc∞ (SM ◦ ) and Re(λ) > 0 (−X ± λ)R± (λ)f, f  = R± (λ)f, (X ± λ)f 

where the pairing uses the measure μL , and the left hand side is equal to f, f 

for Re(λ) > C, thus by using that the right hand side is analytic in Re(λ) > 0, the  right hand side is equal to f, f  in that half-plane. Exercise: Show that the operators R± (λ) extend analytically in λ ∈ C as operators R± (λ) : Cc∞ (SM ◦ \ Γ± ) → C ∞ (SM ). If we assume that μL (Γ± ) = 0, we can then expect to extend R± (0) to some Lp space by some density argument. In fact, observe that |R+ (0)f (x, v)| ≤ ||f ||L∞ + (x, v), thus + ∈ Lp (SM ) implies that R+ (0) : C 0 (SM ) → Lp (SM ) is bounded. 4.2. Santalo formula. The Santalo formula describes the desintegration of the measure μL along flow lines. Lemma 4.2. Assume that μL (Γ± ) = 0 and let f ∈ L1 (SM ), then the following formula holds   + (x,v)  f dμL = f (ϕt (x, v))dt dμν (x, v), SM

∂− SM

0

where dμν (x, v) = v, ν g ι∗∂SM dvG is a measure on ∂SM , dvG being the Riemannian measure of the Sasaki metric viewed as a 3-form. Proof. We first take f ∈ Cc∞ (SM ◦ \ Γ+ ) and write f = −XR+ (0)f with R+ (0)f = 0 on ∂+ SM . Thus using Green’s formula    f dμL = − X(R+ (0)f )dμL = (R+ (0)f )|∂SM X, N G ι∗∂SM dvG SM

SM

∂SM

if N is the unit inward pointing normal to ∂SM for G. The unit normal N satisfies dπ0 (N ) = ν and G(N, V ) = 0 if ν is the exterior pointing unit normal to ∂M for g, since the vertical bundle RV = ker dπ0 is tangent to ∂SM . Then we get

SCATTERING FOR THE GEODESIC FLOW ON SURFACES WITH BOUNDARY

275

N, X G = ν, v g and we use a density argument for the general case to end the  proof. Note that an alternative definition of dμν is dμν = ι∗∂SM iX dμL . 4.3. Boundedness in (weighted) L2 spaces and limiting absorbtion principle. The operators R± (λ) are also bounded on L2 (SM ) for Re(λ) > 0. Indeed, using the Santalo formula and defining f (ϕt (x, v)) := 0 when t ≥ + (x, v), we have   + (x,v)  2 |f (ϕt (x, v))| dμL (x, v) = |f (ϕt+s (x, v))|2 ds dμν (x, v) SM

∂− SM



0



+ (x,v)

= ∂− SM

|f (ϕu (x, v))|2 du dμν (x, v)

t

≤||f ||L2 (SM ) . We can write using Minkowski inequality   +∞  2 1 2 e−Re(λ)t |f (ϕt (x, v))|dt dμL (x, v) ||R+ (λ)f ||L2 (SM ) ≤ SM 0  +∞ ||f ||L2 (SM ) . ≤ e−Re(λ)t ||f ◦ ϕt ||L2 (SM ) dt ≤ Re(λ) 0 The question of the extension of R± (λ) up to the imaginary line in a certain functional space is very similar to the so-called limiting absorbtion principle in quantum scattering theory. To make the parallel, recall that the Laplacian on L2 (R3 ) is selfadjoint as an unbounded operator, its spectrum is [0, ∞) and its resolvent is the operator defined for Im(λ) > 0 by RΔ (λ) = (Δ − λ2 )−1 : L2 (R3 ) → L2 (R3 ) and there is an explicit formula for its integral kernel 

RΔ (λ; x, x ) = C

eiλ|x−x | |x − x |

for some explicit constant C ∈ R. Now when λ ∈ R, this is not anymore a bounded operator on L2 but it makes sense as an operator (4.2)

RΔ (λ) : x −1/2− L2 (R3 ) → x 1/2+ L2 (R3 )

and both RΔ (λ) and RΔ (−λ) are inverses for (Δ − λ2 ) (here x := (1 + x2 )1/2 ). They are called the incoming and outgoing resolvents on the spectrum, and they need to be applied on functions which have some decay near infinity. Such property also holds in higher dimension. A result of Kenig-Ruiz-Sogge [KRS] shows that for λ ∈ R∗ 1 1 2n , + = 1. RΔ (λ) : Lp (Rn ) → Lq (Rn ), p = n+2 p q We will discuss similar properties for the resolvent of the flow vector field −X. The first boundedness property we describe is comparable to (4.2). Lemma 4.3. For each λ ∈ iR and > 0, the resolvent R± (λ) is bounded as a map R± (λ) : + −1/2− L2 (SM ) → + 1/2+ L2 (SM ).

276

COLIN GUILLARMOU

Proof. We do the case R+ (0), the other frequencies λ are similar. First we notice that + (ϕs (y)) = + (y) − s. Then for f ∈ C ∞ (SM ) and f := + −1/2− f we have  |R+ (0)f (y)|2 + (y) −1−2 dμL (y) SM





+ (y)

≤ ∂− SM



0



+ (y)

≤ ∂− SM



+ (y) − s −1−2

0

0





∂− SM

≤C 

+ (y)

dsdμν (y)

0



+ (y)−s

+ (y) − s − t −1−2 dt

+ (y) − s −1−2



+ (y)−s

|f (ϕt+s (y))|2 dtdμν (y)ds

0

+ (y)

+ (y) − s −1−2



0



≤ C2 ∂− SM

f (ϕt+s (y))dt

0



∂− SM

2

+ (y)−s

|f (ϕt+s (y))|2 dtdμν (y)ds

≤C 



0

+ (y)−s

×

+ (y) − s −1−2

+ (y)

|f (ϕu (y))|2 dudμν (y)ds

0

+ (y)

|f (ϕu (y))|2 dudμν (y) ≤ C 2 ||f ||2L2 (SM )

0

where we have used that there is C > 0 depending only on so that  a−s a − s − t −1−2 dt ≤ C. 0

To complete the proof, we use the density of smooth functions in L2 .



A priori, it is not clear that the space + 1/2+ L2 (SM ) can be embedded into the space of distributions on SM , since the function + could explode quite drastically at Γ− . To quantify this, we define the non-escaping mass function: V (T ) := VolμL ({y ∈ SM ◦ ; ϕt (y) ∈ SM ◦ , ∀t ∈ [0, T ]}) = VolμL (−1 + ([T, +∞])) which is like the repartition function of + . The Cavalieri principle gives  ∞   p ||+ ||Lp ≤ C 1 + tp−1 V (t)dt . 1

Recall that + (x, −v) = −− (x, v) thus the Lp norms of + and − are the same since the involution (x, v) → (x, −v) preserves μL . Lemma 4.4. For p ∈ [1, ∞) we have the boundedness properties  ∞ R± (0) : C 0 (SM ) → Lp (SM ) if V (t)tp−1 dt < ∞, 1

and for p > 1



R± (0) : Lp (SM ) → L1 (SM ) if

∞

V (t)t1/(p−1) dt < ∞.

1

Proof. Left as an exercise. Use H¨older and Santalo formula.



SCATTERING FOR THE GEODESIC FLOW ON SURFACES WITH BOUNDARY

277

4.4. Return on X-ray transform. First we remark that the X-ray transform can be defined also in the trapping case as a map I : Cc∞ (SM \ Γ+ ) → C ∞ (∂− SM ); and for a function f ∈ Cc∞ (SM \ Γ+ ) this can be written as If (x, v) = (R+ (0)f )(x, v) (x, v) ∈ ∂− SM. If f = + −1/2−/2 f , we have   |I f (y)|2 dμν (y) = ∂− SM

: : :

∂− SM



+ (y)

0



≤C ∂− SM

:2 : + (y) − t −1/2−/2 f (ϕt (y))dt: dμν (y)

+ (y)

|f (ϕt (y))|2 dtdμν (y) = C||f ||2L2 (SM )

0

and therefore I : + −1/2− L2 (SM ) → L2 (∂− SM, dμν ) is bounded for each > 0. It is a straightforward consequence of Santalo formula that I : L1 (SM ) → L1 (∂− SM, dμν ) is bounded. We get, just as for R± (0), the following boundedness property Lemma 4.5. When p > 2, the X-ray transform is bounded as a map Lp (SM ) →  ∞ p/(p−2) L (∂− SM, dμν ) if 1 t V (t)dt < ∞ 2



Proof. Use the H¨older and Santalo formulas. As a consequence, taking the adjoint, we obtain the boundedness 

I ∗ : L2 (∂− SM, dμν ) → Lp (SM )

under the assumption

∞ 1

tp/(p−2) V (t)dt < ∞ and in general the boundedness of

I ∗ : L2 (∂− SM, dμν ) → + 1/2+ L2 (SM ) holds for all > 0. We would like to characterize what operator is I ∗ . For f, f  smooth supported oustide Γ− ∪ Γ+ , we have   If.f  dμν = R+ (0)f.f  dμν ∂− SM ∂− SM  = −X(R+ (0)f ).E− (f  )dμL = f, E− (f  )

SM 

where E− (f ) solves



XE− (f  ) = 0, E− (f  )|∂− SM = f 

We thus get I ∗ = E− . Notice that E− (f  )(y) = f  (ϕ − (y) (y)) is constant on flow lines.

278

COLIN GUILLARMOU

4.5. The normal operator. Assume that μL (Γ− ∪ Γ+ ) = 0. We define the normal operator on SM as the operator given by the expression Π : + −1/2− L2 (SM ) → + 1/2+ L2 (SM ), Π := I ∗ I. ∞ By Lemma 4.5, we see that if 1 V (t)tp/(p−2) dt < ∞ for some p > 2, then Π extends as a bounded operator 

Π : Lp (SM ) → Lp (SM ). We can relate Π to the operators R± (0). Lemma 4.6. The following identity holds true if μL (Γ− ∪ Γ+ ) = 0 Π = R+ (0) − R− (0). Proof. Note that as operators acting on Cc∞ (SM ◦ ), we have R+ (0)∗ = −R− (0) thus we need to prove I ∗ If, f = 2 R+ (0)f, f . But we have for each f ∈ Cc∞ (SM \ Γ+ )    1 R+ (0)f.f dμL = − X(R+ (0)f ).R+ (0)f dμL = − X((R+ (0)f )2 )dμL 2 SM SM SM   1 1 2 = |R+ (0)f | dμν = |If |2 dμν . 2 ∂SM 2 ∂− SM and we complete the proof using a density argument.



Using this, let us characterize the kernel of I: ∞ Lemma 4.7. Assume 1 V (t)tp/(p−2) dt < ∞ for some p > 2, then f ∈ ker I ∩ C 0 (SM ) if and only there exists a unique u ∈ Lp (SM ) ∩ C 0 (SM \ K) such that Xu = f,

u|∂SM = 0,

K being the trapped set. If K = ∅ and if f ∈ C ∞ (SM ) vanishes to infinite order at ∂SM , then u ∈ C ∞ (SM ) and u vanishes to infinite order at ∂SM . Proof. Assume that If = 0. Set u = −R+ (0)f , then Xu = f , u ∈ C 0 (SM \ Γ− ) and satisfies u|∂+ SM = 0. We have Πf = 0, thus by Lemma 4.6, u = −R+ (0)f = −R− (0)f . Thus u is actually in C 0 (SM \ Γ− ) and vanishes on ∂− SM . Since there is no solution of Xu = 0 with u|∂SM = 0 and u ∈ L1 (SM )∩C 0 (SM \K), we have proved one direction. Conversely, if u ∈ L1 (SM ) ∩ C 0 (SM \ K) satisfies Xu = f in the distribution sense, then u = −R+ (0)f by uniqueness of solutions with u|∂+ SM = 0, and similarly u = −R− (0)f . Thus I ∗ If = Πf = 0 and therefore If = 0 since Πf, f = |If |2L2 . The fact that u is smooth when g has no trapped set and f ∈ C ∞ (SM ) vanishes to infinite order at ∂SM is direct from the expression (4.1). Notice that if f is smooth but does not vanish to infinite order at ∂SM , then  it is not clear that u is smooth at ∂0 SM . We have just seen that f being in ker I can be interpreted in terms of properties of solutions of the transport equation Xu = f . In fact, the regularity of the solutions u to Xu = f will be very important for what follows, and this leads us to define the following Definition 4.8. We shall say that a metric g with strictly convex boundary has the smooth Livsic property if for each f ∈ C ∞ (SM ) satisfying If = 0, there exists a unique u ∈ C ∞ (SM ) such that Xu = f and u|∂SM = 0.

SCATTERING FOR THE GEODESIC FLOW ON SURFACES WITH BOUNDARY

279

Pestov-Uhlmann [PeUh] show the following result Theorem 1. If g is a non-trapping metric on a surface M with and strictly convex boundary, then it satisfies the smooth Livsic property. The main difficulty is the regularity at the glancing region ∂0 SM , which is studied in the work of Pestov-Uhlmann using fold theory - we refer to [PeUh] for the interested reader (where fold theory is recalled). We notice that the presence or absence of conjugate points is not relevant here, since this result is really a property on the flow on SM and has not much to do with the fact that we are working with a geodesic flow. In the recent work [Gu], we show the following result by using techniques of microlocal analysis and anisotropic Sobolev spaces: Theorem 2. If the curvature of g near the trapped set K is negative (or more generally if K is a hyperbolic set for the geodesic flow of g), then g has the smooth Livsic property. We notice that V (t) = O(e−Qt ) for some Q > 0 if K is a hyperbolic set by [BoRu] (see [Gu, Proposition 2.4]), and this implies that ± ∈ Lp for all p < ∞. 5. Injectivity of X-ray transform for tensors In this section, we use the Pestov identity (that we will explain below) as in [PSU1] and the results of previous section to prove the injectivity of X-ray transform on functions and divergence-free 1-forms when the metric has no conjugate points. For 2-tensors, we use the method of [GuKa] to determine the kernel of I2 when the curvature κ is non-positive. First, we denote by D the symmetrized covariant derivative mapping 1-forms to symmetric 2-tensors by Dw(Y1 , Y2 ) := 12 ((∇Y1 w)(Y2 ) + (∇Y2 w)(Y1 )).

(5.1)

Its L2 -adjoint is denoted D∗ and is called the divergence operator on symmetric 2-tensors. We notice that if w is the vector field dual to w, then Dw can be written in terms of Lie derivative of the metric Dw = 12 Lw g.

(5.2)

It will be convenient to use the Fourier decomposition in the circle fibers of SM . Using the vertical vector field, we can decompose each function u ∈ C ∞ (SM ) uniquely as a converging sum (in any C k norms)  uk , with V uk = ikuk u= k∈Z ∞

where uk ∈ C (SM ). This gives an orthogonal (with respect to L2 ) decomposition A C ∞ (SM ) = Ωk k∈Z

where Ωk = ker(V −ik). In isothermal coordinates x = (x1 , x2 ) near a point x0 , one has associated coordinates (x, θ) on SM near Sx0 M with θ ∈ R/2πZ, see Section 2.3. Then the functions uk can be written locally as uk (x, θ) = u ˜k (x)eikθ for some u ˜k smooth on M . In fact, when k ≥ 0, Ωk can be identified to the space of smooth sections of the k-th tensor power of the complex line bundle K := (T ∗ M )1,0 ⊂

280

COLIN GUILLARMOU

CT ∗ M in the sense that uk = πk∗ sk for some section sk ∈ C ∞ (M ; ⊗kS K) where πk∗ is the map (2.3). Similalry when k < 0, Ωk can be identitifed as the space of smooth sections of the k-th tensor power of the bundle K := (T ∗ M )0,1 . There are two natural operators on C ∞ (SM ) called raising and lowering operators, introduced by [GuKa], and defined by η+ := 12 (X + iX⊥ ),

η− := 12 (X − iX⊥ ).

∗ They satisfy η± : Ωk → Ωk±1 , X = η+ + η− and η+ = −η− when acting on smooth functions vanishing at ∂SM .

Theorem 3. Assume that (M, g) has strictly convex boundary and that g has the smooth Livsic property and no conjugate points. Then we have: 1) I0 is injective on C ∞ (M ). 2) If a ∈ C ∞ (M ; T ∗ M )) ∩ ker I1 , then there exists f ∈ C ∞ (M ) such that a = df and f |∂M = 0. 3) Assume that κ ≤ 0. If h ∈ C ∞ (M ; ⊗2S T ∗ M ) ∩ ker I2 , then there exists a 1-form w ∈ C ∞ (M ; T ∗ M ) such that h = Dw and w|∂M = 0, where D is the symmetrized covariant derivative. Proof. If f ∈ ker I0 , by Theorem 2 there is u ∈ C ∞ (SM ) such that u|∂SM = 0 and Xu = π0∗ f . Thus V Xu = 0 since dπ0 (V ) = 0. We have (V X)∗ = XV on smooth functions vanishing at ∂SM by using that V, X preserve μL and that V is tangent to ∂SM . Then we get ||V Xu||2L2 = ||XV u||2L2 + [XV, V X]u, u

and [XV, V X] =XV 2 X − V X 2 V = V XV X + X⊥ V X − V XV X − V XX⊥ =V [X⊥ , X] − X 2 = −X 2 + V κV. This implies the Pestov identity for each u ∈ C ∞ (SM ) vanishing at ∂SM (5.3)

||XV u||2L2 − κV u, V u + ||Xu||2L2 − ||V Xu||2L2 = 0.

We conclude that since V Xu = 0, κ ≤ 0 implies Xu = π0∗ f = 0. In fact, if there are no conjugate points, we claim that for each smooth function h on SM vanishing on ∂SM ||Xh||2L2 − κh, h ≥ 0 and this is equal to 0 only if h = 0. This is proved by using Santalo formula to decompose the integral along geodesics with initial points on ∂− SM \ Γ− and then by using that the index form is non-negative for each of these geodesics, when there is no conjugate points along these geodesics (see for example [PSU2, Lemma 11.2]). This completes the proof of the injectivity of I0 by taking h = V u. Now if I1 a = 0, we have u ∈ C ∞ (SM ) such that Xu = π1∗ a and u|∂SM = 0. We apply the Pestov identity (5.3): since a is a 1-form and V acts as the Hodge operator on pull-backs of 1-forms, we have ||V π1∗ a||2L2 (SM ) = ||π1∗ a||2L2 (SM ) and (5.3) becomes ||XV u||2L2 (SM ) − κV u, V u = 0. This implies that V u = 0 and thus u = π0∗ f for some f ∈ C ∞ (M ) vanishing on ∂M . Then Xu = π1∗ df and this completes the proof since this is equal to π1∗ a.

SCATTERING FOR THE GEODESIC FLOW ON SURFACES WITH BOUNDARY

281

Let h ∈ C ∞ (M ; ⊗2S T ∗ M ), then π2∗ h = h0 + h2 + h−2 with hk ∈ Ωk . If I2 h = 0, there is u ∈ C ∞ (SM ) such that Xu = π2∗ h and, after possibly replacing u by 1 2 (u(x, v) − u(x, −v)), we can always assume that u is odd with respect to the involution A : (x, v) → (x, −v). Indeed, X maps even  functions with respect to A to odd functions, and conversely. We will write u = k uk with uk ∈ Ωk and u2k = 0 for all k ∈ Z. Since X = η+ + η− , we have η+ uk−1 + η− uk+1 = 0 if k∈ / {−2, 0, 2}. Next, using the commutation relation η− η+ = η+ η− − 12 κiV ∗ which follows from (2.2), the fact that uk |∂SM = 0 for all k and η+ = −η− , we obtain

||η+ uk+1 ||2L2 = − η− η+ uk+1 , uk+1 = − η+ η− uk+1 , uk+1 − 12 (k + 1) κuk+1 , uk+1

≥ ||η− uk+1 ||2 for k + 1 ≥ 0, since κ ≤ 0. For k ≥ 3, this implies that ||η+ uk+1 ||L2 ≥ ||η+ uk−1 ||L2 and therefore ck := ||η+ uk ||L2 is a non-decreasing series which converges to 0, that is ck = 0 for all k ≥ 2. A similar argument shows that η− uk = 0 for all k ≤ −2. This shows that u = u1 + u−1 and u = π1∗ w for some smooth 1-form w vanishing  at ∂M . Now it is easy to check that Xu = π2∗ (Dw). Remark that the proof given in 3) actually works as well for ker Im with m ≥ 3, and we obtain that h ∈ ker Im if and only if h = Dw for some w ∈ C ∞ (M, ⊗Sm−1 T ∗ M ) and D is the symmetrized covariant derivative, defined simi∗ ) of an m-symmetric tensor w is larly to (5.1). Acting by X on a pull-back (by πm ∗ equivalent to pull-back Dw on SM by πm+1 . A proof of the injectivity of Im on divergence-free tensors for simple metrics was provided recently in [PSU1], without assuming κ ≤ 0. Combining Theorem 3 with Theorem 2 and Lemma 3.4, we deduce the Corollary 5.1. Let gs be a smooth family of metrics on a surface M with strictly convex boundary, non-positive curvature and with either no trapped set or hyperbolic set. If the lens data (gs , Sgs ) is constant in s, then one has gs = φ∗s g0 where φs is a smooth family of diffeomorphism equal to Identity on ∂M . Proof. Let qs := ∂s gs . By Lemma 3.4, Theorem 3 and Theorem 2, we know that there is ws so that qs = Lws gs (with ws the dual vector field to ws though gs ). We claim that ws = (ΔDs )−1 Ds∗ qs if Ds is the operator D on 1-forms for gs , Ds∗ its adjoint with respect to the L2 -product of gs and ΔDs := Ds∗ Ds with Dirichlet condition. Indeed ΔDs ws = Ds∗ qs = ΔDs (ΔDs )−1 Ds∗ qs and the Laplacian ΔDs with Dirichlet condition has no kernel since ΔDs u, u = ||Ds u||2L2 if u|∂M = 0, and Ds u = 0 with u|∂M = 0 implies u = 0 (check this as an exercise). Then, since gs is smooth in s, it can be shown by elliptic theory that the inverse Δ−1 Ds maps smooth functions of (s, x) to smooth functions of (s, x), if x is the variable on M . This implies that ws is a smooth family in s of smooth 1-forms on M . Integrating the dual vector field ws , we can construct a smooth family of diffeomorphism which are the Identity on ∂M by ∂s φs (x) = ws (φs (x)) and φ0 (x) = x. Then φs satisfies  gs = φ∗s g0 .

282

COLIN GUILLARMOU

6. Some references We haved worked in dimension 2 for simplicity but many results described here are also valid in higher dimension. We provide a few references on the subject, this is not a comprehensive list. We first recommend the lecture notes of Merry-Paternain [Pa] and the lecture notes of Sharafutdinov [Sh], which contain a lot of material on the subject. The survey of Croke [Cr3] also contains a nice overview of the subject (up to 2004). The following articles deal with the boundary rigidity problem or the analysis of X-ray transform. • For simple metrics in a fixed conformal class, Mukhometov [Mu2] proved that the boundary distance function determines the metric, with a stability estimate (see also the previous works [Mu1, MuRo]). This result was proved later with a simpler method by Croke [Cr2]. These works show the injectivity of the X-ray transform on functions for simple metrics (any dimension). • The paper of Michel [Mi] established that simple metrics with constant curvature are boundary rigid in dimension 2. Gromov [Gr] proved the same result in higher dimension for flat metrics. • Croke [Cr1] and Otal [Ot] proved boundary rididity for simple negatively curved surfaces (dimension 2). • Pestov-Uhlmann [PeUh] proved boundary rigidity for all simple surfaces. More particularly, they proved that the scattering data determines the conformal class of the surface by relating the scattering map of the geodesic flow to the Dirichlet-to-Neumann map for the Laplacian. Using Mukhometov result, this shows the full boundary rigidity. • Burago-Ivanov [BuIv] proved that metrics close to flat ones are boundary rigid (any dimension). • Anikonov-Romanov [AnRo] proved injectivity of the X-ray transform on the space of divergence-free 1-forms for simple metrics (any dimension). • Pestov-Sharafutdinov [PeSh] proved the injectivity of the X-ray transform on the space of divergence-free symmetric tensors for simple non-positively curved metrics (any dimension). This uses the so-called Pestov identity. • Stefanov-Uhlmann [StUh] proved injectivity of the X-ray transform on tensors for analytic simple metrics and deduce a local boundary rigidity result for generic metrics. • Paternain-Salo-Uhlmann [PSU1] proved injectivity of the X-ray transform on all divergence-free symmetric tensors for simple surfaces (dimension 2). • Guillarmou [Gu] proved injectivity of the X-ray transform on functions and on the space of divergence-free 1-forms for metrics with strictly convex boundary, hyperbolic trapped set and no conjugate points; this setting contains all negatively curved metrics with strictly convex boundary. When the curvature is in addition non-positive, the injectivity on tensors is also proved (any dimension). The same result as Pestov-Uhlmann is shown also in that class of metrics. • Uhlmann-Vasy [UhVa] proved injectivity of the X-ray transform on functions for manifolds admitting a foliation by convex hypersurfaces (any

SCATTERING FOR THE GEODESIC FLOW ON SURFACES WITH BOUNDARY

283

dimension ≥ 3), and injectivity for the local X-ray transform (i.e. integrals along geodesics almost tangent to boundary) Acknowledgement The author’s thank the anonymous referee for a careful reading and useful comments. References [AnRo] Yu. E. Anikonov and V. G. Romanov, On uniqueness of determination of a form of first degree by its integrals along geodesics, J. Inverse Ill-Posed Probl. 5 (1997), no. 6, 487–490 (1998), DOI 10.1515/jiip.1997.5.6.487. MR1623603 [BoRu] Rufus Bowen and David Ruelle, The ergodic theory of Axiom A flows, Invent. Math. 29 (1975), no. 3, 181–202, DOI 10.1007/BF01389848. MR0380889 [BuIv] Dmitri Burago and Sergei Ivanov, Boundary rigidity and filling volume minimality of metrics close to a flat one, Ann. of Math. (2) 171 (2010), no. 2, 1183–1211, DOI 10.4007/annals.2010.171.1183. MR2630062 [Cr1] Christopher B. Croke, Rigidity for surfaces of nonpositive curvature, Comment. Math. Helv. 65 (1990), no. 1, 150–169, DOI 10.1007/BF02566599. MR1036134 [Cr2] Christopher B. Croke, Rigidity and the distance between boundary points, J. Differential Geom. 33 (1991), no. 2, 445–464. MR1094465 [Cr3] Christopher B. Croke, Rigidity theorems in Riemannian geometry, Geometric methods in inverse problems and PDE control, IMA Vol. Math. Appl., vol. 137, Springer, New York, 2004, pp. 47–72, DOI 10.1007/978-1-4684-9375-7 4. MR2169902 [DKSU] David Dos Santos Ferreira, Carlos E. Kenig, Mikko Salo, and Gunther Uhlmann, Limiting Carleman weights and anisotropic inverse problems, Invent. Math. 178 (2009), no. 1, 119–171, DOI 10.1007/s00222-009-0196-4. MR2534094 [DaSh] Nurlan S. Dairbekov and Vladimir A. Sharafutdinov, Some problems of integral geometry on Anosov manifolds, Ergodic Theory Dynam. Systems 23 (2003), no. 1, 59–74, DOI 10.1017/S0143385702000822. MR1971196 [Gr] Mikhael Gromov, Filling Riemannian manifolds, J. Differential Geom. 18 (1983), no. 1, 1–147. MR697984 [Gu] Colin Guillarmou, Lens rigidity for manifolds with hyperbolic trapped sets, J. Amer. Math. Soc. 30 (2017), no. 2, 561–599, DOI 10.1090/jams/865. MR3600043 [GuKa] V. Guillemin and D. Kazhdan, Some inverse spectral results for negatively curved 2manifolds, Topology 19 (1980), no. 3, 301–312, DOI 10.1016/0040-9383(80)90015-4. MR579579 [KRS] C. E. Kenig, A. Ruiz, and C. D. Sogge, Uniform Sobolev inequalities and unique continuation for second order constant coefficient differential operators, Duke Math. J. 55 (1987), no. 2, 329–347, DOI 10.1215/S0012-7094-87-05518-9. MR894584 [MePa] W.J. Merry, G. Paternain, Inverse Problems in Geometry and Dynamics. Lecture notes. Available at https://www.dpmms.cam.ac.uk/∼gpp24 [Mi] Ren´e Michel, Sur la rigidit´ e impos´ ee par la longueur des g´ eod´ esiques (French), Invent. Math. 65 (1981/82), no. 1, 71–83, DOI 10.1007/BF01389295. MR636880 [Mil] J. Milnor, Morse theory, Based on lecture notes by M. Spivak and R. Wells. Annals of Mathematics Studies, No. 51, Princeton University Press, Princeton, N.J., 1963. MR0163331 [Mu1] R. G. Muhometov, The reconstruction problem of a two-dimensional Riemannian metric, and integral geometry (Russian), Dokl. Akad. Nauk SSSR 232 (1977), no. 1, 32–35. MR0431074 [MuRo] R. G. Muhometov and V. G. Romanov, On the problem of finding an isotropic Riemannian metric in an n-dimensional space (Russian), Dokl. Akad. Nauk SSSR 243 (1978), no. 1, 41–44. MR511273 [Mu2] R. G. Muhometov, On a problem of reconstructing Riemannian metrics (Russian), Sibirsk. Mat. Zh. 22 (1981), no. 3, 119–135, 237. MR621466

284

COLIN GUILLARMOU

Jean-Pierre Otal, Sur les longueurs des g´ eod´ esiques d’une m´ etrique a ` courbure n´ egative dans le disque (French), Comment. Math. Helv. 65 (1990), no. 2, 334–347, DOI 10.1007/BF02566611. MR1057248 [Pa] Gabriel P. Paternain, Geodesic flows, Progress in Mathematics, vol. 180, Birkh¨ auser Boston, Inc., Boston, MA, 1999. MR1712465 [PSU1] Gabriel P. Paternain, Mikko Salo, and Gunther Uhlmann, Tensor tomography on surfaces, Invent. Math. 193 (2013), no. 1, 229–247, DOI 10.1007/s00222-012-0432-1. MR3069117 [PSU2] Gabriel P. Paternain, Mikko Salo, and Gunther Uhlmann, Invariant distributions, Beurling transforms and tensor tomography in higher dimensions, Math. Ann. 363 (2015), no. 1-2, 305–362, DOI 10.1007/s00208-015-1169-0. MR3394381 [PeSh] L. N. Pestov and V. A. Sharafutdinov, Integral geometry of tensor fields on a manifold of negative curvature (Russian), Sibirsk. Mat. Zh. 29 (1988), no. 3, 114–130, 221, DOI 10.1007/BF00969652; English transl., Siberian Math. J. 29 (1988), no. 3, 427–441 (1989). MR953028 [PeUh] Leonid Pestov and Gunther Uhlmann, Two dimensional compact simple Riemannian manifolds are boundary distance rigid, Ann. of Math. (2) 161 (2005), no. 2, 1093–1110, DOI 10.4007/annals.2005.161.1093. MR2153407 [Sh] V.A. Sharafutdinov, Ray transform on Riemannian manifolds, Eight lectures on integral geometry. available at http://www.math.nsc.ru/∼sharafutdinov [StUh] Plamen Stefanov and Gunther Uhlmann, Boundary rigidity and stability for generic simple metrics, J. Amer. Math. Soc. 18 (2005), no. 4, 975–1003, DOI 10.1090/S08940347-05-00494-7. MR2163868 [Ta] Michael E. Taylor, Partial differential equations I. Basic theory, 2nd ed., Applied Mathematical Sciences, vol. 115, Springer, New York, 2011. MR2744150 [UhVa] Gunther Uhlmann and Andr´ as Vasy, The inverse problem for the local geodesic ray transform, Invent. Math. 205 (2016), no. 1, 83–120, DOI 10.1007/s00222-015-0631-7. MR3514959 E-mail address: [email protected] [Ot]

´ DMA, U.M.R. 8553 CNRS, Ecole Normale Superieure, 45 rue d’Ulm, 75230 Paris cedex 05, France

Published Titles in This Subseries 700 Alexandre Girouard, Dmitry Jakobson, Michael Levitin, Nilima Nigam, Iosif Polterovich, and Fr´ ed´ eric Rochon, Editors, Geometric and Computational Spectral Theory, 2017 680 Sergei Gukov, Mikhail Khovanov, and Johannes Walcher, Editors, Physics and Mathematics of Link Homology, 2016 655 A. C. Cojocaru, C. David, and F. Pappalardi, Editors, SCHOLAR—a Scientific Celebration Highlighting Open Lines of Arithmetic Research, 2015 654 Carlo Gasbarri, Steven Lu, Mike Roth, and Yuri Tschinkel, Editors, Rational Points, Rational Curves, and Entire Holomorphic Curves on Projective Varieties, 2015 638 Javad Mashreghi, Emmanuel Fricain, and William Ross, Editors, Invariant Subspaces of the Shift Operator, 2015 630 Pierre Albin, Dmitry Jakobson, and Fr´ ed´ eric Rochon, Editors, Geometric and Spectral Analysis, 2014 622 S. Ejaz Ahmed, Editor, Perspectives on Big Data Analysis, 2014 606 Chantal David, Matilde Lal´ın, and Michelle Manes, Editors, Women in Numbers 2, 2013 605 Omid Amini, Matthew Baker, and Xander Faber, Editors, Tropical and Non-Archimedean Geometry, 2013

CONM

700

American Mathematical Society www.ams.org

ISBN 978-1-4704-2665-1

AMS/CRM

9 781470 426651 CONM/700

Centre de Recherches Mathématiques www.crm.math.ca

Geometric and Computational Spectral Theory • Girouard et al., Editors

The book is a collection of lecture notes and survey papers based on the mini-courses given by leading experts at the 2015 S´eminaire de Math´ematiques Sup´erieures on Geometric and Computational Spectral Theory, held from June 15–26, 2015, at the Centre de Recherches Math´ematiques, Universit´e de Montr´eal, Montr´eal, Quebec, Canada. The volume covers a broad variety of topics in spectral theory, highlighting its connections to differential geometry, mathematical physics and numerical analysis, bringing together the theoretical and computational approaches to spectral theory, and emphasizing the interplay between the two.