Spectral Theory and Mathematical Physics: STMP 2018, Santiago, Chile [1st ed.] 9783030555559, 9783030555566

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Latin American Mathematics Series UFSCar subseries

Pablo Miranda Nicolas Popoff Georgi Raikov  Editors

Spectral Theory and Mathematical Physics STMP 2018, Santiago, Chile

Latin American Mathematics Series

Latin American Mathematics Series – UFSCar subseries Managing Series Editors César R. de Oliveira, Federal University of São Carlos, São Carlos, Brazil Ruy Tojeiro, University of São Paulo, São Carlos, Brazil

Series Editors Shiferaw Berhanu, Temple University, Philadelphia, PA, USA Ugo Bruzzo, SISSA – Scuola Internazionale Superiore di Studi Avanzati, Trieste, Italy Irene Fonseca, Carnegie Mellon University, Pittsburgh, PA, USA

Aims and Scope Published under the Latin American Mathematics Series, which was created to showcase the new, vibrant mathematical output that is emerging from this region, the UFSCar subseries aims to gather high-quality monographs, graduate textbooks, and contributing volumes based on mathematical research conducted at/with the Federal University of São Carlos, a technological pole located in the State of São Paulo, Brazil. Submissions are evaluated by an international editorial board and undergo a rigorous peer review before acceptance.

More information about this subseries at http://www.springer.com/series/15995

Pablo Miranda • Nicolas Popoff • Georgi Raikov Editors

Spectral Theory and Mathematical Physics STMP 2018, Santiago, Chile

Editors Pablo Miranda Departamento de Matemáticas y C. C. Universidad de Santiago de Chile Santiago, Chile

Nicolas Popoff Institut de Mathématiques de Bordeaux Talence, Gironde, France

Georgi Raikov Facultad de Matemáticas Pontificia Universidad Católica de Chile Santiago, Chile

Latin American Mathematics Series ISSN 2524-6755 ISSN 2524-6763 (electronic) Latin American Mathematics Series – UFSCar subseries ISBN 978-3-030-55555-9 ISBN 978-3-030-55556-6 (eBook) https://doi.org/10.1007/978-3-030-55556-6 Mathematics Subject Classification: 35P20, 35P25, 47B35, 47F05, 81Q10 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG. The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

This volume contains the proceedings of the conference Spectral Theory and Mathematical Physics (STMP) that took place at the Pontificia Universidad Católica de Chile (PUC), Santiago, in December 2018. The main purpose of this conference was to bring together a number of young researchers in spectral theory and mathematical physics, and some established specialists in this field, to connect people from different schools and generations, give them the opportunity to exchange ideas and start new collaboration. The conference STMP was organized within the framework of the grant for international networking of young researchers (Grant REDI170562) financed by the Chilean National Commission for Scientific and Technological Research (CONICYT). We were also supported by the French National Centre for Scientific Research (CNRS) international collaboration grant PICS07851. The organizers gratefully acknowledge the financial support of CONICYT, CNRS, the Vice-Rectory of Research and the Faculty of Mathematics of PUC, the International Association of Mathematical Physics, and the Chilean National Fund for Scientific and Technological Development (FONDECYT). Special gratitude is due to the Faculty of Mathematics of PUC for hosting the conference and to its administrative staff for their logistic help in the organization of STMP. This volume contains survey articles and original results presented at the conference. Most of the chapters are dedicated to some of the following topics: • • • • •

Eigenvalues and resonances for quantum Hamiltonians, Spectral shift function and quantum scattering, Spectral properties of random operators, Magnetic quantum Hamiltonians, and Microlocal analysis and its applications in mathematical physics.

As editors, we are grateful to the authors who contributed to this book and to the referees for their professional and time-consuming work. We would also like to thank Liliya Simeonova for handling manuscripts and referee reports and for technical assistance in the preparation of this volume.

v

vi

Preface

It has become a tradition to organize in Santiago de Chile conferences on spectral theory and mathematical physics and to publish their proceedings. The present publication is the third of the series. The proceedings of Spectral Days 2010 and those of STMP 2014 were published by Springer in 2012 and 2016, respectively. We hope to maintain this tradition in the future. Santiago, Chile Talence, France Santiago, Chile

Pablo Miranda Nicolas Popoff Georgi Raikov

Contents

Commutator Methods for N-Body Schrödinger Operators . . . . . . . . . . . . . . . . . Tadayoshi Adachi, Kyohei Itakura, Kenichi Ito, and Erik Skibsted

1

Resolvent Estimates and Resonance Free Domains for Schrödinger Operators with Matrix-Valued Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Marouane Assal

17

One-dimensional Discrete Anderson Model in a Decaying Random Potential: from A.C. Spectrum to Dynamical Localization . . . . . . . . . . . . . . . . . Olivier Bourget, Gregorio R. Moreno Flores, and Amal Taarabt

37

On Non-selfadjoint Operators with Finite Discrete Spectrum . . . . . . . . . . . . . . Olivier Bourget, Diomba Sambou, and Amal Taarabt

63

Pseudo-Differential Perturbations of the Landau Hamiltonian . . . . . . . . . . . . Esteban Cárdenas

87

Semiclassical Surface Wave Tomography of Isotropic Media . . . . . . . . . . . . . . . 105 Maarten V. de Hoop and Alexei Iantchenko Persistence of Point Spectrum for Perturbations of One-Dimensional Operators with Discrete Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . 125 César R. de Oliveira and Mariane Pigossi Resonances for a System of Schrödinger Operators above an Energy-Level Crossing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 Setsuro Fujiié, André Martinez, and Takuya Watanabe Nonexistence Result for Wave Operators in Massive Relativistic System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 Atsuhide Ishida Quantised Calculus for Perturbed Massive Dirac Operator on Noncommutative Euclidean Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 Galina Levitina and Fedor Sukochev vii

viii

Contents

On the Explicit Semiclassical Limiting Eigenvalue (Resonance) Distribution for the Zeeman (Stark) Hydrogen Atom Hamiltonian . . . . . . . . 199 Carlos Pérez-Estrada and Carlos Villegas-Blas The Negative Spectrum of the Robin Laplacian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 Nicolas Popoff On Some Integral Operators Appearing in Scattering Theory, and their Resolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 Serge Richard and Tomio Umeda The Strong Scott Conjecture: the Density of Heavy Atoms Close to the Nucleus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 Heinz Siedentop

Commutator Methods for N -Body Schrödinger Operators Tadayoshi Adachi, Kyohei Itakura, Kenichi Ito, and Erik Skibsted

1 Introduction In this article we report on the author’s recent work [1] on spectral theory of the N-body Schrödinger operators. The main results include Rellich’s theorem, LAP (the Limiting Absorption Principle), microlocal resolvent bounds, and a microlocal Sommerfeld uniqueness result. All of these are formulated in sharp forms employing the Besov-type spaces under minimal assumptions on pair-potentials. We will omit proofs of the results presented in this article, but the basic philosophy is largely dependent on commutator methods proposed by two of the authors [20].

T. Adachi Graduate School of Human and Environmental Studies, Kyoto University, Kyoto, Japan e-mail: [email protected] K. Itakura Research Organization of Science and Technology, Ritsumeikan University, Shiga, Japan e-mail: [email protected] K. Ito () Graduate School of Mathematical Sciences, The University of Tokyo, Tokyo, Japan e-mail: [email protected] E. Skibsted Institut for Matematiske Fag, Aarhus Universitet, Aarhus C, Denmark e-mail: [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 P. Miranda et al. (eds.), Spectral Theory and Mathematical Physics, Latin American Mathematics Series, https://doi.org/10.1007/978-3-030-55556-6_1

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T. Adachi et al.

2 Setting 2.1 N -Body Hamiltonian Let us introduce a generalized N-body model with hard-cores. Let X be a real inner product space of dimension d ∈ N = {1, 2, . . .}, and let {Xa }a∈A be a finite family of subspaces of X. We assume that {Xa }a∈A is closed under intersection, i.e., for any a, b ∈ A there exists a ∨ b ∈ A such that Xa ∩ Xb = Xa∨b . The elements of A are called cluster decompositions, and we write a ⊂ b if Xa ⊃ Xb . In addition, we assume that there exist amin , amax ∈ A such that Xamin = X,

Xamax = {0}.

For any a ∈ A we denote #a = max{k | a = a1  · · ·  ak = amax }, and we say the family {Xa }a∈A is of N-body type if #amin = N + 1. Note that (N + 1) number of particles form an N-body system after separation of the center of mass. We denote by Xa ⊂ X the orthogonal complement of Xa ⊂ X, and decompose x ∈ X as x = x a ⊕ xa ∈ Xa ⊕ Xa . The component xa is called the inter-cluster coordinates, and x a the internal coordinates. Note that for any a, b ∈ A Xa + Xb = Xa∨b . We say V : X → R is a potential of N-body type if there exist Va : Xa → R, a ∈ A, called pair-potentials, such that V (x) =



Va (x a ) for x ∈ X.

a∈A

We may let Vamin = 0 without loss of generality. We also consider hard-cores as follows: For each a ∈ A let Ωa ⊂ Xa be a non-empty open subset with Xa \ Ωa being compact, and we set

Commutator Methods for N -Body Schrödinger Operators

Ω=



3

(Ωa + Xa ).

a∈A

Note Ωamin = Xamin = {0}, since Ωamin = ∅. The complement X \ Ω is the ‘hardcores’ where particles can not penetrate. We refer to [14] for a similar setting. We use the standard notation y = (1 + |y|2 )1/2 . We denote the space of all the bounded operators from a normed space X to another normed space Y by L(X, Y ), and that of all the compact operators by C(X, Y ). The dual space of X is denoted by X∗ . Condition 1 Let δ ∈ (0, 1/2]. For each a ∈ A \ {amin } there exists a splitting Va = Valr + Vasr + Vasi into three real-valued measurable functions in C(H01 (Ωa ), H01 (Ωa )∗ ) such that: 1. Valr has first order distributional derivatives in L1loc (Ωa ) and x a 1+2δ ∂ α Valr ∈ L(H01 (Ωa ), H01 (Ωa )∗ );

|α| = 1.

2. Vasr satisfies x a 1+2δ Vasr ∈ L(H01 (Ωa ), L2 (Ωa )).

(1)

3. Vasi vanishes outside a bounded subset of Ωa . We remark that our assumptions on pair-potentials are very general and may be considered minimal: We impose only a condition on the first order derivatives on long-range parts and have no conditions on derivatives on short-range parts, and we also include singularities of form-bounded type and hard-cores. There is also an alternative version with singularities of operator-bounded type, but we omit it here, see [1]. Now we consider the generalized N-body Hamiltonian H = H0 + V ;

H0 = − 12 Δ,

on H = L2 (Ω).

Here Δ is the Laplace–Beltrami operator on X associated with the inner product. We impose the Dirichlet boundary condition on ∂Ω, i.e., H is defined as the self-adjoint operator associated with the closed quadratic form H˜ given by H˜ ψ = 12 pψ, pψ + ψ, V ψ for ψ ∈ Q(H˜ ) = H01 (Ω);

p = −i∇.

Note that V is infinitesimally H0 -small as a form. Note also that, if we regard H˜ as a bounded operator H01 (Ω) → H01 (Ω)∗ , then the self-adjoint operator H is realized as

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H = H˜ |D(H ) ;

   D(H ) = ψ ∈ H01 (Ω)  H˜ ψ ∈ H ,

(2)

due to construction of the Friedrichs extension, see e.g. [34].

2.2 Sub-Hamiltonians We recall sub-Hamiltonians H a associated with a ∈ A. If a = amin , we define H amin = 0 on Hamin = L2 (Ω amin ) = C. Note V amin = 0 and Ω amin = {0}. If a = amin , for which {Xb ∩ Xa }b⊂a forms a family of subspaces of (#a − 1)-body type in Xa , we let V a (x a ) =



Vb (x b ),

Ωa =

b⊂a

  Ωb + (Xb ∩ Xa ) , b⊂a

similarly to the full Hamiltonian H . Then we define the sub-Hamiltonian H a as H a = − 12 Δx a + V a on Ha = L2 (Ω a ) with the Dirichlet boundary condition on ∂Ω a . We set    σpp (H a )  a ∈ A \ {amax } , T (H ) = and call its element a threshold of H . Under Conditions 1 it is known that that T (H ) is closed and at most countable. Moreover non-threshold eigenvalues are discrete in R \ T (H ), and they can accumulate only at T (H ) from below. See [10, 18, 27]. The so-called HVZ theorem, see e.g. [28], says σess (H ) = [min T (H ), ∞).

2.3 Unique Continuation Property Let us assume the unique continuation property rather than imposing technical sufficient conditions on Va and Ωa . For any a ∈ A \ {amin } we introduce    1 (Ω a ) = ψ ∈ L2loc (Ω a )  χ ψ ∈ H01 (Ω a ) for any χ ∈ Cc∞ (Xa ) . H0,loc

Commutator Methods for N -Body Schrödinger Operators

5

Then H a : H01 (Ω a ) → H01 (Ω a )∗ naturally extends as 1 (Ω a ) → D (Ω a ) H a : H0,loc 1 (Ω a ) a generalized Dirichlet eigenfunction by a partition of unity. We call φ ∈ H0,loc a for H with eigenvalue E ∈ C if it satisfies

H a φ = Eφ in the distributional sense. 1 (Ω a ) is a generalized Dirichlet Condition 2 For all a = amin , if φ ∈ H0,loc a eigenfunction for H and φ = 0 on a non-empty open subset of Ω a , then φ = 0 on Ωa.

One sufficient condition for the unique continuation property here is that V a is locally bounded and Ω a is connected, see e.g. [31]. Roughly, we would not like to allow strong singularities of V a or Ω a to separate the space Xa with a component supporting an isolated eigenfunction. The Coulomb singularity can be treated, see the proof of [18, Corollary 1.8]. See also [11] for the unique continuation property for N -body Schrödinger operators without hard-core interactions.

3 Results 3.1 Exponential Decay and Rellich’s Theorem In order to state our version of Rellich’s theorem we recall definitions of the Besov spaces associated with multiplication operator |x| on H. Set  

 x ∈ Ω  |x| < 1 , 

  Fn = F x ∈ Ω  2n−1 ≤ |x| < 2n for n = 1, 2, . . . , F0 = F

where F (S) demotes the sharp characteristic function of S ⊂ Ω. The Besov spaces B, B ∗ and B0∗ are defined as    B = ψ ∈ L2loc (Ω)  ψB < ∞ ,   B ∗ = ψ ∈ L2loc (Ω)  ψB∗ < ∞ , 

ψB =

2n/2 Fn ψH ,

n=0

ψB∗ = sup 2−n/2 Fn ψH ,



 B0∗ = ψ ∈ B ∗  lim 2−n/2 Fn ψH = 0 ,

∞ 

n≥0

n→∞

respectively. If we denote the standard weighted L2 spaces by

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L2s = x −s L2 (Ω) for s ∈ R, then for any s > 1/2 L2s  B  L21/2  H  L2−1/2  B0∗  B ∗  L2−s . 1 (Ω) be a generalized Theorem 1 Suppose Conditions 1 and 2. Let φ ∈ B0∗ ∩ H0,loc Dirichlet eigenfunction for H with real eigenvalue E ∈ R, and set

   α0 = sup α ≥ 0  eα|x| φ ∈ B0∗ ∈ [0, ∞]. Then one has E + 12 α02 ∈ T (H ) ∪ {∞}. In addition, if α0 = ∞, then φ = 0 on Ω. Corollary 1 There are no nonzero generalized Dirichlet eigenfunctions for H in B0∗ with positive eigenvalues, and there are no positive thresholds for H , neither. Our proof of Theorem 1 unifies previous treatments of Rellich’s theorem [17, 19] and exponential decay estimates of L2 -eigenfunctions [10, 18]. Moreover Theorem 1 and Corollary 1 extend [18, 19]. See [17] and the references therein for an account of history of Rellich’s theorem. We can also extend [27], or [16, Theorem 6.11], by similar arguments.

3.2 LAP Bounds We next provide the sharp LAP bounds. For an interval I ⊂ R we set I± := {z ∈ C | Re z ∈ I, 0 < ± Im z ≤ 1}. Theorem 2 Suppose Condition 1. Let I ⊂ R \ (σpp (H ) ∪ T (H )) be a compact interval. Then there exists C > 0 such that uniformly in z ∈ I± and ψ ∈ B R(z)ψB∗ +

d 

pj R(z)ψB∗ ≤ CψB .

(3)

j =1

Instead of Condition 1 we can also deal with operator-bounded pair-potentials like [3, 30], but here we do not present such a variation, see [1] for precise statements. Theorem 2 and the above-mentioned variation extend many of the results known so far to the general setting of this article, cf. [3–8, 22–25, 30]. Here we do not give precise comparison with them, but see [1] again for it. For the proof

Commutator Methods for N -Body Schrödinger Operators

7

we reformulate the Mourre estimate in terms of a certain ‘zeroth order’ operator B given below. In particular our approach is not based on the differential inequality technique of Mourre, cf. [3–6, 8, 21, 24, 25, 30]. Actually this differential inequality technique (as indicated, used a lot so far) may be considered as somehow indirect, and instead we propose a more direct scheme to bound the resolvent employing a disguised form of the Mourre estimate, elementary commutator estimates and Rellich’s theorem (see Section 4). The latter approach is to some extent similar to [2, 15] for the 1-body case.

3.3 Rescaled Graf Function and the Operator B For a function r1 ∈ C ∞ (X) we denote ω˜ 1 =

1 2

h˜ 1 =

grad r12 ,

1 2

Hess r12 .

We identify the tangent space of X at each x ∈ X with X itself, and we regard ω˜ 1 (x) ∈ X for each x ∈ X. Similarly, using the inner product of X, we regard h˜ 1 (x) : X → X for each x ∈ X. It is known by [13], see also [9, 29], that there exist r1 ∈ C ∞ (X) and a smooth partition of unity {η1,a }a∈A on X such that: 1. There exist c, C > 0 such that for any a, b ∈ A with a ⊂ b and x ∈ supp η1,b |x a | ≥ c,

|x b | ≤ C;

2. There exists C  > 0 such that for any x ∈ X r1 (x) ≥ 1,

  r1 (x) − |x| ≤ C  ;

3. There exists c > 0 such that for any a ∈ A and x ∈ X with |x a | ≤ c ω˜ 1a (x) = 0; 4. For any x, y ∈ X   η1,a (x)|ya |2 ; y, h˜ 1 (x)y ≥



a∈A

5. For any α ∈ Nd0 and k ∈ N0 there exists Cαk > 0 such that for any x ∈ X     ∂ α η1,a (x) + ∂ α (x · ∇)k (ω˜ 1 (x) − x) ≤ Cαk . a∈A

Here we let N0 = {0, 1, 2, . . .}.

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Now we set for large R ≥ 1 rR (x) = Rr1 (x/R),

ωR = grad rR ,

ηR (x) = η1 (x/R),

and define the self-adjoint operator BR on H as BR = 12 (ωR · p + p · ωR ). In the following let us suppress the dependence on R ≥ 1, writing simply r = rR ,

ω = ωR ,

B = BR ,

η = ηR .

In our theory the operator B is employed in a systematic manner. See [12], where B was also employed although not systematically.

3.4 Microlocal Resolvent Bounds and Applications Here we present microlocal resolvent bounds. Define d : R → R as  d(λ) =

   inf λ − τ  τ ∈ T (H ) ∩ (−∞, λ]

if T (H ) ∩ (−∞, λ] = ∅,

1

otherwise,

and introduce for any λ ∈ R and I ⊂ R γ (λ) =

 2d(λ),

γ− (I ) = inf γ (λ) = inf λ∈I



λ∈I

2d(λ).

We let δ ∈ (0, 1/2] be the parameter from Condition 1, and we then set κ = δ/(1 + 2δ) ∈ (0, 1/4].

(4)

Theorem 3 Suppose Condition 1. Let I ⊂ R \ (σpp (H ) ∪ T (H )) be a compact interval, and take R ≥ 1 sufficiently large. Then for any β ∈ (0, κ) and F ∈ C ∞ (R) with supp F ⊂ (−∞, γ− (I )) and F  ∈ Cc∞ (R), there exists C > 0 such that uniformly in z ∈ I± and ψ ∈ L21/2+β F (±B)R(z)ψL2

−1/2+β

respectively.

≤ CψL2

1/2+β

,

Commutator Methods for N -Body Schrödinger Operators

9

The first application of Theorem 3 is the Hölder continuity of R(z), and in particular LAP. (We distinguish between ‘LAP bounds’ and ‘LAP’. The former refers to locally uniform boundedness of the resolvent near the real axis, while the later refers to existence of limits at the real axis, cf. the terminologies ‘LAP’ and ‘strong LAP’ used in [3].) Corollary 2 Suppose Condition 1. Let I ⊂ R \ (σpp (H ) ∪ T (H )) be a compact interval, and let s > 1/2 and β ∈ (0, min{κ, s − 1/2}). Then there exists C > 0 such that for all j ∈ {1, . . . , d}, k ∈ {0, 1}, z ∈ I± and z ∈ I± , respectively, pjk R(z) − pjk R(z )L(L2 ,L2 s

−s )

≤ C|z − z |β .

In particular, for any E ∈ I and s > 1/2 the following boundary values exist: pjk R(E ± i0) := lim pjk R(E ± i) in L(L2s , L2−s ), →0+

respectively. The same boundary values are realized (in an extended form) as pjk R(E ± i0) = s-w -lim pjk R(E ± i) in L(B, B ∗ ), →0+

respectively. The second application is a microlocal Sommerfeld uniqueness result. It characterizes R(E ± i0) by the Helmholtz equation and microlocal radiation conditions. 1 (Ω) is a generalized Dirichlet solution to Given ψ ∈ L2loc (Ω), we say φ ∈ H0,loc (H − E)φ = ψ if it satisfies (H − E)φ = ψ in the distributional sense. Corollary 3 Suppose Condition 1. Let E ∈ R \ (σpp (H ) ∪ T (H )), and take R ≥ 1 sufficiently large. Let ψ ∈ r −β B with β ∈ [0, κ). Then φ = R(E ± i0)ψ ∈ B ∗ ∩ 1 (Ω) satisfies H0,loc 1. φ is a generalized Dirichlet solution to (H − E)φ = ψ, 2. for any F ∈ C ∞ (R) with supp F ⊂ (−∞, γ (E)) and F  ∈ Cc∞ (R) one has F (±B)φ ∈ r −β B0∗ , 1 (Ω), s ∈ R, satisfies respectively. Conversely, if φ  ∈ L2s ∩ H0,loc

1. φ  is a generalized Dirichlet solution to (H − E)φ  = ψ, 2. there exists γ > 0 such that for any F ∈ C ∞ (R) with

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supp F ⊂ (−∞, γ ) and F  ∈ Cc∞ (R) one has F (±B)φ  ∈ B0∗ , then φ  = R(E ± i0)ψ, respectively. Corollary 3 generalizes versions [17, 26] of microlocal Sommerfeld uniqueness results in several ways. We also refer to [32, 33] and the references therein.

4 Commutators The proofs of the main results are heavily dependent on commutator arguments. We construct an appropriate ‘propagation observable’ P , and compute and bound the commutator DP := i[H, P ]

(5)

from below by a non-negative quantity. Although the propagation observable P changes according to the context, it is basically made up of functions of H , r and B. For example, to prove Theorem 1 we first choose P = Θ(r)f (H )ζ (B)f (H )Θ(r) ∈ L(H), where Θ, f and ζ are certain functions for which we do not give precise definitions in this paper. It is observed that the computation of (5) reduces to those of commutators of functions of H , r and B. In this section we present formal computations of these commutators without proofs. More precisely, there are two ingredients. First, to bound (5) from below we need to bound DB := i[H, B] from below. This seems very similar to the socalled Mourre commutator, but our lower bound is not uniformly positive but decays at infinity. Nevertheless our theory works and provides refined results. Second, to control errors we need to bound commutators of functions of H , r and B from above. Particularly a control of the second commutator i[i[H, B], B] is a key issue under our minimal assumptions. In these computations there always appears the domain problem making proofs slightly technical, here largely ignored though. A rigorous treatment is of course possible, and we refer to [1] for it along with the full proofs of our results.

4.1 Bound from below Here we bound the commutator DB = i[H, B] from below. It would be convenient to first have a comparison with the Mourre commutator considered with respect to

Commutator Methods for N -Body Schrödinger Operators

A = 12 (ω˜ · p + p · ω); ˜

11

ω˜ =

1 2

grad r 2 .

In fact, then we have A = 2i [H, r 2 ],

B = i[H, r],

A = r 1/2 Br 1/2 ,

and hence ˜ − 1 (Δ2 r 2 ) − ω˜ · (∇V ), i[H, A] = p · hp 8

 −1/2 i[H, B] = r i[H, A] − B 2 r −1/2 + 14 r −2 ω · hω, where h˜ =

1 2

Hess r 2 ,

h = Hess r.

Now the Mourre estimate, e.g. [18], suggests the following bound. Proposition 1 Let λ ∈ R \ T (H ) and σ ∈ (0, γ (λ)), and take R ≥ 1 large enough and a neighborhood U ⊂ R of λ small enough. Then for any real-valued function f ∈ Cc∞ (U ) there exists C > 0 such that

 f (H )(DB)f (H ) ≥ f (H )r −1/2 σ 2 − B 2 r −1/2 f (H ) − Cr −2 . Remark 1 Unlike the usual Mourre estimate the first term on the right-hand side is not uniformly positive since we have the weights r −1/2 at both sides. In addition, we have a negative effect from −B 2 . Nevertheless commutator arguments still work. We could say that this is because we actually have a better balance of positivity and error control than the usual Mourre estimate does. The advantage of using B rather than A is the different ‘order’, in fact B is of ‘zeroth order’ and therefore relatively bounded to the Hamiltonian H .

4.2 Bound from above Next we bound commutators of functions of H , r and B from above, cf. [12]. Throughout the subsection let us fix sufficiently large R ≥ 1. Let us introduce the following terminology. Definition 1 Let T be a linear operator on H with L2∞ := ∩s∈R L2s ⊂ D(T ), and let t ∈ R. We say T is an operator of order t, and write T = O(r t ), if for each s ∈ R there exists Cs > 0 such that

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r s−t T r −s f  ≤ Cs f  for all f ∈ L2∞ . Note that, if T = O(r t ) and S = O(r s ), then T ∗ = O(r t ) and T S = O(r t+s ). Let us also recall the Helffer–Sjöstrand formula. For any t ∈ R set    F t = f ∈ C ∞ (R)  |f (k) (x)| ≤ Ck x t−k for any k ∈ N0 and x ∈ R . It is known that for any f ∈ F t there exists an almost analytic extension f˜ ∈ C ∞ (C) such that f˜|R = f,

|f˜(z)| ≤ Cz t ,

  (∂¯ f˜)(z) ≤ Ck | Im z|k z t−k−1 for any k ∈ N0 .

Lemma 1 Let T be a self-adjoint operator on H, and let f ∈ F t with t ∈ R. Take an almost analytic extension f˜ ∈ C ∞ (C) of f , and set dμf (z) = π −1 (∂¯ f˜)(z) dudv;

z = u + iv.

Then for any k ∈ N0 with k > t the operator f (k) (T ) ∈ L(H) is expressed as  f

(k)

(T ) = (−1) k! k

C

(T − z)−k−1 dμf (z).

Now, using the Helffer–Sjöstrand formula, we can express and bound the first commutators of functions of H , r and B as follows. Proposition 2 1. Let f ∈ F 0 . Then f (H ) = O(r 0 ). 2. Let f ∈ F t with t < 1/2, and let s ∈ R. Then one can write  i[f (H ), r s ] = −s

C



(H − z)−1 Re(r s−1 ω · p) (H − z)−1 dμf (z),

(6)

and, in particular, i[f (H ), r s ] = O(r s−1 ). Proposition 3 1. Let F ∈ F 0 . Then F (B) = O(r 0 ). 2. Let F ∈ F t with t < 1, and let s ∈ R. Then one can write 

 s i[F (B), r ] = −s (B − z)−1 ω2 r s−1 (B − z)−1 dμF (z), C

and, in particular, i[F (B), r s ] = O(r s−1 ). 

Proposition 4 Let f ∈ F t and F ∈ F t with t < −1/2 and t  < 1. Then one can write

Commutator Methods for N -Body Schrödinger Operators

 i[f (H ), B] = −  i[f (H ), F (B)] = −

C

C

13

 (H − z)−1 DB (H − z)−1 dμf (z),

 (B − z)−1 i[f (H ), B] (B − z)−1 dμF (z),

and, in particular, i[f (H ), B] = O(r −1 ),

i[f (H ), F (B)] = O(r −1 ).

Finally we present a bound for the second commutator i[DB, B], or more generally for i[i[f (H ), B], B]. Since we do not assume any additional differentiability for the potential V , the realization of i[DB, B] itself is highly non-trivial. Actually its verification requires some technical decompositions of the operator B, which we are going to omit in this paper. We refer to [1] for the precise statements and proofs. We remark that as in the usual Mourre theory at present we can not avoid these second commutators, cf. [4]. Proposition 5 Let κ = δ/(1 + 2δ) as in (4). Then

 (H − i)−1 i[DB, B] (H + i)−1 = O(r −1−2κ ). Corollary 4 Let f ∈ F t with t < −1. Then one can write   i i[f (H ), B], B = −

 C

 (H − z)−1 i[DB, B] (H − z)−1 dμf (z)



+2

C

(H − z)−1 (DB)(H − z)−1 (DB)(H − z)−1 dμf (z),

and, in particular,   i i[f (H ), B], B = O(r −1−2κ ). Acknowledgments K. Ito would like to thank the anonymous referee for valuable comments that improved presentation of this paper. T.A. is supported by JSPS KAKENHI, grant nr. 17K05319. K. Ito is supported by JSPS KAKENHI, grant nr. 17K05325. E.S. is supported by the Danish Council for Independent Research | Natural Sciences, grant nr. DFF-4181-00042.

References 1. T. Adachi, K. Itakura, K. Ito, E. Skibsted, New Methods in Spectral Theory of N-Body Schrodinger Operators. arXiv:1804.07874 [math-ph] 2. S. Agmon, L. Hörmander, Asymptotic properties of solutions of differential equations with simple characteristics. J. d’Analyse Math. 30, 1–38 (1976)

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3. W. Amrein, A. Boutet de Monvel-Bertier, V. Georgescu, On Mourre’s approach to spectral theory. Helv. Phys. Acta 62(1), 1–20 (1989) 4. W. Amrein, A. Boutet de Monvel-Bertier, V. Georgescu, C0 -groups, Commutator Methods and Spectral Theory of N -Body Hamiltonians (Birkhäuser, Basel–Boston–Berlin, 1996) 5. A. Boutet de Monvel, V. Georgescu, M. Mantoiu, Mourre theory in a Besov space setting. C. R. Acad. Sci. Paris, Ser. I 310, 233–237 (1990) 6. A. Boutet de Monvel, V. Georgescu, M. Mantoiu, Locally smooth operators and the limiting absorption principle for N-body Hamiltonians. Rev. Math. Phys. 5(1), 105–189 (1993) 7. A. Boutet de Monvel-Bertier, V. Georgescu, A. Soffer, N -body Hamiltonians with hard-core interactions. Rev. Math. Phys. 6(4), 515–596 (1994) 8. A. Boutet de Monvel-Bertier, D. Manda, R. Purice, The commutator method for form-relatively compact perturbations. Lett. Math. Phys. 22, 211–223 (1991) 9. J. Derezi´nski, Asymptotic completeness for N -particle long-range quantum systems. Ann. Math. 38, 427–476 (1993) 10. R. Froese, I. Herbst, Exponential bounds and absence of positive eigenvalues for N -body Schrödinger operators. Comm. Math. Phys. 87(3), 429–447 (1982/83) 11. V. Georgescu, On the unique continuation property for Schrödinger Hamiltonians. Helv. Phys. Acta 52, 655–670 (1979) 12. C. Gérard, H. Isozaki, E. Skibsted, N -body resolvent estimates. J. Math. Soc. Jpn. 48(1), 135– 160 (1996) 13. G.M. Graf, Asymptotic completeness for N -body short-range quantum systems: a new proof. Commun. Math. Phys. 132, 73–101 (1990) 14. M. Griesemer, N -body quantum systems with singular potentials. Ann. Inst. Henri Poincaré 69(2), 135–187 (1998) 15. L. Hörmander, The Analysis of Linear Partial Differential Operators. IV (Springer, Berlin, 1983–1985) 16. W. Hunziker, I.M. Sigal, The quantum N-body problem. J. Math. Phys. 41(6), 3448–3510 (2000) 17. H. Isozaki, A generalization of the radiation condition of Sommerfeld for N -body Schrödinger operators. Duke Math. J. 74(2), 557–584 (1994) 18. K. Ito, E. Skibsted, Absence of positive eigenvalues for hard-core N -body systems. Ann. Inst. Henri Poincaré 15, 2379–2408 (2014) 19. K. Ito, E. Skibsted, Rellich’s theorem and N -body Schrödinger operators. Rev. Math. Phys. 28(5), 12 pp. (2016) 20. K. Ito, E. Skibsted, Radiation condition bounds on manifolds with ends. J. Funct. Anal. 278(9), 108449 (2020) 21. A. Jensen, P. Perry, Commutator methods and Besov space estimates for Schrödinger operators. J. Oper. Theory 14, 181–188 (1985) 22. R. Lavine, Absolute continuity of Hamiltonian operators with repulsive potential. Proc. Am. Math. Sot. 22, 55–60 (1969) 23. R. Lavine, Absolute continuity of positive spectrum for Schrödinger operators with long-range potentials. J. Funct. Anal. 12, 30–54 (1973) 24. É. Mourre, Absence of singular continuous spectrum for certain selfadjoint operators. Commun. Math. Phys. 78(3), 391–408 (1980/81) 25. É. Mourre, Operateurs conjugués et propriétés de propagation. Commun. Math. Phys. 91, 297– 300 (1983) 26. J.S. Møller, An abstract radiation condition and applications to N -body systems. Rev. Math. Phys. 12(5), 767–803 (2000) 27. P. Perry, Exponential bounds and semifiniteness of point spectrum for N -body Schrödinger operators. Commun. Math. Phys. 92, 481–483 (1984) 28. M. Reed, B. Simon, Methods of Modern Mathematical Physics I-IV (Academic Press, New York, 1972-1978) 29. E. Skibsted, Propagation estimates for N -body Schrödinger operators. Commun. Math. Phys. 142, 67–98 (1991)

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30. H. Tamura, Principle of limiting absorption for N -body Schrödinger operators - a remark on the commutator method. Lett. Math. Phys. 17, 31–36 (1989) 31. T. Wolff, Recent work on sharp estimates in second-order elliptic unique continuation problems. J. Geom. Anal. 3(6), 621–650 (1993) 32. D. Yafaev, Eigenfunctions of the Continuous Spectrum for the N-Particle Schrödinger Operator. Spectral and Scattering Theory (Sanda, 1992), pp. 259–286. Lecture Notes in Pure and Appl. Math., vol. 161 (Dekker, New York, 1994) 33. D. Yafaev, Radiation conditions and scattering theory for N -particle Hamiltonians. Commun. Math. Phys. 154(3), 523–554 (1993) 34. K. Yosida, Functional Analysis. Reprint of the Sixth (1980) Edition. Classics in Mathematics (Springer, Berlin, 1995). xii+501

Resolvent Estimates and Resonance Free Domains for Schrödinger Operators with Matrix-Valued Potentials Marouane Assal

1 Introduction and Background In this note we are interested in resolvent estimates and quantum resonances for Schrödinger operators with long-range matrix-valued potentials in the semiclassical regime. The results established here are a part of a work in progress [4] which will be published elsewhere. In this forthcoming work we will also give some applications of our results to scattering theory for matrix Schrödinger operators.

1.1 Preliminaries Consider the semiclassical Schrödinger operator on the Hilbert space L2 (Rd ; CN ), d ≥ 1, P (h) := −h2  · IN + V (x),

(1.1)

where IN is the identity N × N matrix and V : Rd → HN is a smooth N × N Hermitian matrix-valued potential, i.e., V (x) = (Vij (x))1≤i,j ≤N ,

Vij (x) = Vj i (x),

with long-range behavior at infinity, i.e., there exists a constant matrix V∞ ∈ HN and ρ0 > 0 such that M. Assal () Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Santiago, Chile e-mail: [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 P. Miranda et al. (eds.), Spectral Theory and Mathematical Physics, Latin American Mathematics Series, https://doi.org/10.1007/978-3-030-55556-6_2

17

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M. Assal

 ∂xα V (x) − V∞ N ×N = Oα (x −ρ0 −|α| ),

∀x ∈ Rd , α ∈ Nd ,

(1.2)

with x := (1 + |x|2 )1/2 . Here HN denotes the space of N × N Hermitian matrices endowed with the norm  · N ×N defined by (2.1) and h > 0 is the semiclassical parameter. Such operators arise as important models in molecular physics and quantum chemistry, for instance in the Born-Oppenheimer approximation which allows for a drastic reduction of problem size when dealing with molecular systems. In this case, the semiclassical parameter h represents the square root of the quotient between the electronic and nuclear masses (see e.g. [22, 24]). Set P∞ (h) := −h2  · IN + V∞ and we denote λ1 ≤ · · · ≤ λN the eigenvalues of the constant matrix V∞ . Without loss of generality, we can assume that V∞ is diagonal, i.e., V∞ = diag (λ1 , . . . , λN ). The operator P∞ (h) is self-adjoint in L2 (Rd ; CN ) with domain the Sobolev space H 2 (Rd ; CN ), and its spectrum coincides with [λ1 , +∞). Since V − V∞ is -compact, it follows by the Weyl perturbation Theorem that the operator P (h) admits a unique self-adjoint realization in L2 (Rd ; CN ) with domain H 2 (Rd ; CN ) and for any fixed h > 0, σess (P (h)) = σess (P∞ (h)) = [λ1 , +∞). Thus, the operator P (h) may have discrete eigenvalues in (−∞, λ1 ) and embedded ones in the interval [λ1 , λN ]. Let Rh (z) := (P (h) − z)−1 ∈ L(L2 (Rd ; CN )), z = 0, denotes the resolvent of P (h). It follows by the Limiting Absorption Principle (see e.g. [1]), using the dilation generator as a scalar conjugate operator, that for any E > V∞ N ×N and any s > 12 , the boundary value of the resolvent Rh (E ± i0) := lim Rh (E ± iε) ε→0+

exists as a bounded operator in L(L2,s (Rd ; CN ), L2,−s (Rd ; CN )). Here L2,s (Rd ; CN ) denotes the space of CN -valued functions that are square integrable on Rd with respect to the measure x s dx, equipped with its natural norm  uL2,s (Rd ;CN ) := x uL2 (Rd ;CN ) = s

Rd

1 |u(x)|2CN x 2s dx

2

.

A natural question is to study the behavior of Rh (E ± i0) as h → 0+ , more precisely, to estimate the size of the norm Rh (E ± i0)L2,s (Rd ;CN )→L2,−s (Rd ;CN ) ,

s>

1 , 2

(1.3)

Resolvent Estimates and Resonance Free Domains for Systems

19

with respect to h. This problem is important in scattering theory, for instance, for studying the behavior of observables like the scattering matrix and the total cross section (see e.g., [26, 27, 31]). Moreover, it is well known that the semiclassical behavior of the resolvent near a given energy-level have a deep relationship with the existence or the absence of resonance near this level. As we shall see later, getting estimates on the size of the above norm entails important results on the location of resonances of the operator P (h) near the energy-level E. Let Res (P (h)) denote the set of resonances of P (h) which we will define rigorously in Sect. 2.

1.2 Background on the Scalar Case In the scalar case N = 1 (and V∞ = 0), it is well known that the size of (1.3) is O(h−1 ) in non-trapping situations, that is Rh (E ± i0)L2,s (Rd )→L2,−s (Rd )  h−1 , ∀s >

1 , 2

(1.4)

provided that E > 0 is non-trapping for the classical Hamiltonian p(x, ξ ) := ξ 2 + V (x), (x, ξ ) ∈ T ∗ Rd , associated with P (h). We recall that an energy E > 0 is said to be non-trapping for p if the set of trapped trajectories at E defined by

T (E) := (x, ξ ) ∈ p−1 (E); exp (tHp )(x, ξ )  ∞ as t → ±∞ is empty. Here exp (tHp ) : T ∗ Rd → T ∗ Rd is the flow generated by the Hamiltonian vector field Hp = 2ξ · ∂x − ∇x V · ∂ξ . This result was proved by Robert and Tamura [31] using Mourre theory and Fourier integral methods. In particular, estimate (1.4) is the key ingredient in the estimation of the behavior of the scattering cross-section and the complete asymptotic expansion in powers of h for the spectral shift function associated to the operator pair (−h2  + V (x), −h2 ). A shorter proof using the construction of a global escape function and Mourre theory was given later by Gérard and Martinez [15]. Another consequence of the non-trapping hypothesis on the energy E > 0 is the following absence of resonances result due to Martinez [25]   Res(P (h)) ∩ z ∈ C; Re z ∈ [E − ε0 , E + ε0 ] and Im z ≥ −Ch| ln h| = ∅, for some ε0 > 0 and all C > 0, h ∈ (0, hC ]. Martinez’s approach is based on some microlocal weighted estimates combined with the construction of a global escape function associated to the classical Hamiltonian p. An alternative approach to conjugated operators was introduced later by Sjöstrand and Zworski [36].

20

M. Assal

Now, without any assumption on the set of trapped trajectories (trapping situations), we have the following estimates for any s > 12 and E > 0, C

Rh (E ± i0)L2,s (Rd )→L2,−s (Rd )  e h ,  1

{x∈Rd ;|x|≥R0 } Rh (E

 ± i0)1{x∈Rd ;|x|≥R0 } L2,s (Rd )→L2,−s (Rd )  h−1 ,

(1.5) (1.6)

for some constants C, R0 > 0 and h > 0 small enough. Here 1{x∈Rd ;|x|≥R0 } is the characteristic function of the set {x ∈ Rd ; |x| ≥ R0 }. These estimates are originally due to Burq [6] in a more general framework including perturbations of secondorder elliptic operators with smooth coefficients, obstacle scattering and metric scattering. Others proofs was given later by Vodev [37] and Sjöstrand [34]. Cardoso and Vodev [7] and recently Rodianski and Tao [32] established generalizations of these estimates to Schrödinger operators on manifolds. Carleman estimates are the main tool in all these works. Recently, Datchev [8] provided an elementary proof of these estimates for scalar Schrödinger operators on Rd , d = 2 (see [33] for the case d = 2). The proof in [8] is based on a global weighted Carleman estimate which has the advantage to be valid under low regularity assumption on the potential V and where the construction of the weight function is simple and explicit. More precisely, Datchev’s method only requires that the potential V is bounded on Rd together with its radial derivative, and they decay like (1 + |x|)−ρ0 and (1 + |x|)−ρ0 −1 at infinity, respectively. The h-dependence in (1.5) and (1.6) is optimal in general, that is without any assumption on the underlying classical dynamics. Moreover, estimate (1.6) is not true in general when removing one of the characteristic function 1{x∈Rd ;|x|≥R0 } (see [9]). As a consequence of estimate (1.5), Burq [6] proved the following resonance free region   Res(P (h)) ∩ z ∈ C; Re z ∈ J and Im z ≥ −Ce−C/ h = ∅, for any compact interval J ⊂ (0, +∞) and some constant C > 0, and h > 0 small enough.

1.3 Comments on the Matrix-Valued Case In the matrix-valued case, that is, when the considered operator is of the form (1.1), the situation is more complicated. Notice that since the eigenvalues are not enough regular in general, the usual definition of the Hamiltonian flow for a matrix-valued Hamiltonian function does not make sense (see [23]). The non-trapping condition in this case is usually characterized by the existence of a global escape function

Resolvent Estimates and Resonance Free Domains for Systems

21

associated with the matrix-valued symbol of the operator P (h) at the considered energy-level. Let p(x, ξ ) := |ξ |2 IN +V (x), (x, ξ ) ∈ T ∗ Rd , be the matrix-valued semiclassical symbol of P (h). We denote λ1 (x) ≤ λ2 (x) ≤ · · · ≤ λN (x) the (real) eigenvalues of V (x), x ∈ Rd . In general, the functions x → λj (x) are continuous on Rd , j = 1, . . . , N . For an energy E ∈ R, we denote E the corresponding energy surface defined by E :=

N

(x, ξ ) ∈ T ∗ Rd ; |ξ |2 + λj (x) = E . j =1

A smooth real-valued function G ∈ C ∞ (T ∗ Rd ; R) is a global escape function associated with the classical Hamiltonian p at E if there exists a constant c > 0 such that {p, G}|E ≥ cIN in the sense of Hermitian matrices, i.e.,

 {p, G}(x, ξ )w, w CN ≥ c|w|2 ,

∀(x, ξ ) ∈ E , ∀w ∈ CN .

(1.7)

Here {p, G} := ∂ξ p · ∂x G − ∂x p · ∂ξ G denotes the Poisson bracket of p and G, and (·, ·)CN denotes the inner product in CN . In the scalar case N = 1, the existence of a global escape function associated with the symbol p at an energy E > 0 is equivalent to the non-trapping condition (see for instance [15]). In [20], under the existence condition of a global escape function, Jecko proved that estimate (1.4) still holds in the case of matrix-valued potentials. With this result at hand, the main challenge consists in the construction of a global escape function which may be quite complicated. This question has been the subject of many works (see [11, 19–21] and the references therein) for different type of eigenvalue crossings. In this note, we provide generalizations of Burq’s estimates (1.5) and (1.6) to the case of Schrödinger operators with matrix-valued potentials using the approach developed in [8] and we prove related results on the absence of resonances near the real axis. We refer to [4, 5] for applications of these results to scattering theory. We also refer to ([3, 12–14, 17] and the references therein) for some recent works on the widths of resonances for systems of coupled Schrödinger operators for different potentials and at different energy levels.

2 Statement of the Results Let HN denotes the space of N × N Hermitian matrices endowed with the norm  · N ×N , where for M ∈ HN , MN ×N :=

sup {w∈RN ;|w|≤1}

|Mw|.

(2.1)

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M. Assal

2.1 Resolvent Estimates Consider the semiclassical Schrödinger operator on L2 (Rd ; CN ), d = 2, P (h) := −h2  · IN + V (x), where IN is the identity N × N matrix and h > 0 is the semiclassical parameter. Using the polar coordinates Rd  x = (r, ω) ∈ R+ × Sd−1 , where Sd−1 denotes the unit sphere on Rd , we assume the following conditions on the potential V . Assumption (A1) V : R+ × Sd−1 → HN and its distributional derivative ∂r V are bounded on R+ × Sd−1 , i.e., V , ∂r V ∈ L∞ (R+ × Sd−1 ; HN ),

(2.2)

and has the following long-range behavior at infinity: Assumption (A2) There exist a constant Hermitian matrix V∞ ∈ HN and ρ0 > 0 such that V (r, ω) − V∞ N ×N ≤ (1 + r)−ρ0 ,

∂r V (r, ω)N ×N ≤ (1 + r)−ρ0 −1 ,

(2.3)

for all (r, ω) ∈ R+ × Sd−1 . We have the following resolvent estimates. Theorem 2.1 Assume (A1) and (A2). For any E > V∞ N ×N and s > 12 , there exist C, R0 , h0 > 0 such that for all h ∈ (0, h0 ] and ε > 0, the following estimates hold C

Rh (E ± iε)L2,s (Rd ;CN )→L2,−s (Rd ;CN ) ≤ e h ,

(2.4)

 ± iε)1{x∈Rd ;|x|≥R0 } L2,s (Rd ;CN )→L2,−s (Rd ;CN ) ≤ Ch−1 . (2.5) Here 1{x∈Rd ;|x|≥R0 } is the characteristic function of the set {x ∈ Rd ; |x| ≥ R0 }.  1

{x∈Rd ;|x|≥R0 } Rh (E

Remark 2.2 Estimates on resolvent truncated outside a large compact set are important in scattering theory. Indeed, the operator 1{x∈Rd ;|x|≥R0 } Rh (z)1{x∈Rd ;|x|≥R0 } appears for instance in the representation of the scattering amplitude for compactly supported perturbations (see [30]). In [4], we apply (2.5) to prove estimates on the scattering amplitude for Schrödinger operators with matrix-valued potentials.

Resolvent Estimates and Resonance Free Domains for Systems

23

2.2 Resonance Free Domains Now, we state our results on the resonances of P (h). To define the resonances of P (h), we need the following assumption on the potential V . Assumption (Hol∞ ) V ∈ C ∞ (Rd ; HN ) and extends to an analytic function on S ⊂ Cd S := {x ∈ Cd ; |Im x| ≤ c0 Re x , |Re x| > κ},

(2.6)

for some constants c0 , κ > 0. Moreover, there exist ρ0 > 0 and a constant C > 0 such that for all x ∈ S V (x) − V∞ N ×N ≤ Cx −ρ0 ,

1

x := (1 + |x|2 ) 2 .

(2.7)

Under this assumption, we can define the resonances of P (h) near the real axis by the method of complex distortion as it was done in [18, 29] (see also [2, 35] for an alternative approach). Let A  1 be a large constant, and let F : Rd → Rd be a smooth vector-field such that  0 for |x| ≤ A F (x) = (2.8) x for |x| ≥ A + 1. We introduce the one-parameter family of unitary distortion 1

C0∞ (Rd ; CN )  f −→ Uω f (x) := |Jφω (x) | 2 f (φω (x)),

ω ∈ R,

where φω (x) := x + ωF (x) and Jφω (x) := det(1 + ω∇F (x)) is the Jacobian of φω (x). For ω ∈ R small enough, Uω extends to a unitary operator on L2 (Rd ; CN ). We define Pω (h) := Uω P (h)(Uω )−1 .

(2.9)

Under the Assumption (Hol∞ ) on the potential V , the operator Pω (h) is a differential operator with analytic coefficients with respect to ω, and can therefore be continued in a unique way to small enough complex values of ω. Therefore, the distorted operator Pθ (h) := Uiθ P (h)(Uiθ )−1

(2.10)

is well defined for θ > 0 small enough. By Weyl perturbation theorem its essential spectrum is given by

24

M. Assal

σess (Pθ (h)) =

N

(λj + e−2iθ R+ ).

j =1

Hence, the spectrum of Pθ (h) in the complex sector     −2iθ (λ + e R ) Sθ := λ1 + e−2i[0,θ) R∗+ \ ∪N j + j =1 is discrete, consists on isolated eigenvalues with finite multiplicities. Moreover, standard arguments (see e.g. [16]) show that σ (Pθ (h)) ∩ Sθ does not depends on the particular choice of the vector-field F and for any 0 < θ < θ  small enough and h > 0 fixed, we have σ (Pθ (h)) ∩ Sθ = σ (Pθ  (h)) ∩ Sθ . The resonances of P (h) in Sθ are defined as the eigenvalues of Pθ (h) in Sθ , or equivalently as the eigenvalues of Pθ  (h) in Sθ for all 0 < θ < θ  small enough. In the following, we denote Res (P (h)) the set of resonances of P (h). The estimate (2.4) entails the following result on the absence of resonances in an exponentially small band below (V∞ N ×N , +∞). Theorem 2.3 Assume (Hol∞ ). For any compact interval J ⊂ (V∞ N ×N , +∞), there exist a constant C > 0 and h0 ∈ (0, 1] such that for all h ∈ (0, h0 ], we have

Res(P (h)) ∩ z ∈ C; Re z ∈ J and Im z ≥ −Ce−C/ h = ∅. Remark 2.4 Notice that Assumption (Hol∞ ) and the Cauchy formula imply Assumptions (A1) and (A2). In our proof of Theorem 2.3, we shall use the following result which generalizes the well-known result of [25] to the case of Schrödinger operators with matrixvalued potentials. Theorem 2.5 Assume (Hol∞ ). Let E0 > V∞ N ×N and suppose that there exists G ∈ C ∞ (T ∗ Rd ; R) such that (1.7) holds on E0 . Then there exists ε0 > 0 such that for all C > 0, there exists hC ∈ (0, 1] such that for all 0 < h ≤ hC , we have   Res(P (h)) ∩ z ∈ C; Re z ∈ [E0 − ε0 , E0 + ε0 ] and Im z ≥ −Ch| ln h| = ∅. Remark 2.6 In the scalar case N = 1, a quantitative version of the previous result in terms of an estimate on the distorted resolvent was proved in [28] (see also [36]). More precisely, if Pθ (h) denotes the distorted operator with θ = θ (h) = C  h| ln h|, C   C, then for any C > 0 there exist a constant C  > 0 and hC > 0 such that

Resolvent Estimates and Resonance Free Domains for Systems

(Pθ (h) − z)−1  ≤

25

C  exp(C  |Im z|/ h), h

(2.11)

uniformly for z ∈ {z ∈ C; Re z ∈ [E0 − ε0 , E0 + ε0 ] and Im z ≥ −Ch| ln h|} and h ∈ (0, hC ]. In Lemma 4.1, we prove a weaker estimate (see (4.2)) on the distorted resolvent which coincides with (2.11) for Im z = Ch| ln h|. As in [28], using a version of the non-trapping estimate and the semiclassical maximum principle, we can prove the same estimate (2.11) for our matrix-valued operator. This will be proved in details in [4].

3 Resolvent Estimates In this section, we present the main ideas of the proof of Theorem 2.1 referring to [4] for the details. We follow the approach developed in [8] which we adapt in our context of matrix-valued operator. The main step in the proof of estimates (2.4) and (2.5) is the following global weighted Carleman estimate. In the following s > 12 and E > V∞ N ×N are fixed. Proposition 3.1 There exist R0 , h0 , C > 0 and a positive radial function ϕ = ϕ(r) ∈ C ∞ (R+ ; R+ ) with ϕ  ≥ 0 and supp ϕ  = [0, R0 ] such that the following estimate holds 2  h2 eϕ/ h v L2,−s (Rd ;CN )   2

 2 ≤ C eϕ/ h P (h) − (E + iε)IN v L2,s (Rd ;CN ) + Cεheϕ/ h v L2 (Rd ;CN ) , (3.1) for all ε ≥ 0, h ∈ (0, h0 ] and v ∈ C0∞ (Rd ; CN ). The proof of this result relies mainly on two steps. Step 1.

The first step consists in the construction of a weight function ϕ ∈ C ∞ (R+ ; R+ ) such that for some 0 < R < R0 , ϕ increases linearly on [0, R], ϕ increases slowly on [R, R0 ] and ϕ = Cste on [R0 , +∞) (see Fig. 1), so that it satisfies a nice estimate in relation with the potential V , more precisely, 



 1 ∂r m(r) EIN − Vϕ (r, ω; h) w, w  m (r)

 CN

≥ C|w|2CN

(3.2)

for some constant C > 0, uniformly with respect to (r, ω) ∈ (0, +∞) × Sd−1 , h ∈ (0, h0 ] and w ∈ CN , where m(r) := 1 − (1 + r)1−2s and

26

M. Assal

ϕ(r)

Fig. 1 The weight function ϕ

ϕ(R0 )

ϕ

0

R

r

R0

 Vϕ (r, ω; h) := V (r, ω) − (ϕ  (r))2 − hϕ  (r) IN .

Step 2.

(3.3)

Here (·, ·)CN and | · |CN denote the Hermitian inner product and norm in CN . Now, using the weighted function ϕ constructed above, we introduce the conjugated operator

 Pϕ = Pϕ (h) := eϕ/ h r (d−1)/2 P (h) − (E + iε)IN r −(d−1)/2 e−ϕ/ h . We have

 Pϕ = − h2 ∂r2 + 2hϕ  (r)∂r + Q · IN + Vϕ (r, ω; h) − (E + iε)IN , where Vϕ is the effective potential defined by (3.3) and  Q = Q(h) :=

0

h2 r −2 − Sd−1 +

(d−1)(d−3)  4

(d = 1), (d ≥ 3).

Here Sd−1 denotes the Laplacian on the unit sphere Sd−1 . Passing to polar coordinates using that L2 (Rd , dx) = L2 (R+ × Sd−1 , the estimate (3.1) is equivalent to the following one

r d−1 drdω), 

r,ω

m (r)|u(r, ω)|2CN drdω ≤



r,ω

C h2

 r,ω

|Pϕ u(r, ω)|2CN m (r)

drdω +

|u(r, ω)|2CN drdω,

Cε h (3.4)

for all u ∈ eϕ/ h r (d−1)/2 C0∞ (Rd ; CN ). Let ·, · Sd−1 and  · Sd−1 denote the inner product and norm in L2 (Sd−1 ; CN ), and we introduce the functional   Lh (r) := hu (r, ω)2Sd−1 − (Q · IN + Vϕ (r, ω; h) − EIN )u, u Sd−1 ,

r > 0,

where here and in the sequel prime notation always denote differentiation with respect to r, for instance u := ∂r u. Using the self-adjointness of

Resolvent Estimates and Resonance Free Domains for Systems

27

 Q · IN + Vϕ − EIN = Pϕ − − h2 ∂r2 + 2hϕ  (r)∂r − iε · IN , we get Lh (r) = − 2 Re Pϕ u, u Sd−1 + 4hϕ  (r)u 2Sd−1 + 2ε Im u, u Sd−1 + 2 r −1 Q · IN u, u Sd−1 − Vϕ u, u Sd−1 . It follows that (mLh ) (r) = − 2m Re Pϕ u, u Sd−1 + (4h−1 mϕ  + m )hu 2Sd−1 + 2mε Im u, u Sd−1     + (2mr −1 − m )Q · IN u, u Sd−1 + ∂r m(EIN − Vϕ ) u, u Sd−1 . First, using the fact that mϕ  ≥ 0, m > 0, Q ≥ 0, 2mr −1 − m > 0, we obtain   (mLh ) (r) ≥ − 2mRe Pϕ u, u Sd−1 + m hu 2Sd−1 + ∂r m(EIN − Vϕ ) u, u Sd−1 + 2mε Im u, u Sd−1 . Then, using the inequality −2 Re a, b + b2 ≥ −a2 , we get (mLh ) (r) ≥ −

1

Pϕ u2Sd−1 + 2mε Im u, u Sd−1   + ∂r m(EIN − Vϕ ) u, u Sd−1 . h2 m

By integrating with respect to r using the fact that Cauchy-Schwarz inequality, we obtain 

   ∂r m(EIN − Vϕ ) u, u CN drdω

 +∞ 0

(mLh ) (r)dr = 0 and the

r,ω

≤

1 h2

 r,ω



|Pϕ u|2CN

drdω + 2ε

m (r)

r,ω

|u|CN |u |CN drdω.

It follows from estimate (3.2) that  C

m r,ω



(r)|w|2CN drdω

1 ≤ 2 h

 r,ω

|Pϕ u|2CN m (r)

 drdω+2ε r,ω

|u|CN |u |CN drdω,

(3.5) for some constant C > 0. On the other hand, it is not difficult to prove that there exists a constant C  > 0 such that

28

M. Assal

 2ε

Cε |u|CN |u |CN drdω ≤ h r,ω 

 r,ω

|u|2CN drdω

Cε + h

 r,ω

|Pϕ u|2CN drdω.

(3.6) Putting together (3.5) and (3.6) we obtain (3.4). This ends the proof of Proposition 3.1.  Now, using the fact that ϕ(r) = C20 for r ≥ R0 , with C0 = 2 max(0,+∞) ϕ, and estimate (3.1), we obtain e−C0 / h 1{x∈Rd ;|x|≤R0 } v2L2,−s + 1{x∈Rd ;|x|≥R0 } v2L2,−s ≤  

  C  P − (E + iε)IN v 2 2,s + Cε v 2 2 , L L 2 h h

(3.7)

for some constant C > 0, uniformly for v ∈ C0∞ (Rd ; CN ), ε ≥ 0 and h > 0 small enough. On the other hand, using the selfadjointness of P and the Cauchy-Schwarz inequality, we get, for any ε ≥ 0, C  > 0 and h > 0 small enough, 

2εv2L2 = −2 Im  P − (E + iε)IN v, v L2

 C 1{x∈Rd ;|x|≥R0 } P − (E + iε)IN v2L2,s h h +  1{x∈Rd ;|x|≥R0 } v2L2,−s C

 +e2C0 / h 1{x∈Rd ;|x|≤R0 } P − (E + iε)IN v2L2,s ≤

+e−2C0 / h 1{x∈Rd ;|x|≤R0 } v2L2,−s .

(3.8)

Combining (3.7) and (3.8), we get e−C/ h  1{x∈Rd ;|x|≤R0 } v2L2,−s + 1{x∈Rd ;|x|≥R0 } v2L2,−s ≤ 

 2 C eC/ h 1{x∈Rd |x|≤R0 } P − (E + iε)IN v L2,s + 2 h 

 2  1 {x∈Rd ;|x|≥R0 } P − (E + iε)IN v L2,s , uniformly for v ∈ C0∞ (Rd ; CN ), ε ≥ 0 and h > 0 small enough. Using this estimate, the proof can be finished by the density argument of [8]. 

4 Resonance Free Domains This section is devoted to the proofs of Theorems 2.3 and 2.5. We use the ordinary notations and some basic results of semiclassical analysis referring for example to the textbooks [10, 38] for a clear presentation of this theory. We first prove Theorem 2.5 using the approach to conjugate operators developed in [36].

Resolvent Estimates and Resonance Free Domains for Systems

29

4.1 Proof of Theorem 2.5 Fix E0 > V∞ N ×N , and let p(x, ξ ) := |ξ |2 IN + V (x),

(x, ξ ) ∈ T ∗ Rd ,

be the semiclassical symbol of P (h). Let G ∈ C ∞ (T ∗ Rd ; R) be an escape function associated with p at E0 , i.e., G satisfies (1.7) on E0 . For |(x, ξ )| large enough, the function T ∗ Rd  (x, ξ ) → x · ξ is an escape function associated with p at E0 . Indeed, we have {p, x · ξ }(x, ξ ) = 2|ξ |2 IN − x · ∇x V (x),

(x, ξ ) ∈ T ∗ Rd .

The Assumption (Hol∞ ) and the Cauchy formula imply that x · ∇x V (x) → 0 as |x| → +∞. On the other hand, by the Assumption (Hol∞ ), |ξ |2 ≥ (E0 − V∞ N ×N )/2 on the energy surface E0 for |(x, ξ )| large enough. Therefore, {p, x · ξ }(x, ξ ) ≥ (E0 − V∞ N ×N )/2, for all (x, ξ ) ∈ E0 ∩ {|(x, ξ )|  1}. Thus, without any loss of generality, we may assume that G(x, ξ ) = x · ξ for |(x, ξ )| large enough. We set G(x, ξ ) := G(x, ξ ) − F (x) · ξ ∈ C0∞ (R2d ; R), where F is the vector field used in the complex distortion (see (2.8)). Let Gh := Opw h (G) be the pseudodifferential operator with symbol G. Let M > 0 be independent of h and set θ1 = θ1 (h) = iMκ(h) with κ(h) := h| ln h|. We introduce the conjugated operator θ1 (h) := e− P

Mκ(h) h Gh

Pθ1 (h)e

Mκ(h) h Gh

.

Since G is compactly supported it follows by the Calderón-Vaillancourt theorem (see for instance [10, Chapter 7]) that Gh is bounded in L2 (Rd ; CN ) and then the Mκ(h) operators e± h Gh are well defined.  Let ε0 > 0 be small enough such that (1.7) holds on Iε0 := E∈Iε E , i.e., 0 there exists C > 0 such that {p, G}(x, ξ ) ≥ C,

∀(x, ξ ) ∈ Iε0 ,

(4.1)

in the sense of Hermitian matrices, where Iε0 := [E0 − ε0 , E0 + ε0 ]. For η > 0, we introduce the complex region

30

M. Assal

η := Iε0 − i[0, ηκ(h)]. Our objective is to prove the following Lemma from which Theorem 2.5 follows. Lemma 4.1 There exists a constant c > 0 such that for all M > 0, there exists θ1 (h) − z is invertible for every z ∈ cM and hM ∈ (0, 1] such that the operator P h ∈ (0, hM ], and we have θ1 (h) − z)−1  = O(κ(h)−1 ), (P uniformly for z ∈ cM and h ∈ (0, hM ]. Remark 4.2 In particular, from the above Lemma we get immediately that the operator Pθ1 (h) − z is invertible for every z ∈ cM and h ∈ (0, hM ], hence P (h) has no resonances in cM for all M > 0 and h ∈ (0, hM ]. Furthermore, we have the following estimate on the distorted resolvent (Pθ1 (h) − z)−1  = O(h−C ),

(4.2)

uniformly for z ∈ cM and h ∈ (0, hM ], for some constant C > 0. Proof We have θ1 (h) = e− P

Mκ(h) h ad Gh

Pθ1 (h) ∼

+∞  (−Mκ(h))k 1 k=0

k!

h

k adGh Pθ1 (h),

where here we use the usual notation adA B := [A, B] for the commutator. The fact that G is scalar-valued and compactly supported ensures that adGh Pθ1 (h) = [Gh , Pθ1 (h)] = O(h) (in norm L(L2 )) and then the previous asymptotic expansion makes sense since κ(h) → 0 as h tends to 0. In particular, we have θ1 (h) = Pθ1 (h) − Mκ(h) [Gh , Pθ1 (h)] + O(M 2 κ(h)2 ). P h θ1 (h) Let pθ1 , p θ1 be the semiclassical symbols corresponding to Pθ1 (h) and P respectively. By the h-pseudodifferential symbolic calculus (see for instance [10, 38]), we have p θ1 (x, ξ ) = pθ1 (x, ξ ) − iMκ(h){G, pθ1 }(x, ξ ) + O(M 2 κ(h)2 ).

(4.3)

On the other hand, by Taylor’s expansion of pθ1 with respect to θ1 , we get pθ1 (x, ξ ) = p(x, ξ ) − iMκ(h){p, F (x) · ξ }(x, ξ ) + O(M 2 κ(h)2 ). Combining (4.3) and (4.4), we obtain

(4.4)

Resolvent Estimates and Resonance Free Domains for Systems

31

Im p θ1 (x, ξ ) = −Mκ(h){p, G + F (x) · ξ }(x, ξ ) + O(M 2 κ(h)2 ),

(4.5)

Re p θ1 (x, ξ ) = p(x, ξ ) + O(Mκ(h)). According to (4.1), there exists C > 0 such that − Im p θ1 (x, ξ ) ≥ CMκ(h),

∀ (x, ξ ) ∈ Iε0 .

(4.6)

θ1 (h) − z = Aθ1 (h) − Re z + i(Bθ1 (h) − Im z), where Aθ1 (h) and Bθ1 (h) We write P are the self-adjoint operators given by Aθ1 (h) :=

 1  θ1 (h))∗ , Pθ1 (h) + (P 2

Bθ1 (h) :=

 1  θ1 (h))∗ . Pθ1 (h) − (P 2i

Let ψ1 , ψ2 ∈ C ∞ (R2d ; R) be such that, for I  Iε0 , 

ψ12 + ψ22 = 1 on R2d ψ1 = 1 on I and supp(ψ1 ) ⊂ Iε0 .

(4.7)

According to Lemma 3.2 in [36], there exist two self-adjoint operators 1 and 2 with principal symbols respectively ψ1 and ψ2 such that (1 )2 + (2 )2 = Id + O(h∞ )

in L(L2 (Rd ; CN )).

(4.8)

We denote by the same letters the operators i := i IN , i = 1, 2. On the support of ψ1 , we see from (4.6) that the principal symbol of −Bθ1 (h) is bounded from below by CMκ(h). Thus, by Gårding’s inequality (see e.g. [10, 38]), we have for all u ∈ L2 (Rd ; CN ) θ1 (h) − z)1 u, 1 u θ1 (h) − z)1 u · 1 u ≥ (P (P θ1 (h) − Im z)1 u, 1 u ≥ (Im P = (Im z − Bθ1 (h))1 u, 1 u ≥ (Im z + CMκ(h) − O(h))1 u2 ≥

C Mκ(h)1 u2 , 3

(4.9)

uniformly for z > − C3 Mκ(h). On the other hand, since Aθ1 (h) − Re z is uniformly elliptic on the support of ψ2 and Re z ∈ Iε0 , the symbolic calculus permits us to construct a parametrix R ∈ S 0 (ξ −2 ) of Aθ1 (h) − Re z such that, in the sense of corresponding symbols, R#(Aθ1 (h) − Re z)ψ2 = ψ2 + O(h∞ ),

32

M. Assal

where # stands for the Weyl composition of symbols. As a consequence, we obtain θ1 (h) − z)2 u ≥ (P

1 2 u − O(h∞ )u2 . C

(4.10)

for all u ∈ L2 (Rd ; CN ). Furthermore, by means of standard elliptic arguments, one can easily prove the following semiclassical inequality, for i = 1, 2, θ1 (h)u + u), θ1 (h), i ]u ≤ C2 h(P [P

∀ u ∈ H 2 (Rd ; CN ).

(4.11)

Combining (4.8)–(4.11) with the estimate θ1 (h) − z)u2 = (P

2 

θ1 (h) − z)u2 − O(h∞ )(P θ1 (h) − z)u2 i (P

(4.12)

i=1

≥

 1  θ1 (h), i ]u2 − O(h∞ )(P θ1 (h) − z)u2 , (Pθ1 (h) − z)i u2 − [P 2 2

2

i=1

i=1

we deduce, for z ∈ cM := {z ∈ C; Re z ∈ Iε0 and Im z ≥ −cMκ(h)} (with c > 0 independent of M and h) and sufficiently small h, θ1 (h) − z)u ≥ (P

κ(h) u. C

(4.13)

By the same arguments, we prove an estimate similar to (4.13) for the adjoint

 θ1 (h) ∗ − z and we conclude that P θ1 (h) − z is invertible for every operator P  z ∈ cM . Hence, Pθ1 (h) has no spectrum in cM and we have the estimate θ1 (h) − z)−1  ≤ Cκ(h)−1 , (P

(4.14)  

uniformly for z ∈ cM .

4.2 Proof of Theorem 2.3 We start by proving the following estimate on the distorted resolvent. Lemma 4.3 Assume (Hol∞ ) and let V∞ N ×N < α < β < +∞, η > 0 and θ = h| ln h|. Then, there exist a constant C > 0 and h0 ∈ (0, 1] such that (Pθ (h) − z)−1  = O(eC/ h ), uniformly for z ∈ [α, β] − i[0, ηh| ln h|], z ∈ / Res (P (h)) and h ∈ (0, h0 ].

(4.15)

Resolvent Estimates and Resonance Free Domains for Systems

33

Proof Let ψ ∈ C0∞ (Rd ; [0, 1]) be such that ψ(x) = 1 for |x| ≤ 1 and ψ(x) = 0 for |x| ≥ 2. We introduce the Schrödinger operator A(h) := −h2  · IN + Vr (x),

  x  V (x), with Vr (x) := 1 − ψ r

r > 0.

Let ar (x, ξ ) := |ξ |2 IN + Vr (x) be the semiclassical symbol of A(h) and set Er

N   x 

r λj (x) = E , [α,β] (x, ξ ) ∈ R2d ; |ξ |2 + 1 − ψ := := r j =1



Er .

E∈[α,β]

where we recall that the λj (x)’s are the eigenvalues of V (x), x ∈ Rd . From (Hol∞ ), r for r > 0 large enough, we have |ξ |2 > α/2 on πξ ([α,β] ). Here πξ denotes the spatial projection (x, ξ ) → ξ . On the other hand, using that x · ∇x V (x) → +∞ as |x| → +∞ according to (Hol∞ ) and the Cauchy formula, we get {ar , x · ξ }(x, ξ ) = 2|ξ |2 IN +

  x  x  x ∇x ψ V (x) − 1 − ψ x∇x V (x) r r r

≥ α/2, r and r > 0 large enough. Thus, we deduce that R2d  (x, ξ ) → for (x, ξ ) ∈ [α,β] r x · ξ is an escape function associated with ar on [α,β] for r > 0 large enough. Let Aθ (h) be the distorted operator associated with A(h), obtained by replacing P (h) with A(h) in (2.10). Let χ ∈ C0∞ (Rd ; [0, 1]) such that χ (x) = 1 for |x| < 2r. In particular, we have

χ (V − Vr ) = χ ψr V = (V − Vr ). Using the resolvent identity, we get for z ∈ / Res (P (h)), (Pθ (h) − z)−1 =(Aθ (h) − z)−1 − (Pθ (h) − z)−1 (V − Vr )(Aθ (h) − z)−1 =(Aθ (h) − z)−1 − (Aθ (h) − z)−1 (V − Vr )(Aθ (h) − z)−1 +(Aθ (h) − z)−1 (V − Vr )(Pθ (h) − z)−1 (V − Vr )(Aθ (h) − z)−1 =(Aθ (h) − z)−1 − (Aθ (h) − z)−1 (V − Vr )(Aθ (h) − z)−1 +(Aθ (h) − z)−1 (V − Vr )χ (P (h) − z)−1 χ (V − Vr )(Aθ (h) − z)−1 . According to the non-trapping estimate (4.2), there exists a constant C  > 0 such that 

(Aθ (h) − z)−1  = O(h−C ),

(4.16)

34

M. Assal

uniformly for z ∈ [α, β] − i[0, ηh| ln h|] and h > 0 small enough. It follows that

   (Pθ (h) − z)−1  = O h−2C + h−2C χ (P (h) − z)−1 χ  ,

(4.17)

uniformly for z ∈ [α, β] − i[0, ηh| ln h|], z ∈ / Res (P (h)) and h > 0 small enough. On the other hand, the weighted estimate (2.4) clearly implies the same estimate for the truncated resolvent, that is χ (P (h) − z)−1 χ  = O(eC

 / h

(4.18)

),

for some constant C  > 0, uniformly for z ∈ [α, β]−i[0, ηh| ln h|] and h > 0 small enough. Putting together (4.17) and (4.18), we obtain the desired estimate (4.15).   End of the Proof of Theorem 2.3 Let J = [α, β] ⊂ (V∞ N ×N , +∞) and let θ = h| ln h|. For z ∈ C with Re z ∈ J , we write Pθ (h)−z = (Pθ (h)−Re z)(I −K(z; h))

with

K(z; h) := iIm z(Pθ (h)−Re z)−1 .

According to Lemma 4.3, there exists a constant C > 0 such that for h small enough, (Pθ (h) − Re z)−1  ≤ CeC/ h . It follows that for h small enough, Re z ∈ J and |Im z|
0. For large values of α and d ≥ 3, scattering methods can be applied, leading to the proof of absolutely continuous spectrum [33]. Purely absolutely continuous spectrum was showed in [27]. A wider range of values of α was considered by Bourgain in dimension 2 [5] and higher [6]. Point spectrum was also showed to hold outside the essential spectrum of the operator in [30]. In one dimension, transfer matrix analysis can be applied, leading to a complete understanding of the spectrum of the operator [17, 31]. This time however, the phase diagram of the model remains non-trivial and absolutely continuous spectrum can still be observed for large values of α. As it is natural to expect, small values of α lead to pure point spectrum. Interestingly, there is a critical value of α for which a transition from pure point to singular continuous spectrum is observed as a function of the coupling constant. The three above regimes correspond to α > 12 , α < 12 and α = 12 respectively. A complete study of the spectral behaviour of the onedimensional model is given in [31]. From the dynamical point of view, it is standard to show that the system propagates α > 12 . For the critical case α = 12 , no transition occurs at the dynamical level, despite of the spectral transition: there are non-trivial transport exponents for all values of the coupling constant [24]. This provides yet another example of a model where spectral localization and transport coexist. Dynamical localization in the regime 0 < α < 12 was showed in [36] providing, as a by-product, a proof of point spectrum in this regime. In [7], we provided a different proof of this behaviour for a continuum version of the model which can be easily adapted to the discrete case. Structure of the article We introduce the model and state the main results in Sect. 2. The transfer matrix analysis of [31] is presented in details in Sect. 3, some technical estimates being deferred to the Appendix. We show absolutely continuous spectrum for the super-critical case in Sect. 4. In Sect. 5, we discuss the spectral transition and the absence of dynamical localization in the critical case. Finally, we outline the proof of dynamical localization for the sub-critical case in Sect. 6.

Anderson Model in a Decaying Potential

39

2 Model and Results We consider a one dimensional Schrödinger random operator Hω,λ defined by Hω,λ =  + λVω

on

2

(N),

(2.1)

with boundary condition x0 = 0 and where (x)(n) = xn+1 + xn−1 is the discrete Laplacian, Vω is a random potential described below and λ ∈ R is a coupling parameter. Denoting by δn the n-th canonical vector, the action of the random potential Vω is given by Vω δn = an ωn δn ,

n ∈ N,

(2.2)

where (ωn )n≥0 are i.i.d. bounded centered random variables defined on a probability space (!, F, P) with bounded density ρ. We will assume that all ωn ’s have variance equal to 1. The envelope on the environment is given by a positive sequence an such that lim nα an = 1 where α > 0 denote the decay rate. Under these n→∞

hypotheses (Hω )ω is non-ergodic family of self-adjoint operators. Note that the random variables are assumed to be bounded so that the density ρ has a bounded support.

2.1 Spectral Transition We recall the spectral results of [31] which give the characterization of the spectrum of Hω,λ in terms of the parameters. Theorem 2.1 Under the hypothesis above, the essential spectrum of Hω,λ is P-a.s. equal to [−2, 2]. Furthermore, (1) Super-critical case. If α > 12 then for all λ ∈ R, the spectrum of Hω,λ is almost surely purely absolutely continuous in (−2, 2) (2) Critical case. If α = 12 then for all λ = 0, the a.c. spectrum of Hω,λ is almost surely empty. Moreover, a. If |λ| ≥ 2, the spectrum of Hω,λ is almost surely pure point in [−2, 2]. b. If √ |λ| < 2, the spectrum of H √ω,λ is purely singular continuous in {|E| < 2 4 − λ } and pure point in { 4 − λ2 < |E| < 2}, almost surely. (3) Sub-critical case. If α < 12 then for all λ = 0, the spectrum of Hω,λ is almost surely pure point in [−2, 2]. We will provide complete proofs of Parts 1 and 2 in Sects. 4 and 5 respectively. Part 3 is a consequence of the dynamical localization result discussed below.

40

O. Bourget et al.

2.2 Dynamical Behavior Delocalization, or spreading of wave packets, for α > 12 follows from the RAGE theorem. The situation is particularly interesting for α = 12 : non-trivial transport occurs regardless of the precise nature of the spectrum. In particular, this provides an example of an operator displaying pure point spectrum but no dynamical localization. To describe the dynamics, we consider the random moment of order p ≥ 0 at time t for the time evolution, initially spatially localized on the origin and localized in energy by a positive function f ∈ C0∞ (R), Mω (p, f, T ) :=

2 T



∞

0

 p 2 2t   e− T |X| 2 e−itHω,λ f (Hω,λ )δ0  dt,

(2.3)

where |X| denotes the position operator and C0∞ (R) is the space of infinitely differentiable compactly supported functions. The following result is proved in [24] and establishes the absence of dynamical transition in the critical case. Theorem 2.2 Let α = 12 and λ ∈ R. The following holds P-amost surely: for all positive f ∈ C0∞ (R) constantly equal to 1 on a compact interval J ⊂ (−2, 2), for any ν > 0 and all p > 2γJ + ν where γJ = inf{λ(8 − 2E 2 )−1 : E ∈ J }, there exists Cω (p, J, ν) such that Mω (p, f, T ) ≥ Cω (p, J, ν)T p−2γJ −ν .

(2.4)

In Proposition 5.7, we establish a weaker result that already implies the absence of dynamical localization in the critical regime where α = 12 . To characterize the dynamical localization, we define the eigenfunction correlator Qω (m, n; I ) =

  sup δm , PI (Hω,λ )f (Hω,λ )δn  ,

f ∈C0 (I ) f ∞ ≤1

(2.5)

where PI (Hω ) denotes the spectral projection of Hω,λ on the interval I and C0 (I ) is the space of bounded measurable compactly supported functions in I . We say that Hω,λ exhibits dynamical localization in an interval I ⊂ R if we have    E Qω (m, n; I )2 < ∞, n

for all m ∈ Z. We state the main result of [36].

(2.6)

Anderson Model in a Decaying Potential

41

Theorem 2.3 Let 0 < α < 12 and λ = 0. Then, for each m ∈ Z and each compact energy interval I ⊂ (−2, 2), there exist constants C = C(m, I ) > 0, c(m, I ) > 0 such that E[Qω (m, n; I )2 ] ≤ Ce−cn

1−2α

(2.7)

,

for all m, n ∈ Z. In particular, Hω,λ exhibits dynamical localization in the interval I. Although the lack of ergodicity of the model induces the dependence of (2.7) on the base site m, it can be showed that the bound (2.6) still implies pure point spectrum and finiteness of the moments such that for all p > 0,   p 2    E sup |X| 2 e−itHω PI (Hω )ϕ0  < ∞, t∈R

for all ϕ0 ∈ l 2 (Z) with bounded support. A proof of these simple facts can be found in [8] in a related discrete model or in [7] in the continuum. The proof of Theorem 2.3 in [36] uses the Kunz-Souillard method (KSM) [12, 34]. In [8], we prove localization for the one-dimensional Dirac operator in a subcritical potential by means of the fractional moments method instead. This approach allowed us to greatly generalize the hypothesis required in [36]. For instance, we can handle unbounded potentials under mild regularity assumptions on their law, a context which is out of the scope of the KSM. Our approach, which is discussed in Sect. 6 below, can be easily adapted to the Anderson model providing an alternative proof of Theorem 2.3 under more general assumptions. Furthermore, in [7], we proved the corresponding result for a continuum version of the model, a problem which was left open in [13] where the authors develop a continuum version of the KSM. The analysis in [7] provides a control on the eigenfunctions of Hω,λ . Let xω,E = (xω,E,n )n denote the eigenfunction of Hω corresponding to the eigenvalue E. In [31, Theorem 8.6], it is showed that lim

n→∞

1 n1−2α

 (1 − 2α)λ2 , log |xω,E,n |2 + |xω,E,n−1 |2 = − 2(4 − E 2 )

P − a.s.,

for almost every fixed E ∈ (−2, 2). In particular, this shows that for almost every E ∈ (−2, 2), P-almost surely, there exists a constant Cω,E such that 

2

− (1−2α)λ2 n1−2α

|xω,E,n |2 + |xω,E,n−1 |2 ≤ Cω,E e

2(4−E )

.

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O. Bourget et al.

It is known that certain types of decay of eigenfunctions are closely related to dynamical localization [14, 15, 23]. Such criteria usually require a control on the localization centres of the eigenfunctions, uniformly in energy intervals. This information is missing in the above bound. In [7] (see also [8]), it is showed that: Proposition 2.4 Let 0 < α < 12 . For all compact energy interval I ⊂ (−2, 2), there exists two deterministic constants c1 = c1 (I ), c2 = c2 (I ) and positive random quantities cω = cω (I ), Cω = Cω (I ) such that cω e−c1 n

1−2α

≤

 1−2α |xω,E,n |2 + |xω,E,n−1 |2 ≤ Cω e−c2 n ,

P − a.s.,

(2.8)

for all E ∈ I ∩ σ (Hω,λ ) and all n ∈ Z. This asymptotics, although less precise on the exact rate of decay, is uniform in energy intervals. The upper bound (2.8) can be seen as a form of the condition SULE where the localization centres are all equal to 0 (see [14, 22], equation (2)). As a corollary of the lower bound above, we can show that certain stretched exponential moments diverge, a fact that characterizes in some sense the strength of the localization. Proposition 2.5 Let 0 < α < 12 and λ > 0. Let I ⊂ (−2, 2) be a compact interval contained in the interior of σpp (Hω,λ ). Then, for P-almost every ω,  2   lim sup exp{|X|p } e−itHω,λ ψ  = ∞,

(2.9)

t→∞

for all p > 1 − 2α and ψ ∈ RanPI (Hω,λ ).

3 Transfer Matrix Analysis Let |E| ≤ 2 and consider the eigenvalue equation xn+1 + xn−1 + Vω,n xn = Exn ,

n ≥ 1,

(3.1)

with some initial condition x0 = a, x1 = b. The  solution of this equation  formal  can  xn b for n ≥ 1 where X1 = be obtained via transfer matrices. Let Xn = . xn−1 a Then Xn+1 = Tω,n Xn where  Tω,n = Tω,n (E) =

 E − Vω,n −1 . 1 0

(3.2)

Anderson Model in a Decaying Potential

43

Iterating this relation yields Xn = Tω,n−1 · · · Tω,2 Tω,1 X1 . The free system xn+1 + xn−1 = Exn ,

n≥1

x1 = Ex0 , with x0 = a, x1 = b, has a basis of solutions given by ϕn+ = cos(kn) and ϕn− =   1 sin(kn) where 2 cos k = E, which correspond to initial conditions and cos k     ϕn 0 can be given respectively. Once more, the general solutions #n = ϕn−1 sin k in terms of transfer matrices so that #n = T n #0 where T =

  E −1 , 1 0

  a . 0

#0 =

In the case of decaying potentials, T corresponds to the asymptotic transfer matrix for vanishing potentials. We seek for a new basis in which the transfer matrix of the perturbed system is a perturbation of the identity. Hence, it is natural to express the system in the basis given by the free solutions. Let  n =



cos(kn) sin(kn) cos(k(n − 1)) sin(k(n − 1))

for

n ≥ 1,

(3.3)

and define new coordinates (Yn )n such that Xn = n Yn . This is usually known as the Prüfer transform. The recurrence relation for this new representation is given −1 by Yn+1 = Mω,n Yn where Mω,n = n+1 Tω,a n . Noting that n+1 = T n , we factorize this in a convenient way as −1 −1 (Tω,n T −1 )T n = n+1 (Tω,n T −1 )n+1 . Mω,n = n+1

Vω,n An where sin k   cos(nk) sin(nk) sin2 (nk) . An = − cos2 (nk) − cos(nk) sin(nk)

A simple computation shows that Mω,n = I +

Summarizing, our new recurrence corresponds to   Vω,n An Yn . Yn+1 = I + sin k

(3.4)

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O. Bourget et al.

 yn and define new sequences (ρn )n , (Rn )n and (θn )n through the Write Yn = yn−1 relation 

ρn = yn + iyn−1 = Rn eiθn

for

n ≥ 1,

where we adopt the convention θ1 ∈ [0, 2π ) and θn − θn−1 ∈ (0, π ) for n ≥ 2. Note that all the above quantities depend on the disorder ω which we will hide from the notation. Note also that the original variables can be easily recovered from the Prüfer variables:     cos(θ¯n ) xn = Rn with θ¯n = nk − θn . (3.5) xn−1 cos(θ¯n − k) Remark 3.1 Let Fn be the σ -algebra generated by the random variables (ωk )0≤k≤n so that Fn = σ (ω0 , · · · , ωn ). We note that Tω,j ∈ Fn for j ∈ {0, · · · , n} and that Tω,j is independent of Fn for j > n. In particular, Xn , Yn , ρn , Rn , θn ∈ Fn−1 and are hence independent of ωn . Similar considerations hold for negative values of n. These simple facts will turn certain objects into martingales. We seek for a more convenient form of the recursion (3.4) identifying An as a linear transform on the complex plane such that An eiθ = −i cos(nk − θ ) eink = −i cos(nk − θ ) ei(nk−θ) eiθ . We can see that (3.4) takes the simple form   Vω,n ¯ cos(θ¯n ) ei θn ρn , ρn+1 = 1 − i sin k

(3.6)

where θ¯n = nk − θn . The recurrence in terms of (Rn , θn )n becomes 2 Rn+1

=

Rn2

! 2 Vω,n Vω,n 2 ¯ ¯ sin(2θn ) + cos (θn ) , 1− sin k sin2 k

(3.7)

which can be iterated to yield 2 Rn+1

=

R12

n " j =1

! 2 Vω,j Vω,j 2 sin(2θ¯j ) + cos (θ¯j ) , 1− sin k sin2 k

(3.8)

where here and below we always assume R1 = 1. One can show that the phases (θ¯n )n satisfy the recursion

Anderson Model in a Decaying Potential

45

tan(θ¯n+1 − k) = tan(θ¯n ) +

Vω,n . sin k

(3.9)

The recursion (3.7) will be the starting point of our analysis but first, we have to show that the asymptotics of the (Xn )n and (Yn )n systems are equivalent. We start noticing that Tr(n∗ n ) ≤ 4,

det(n∗ n ) = sin2 k.

Hence, if 0 < λ1 < λ2 are the eigenvalues of n∗ n then λ1 + λ2 ≤ 4,

λ1 λ2 = sin2 k,

and thus λ1 =

sin2 k sin2 k sin2 k . = ≥ λ2 Tr(n∗ n ) − λ1 4

Since n 2 ≤ 4, we obtain sin2 k 2 Rn ≤ Xn 2 ≤ 4Rn2 , 4

(3.10)

where we recall that Rn = Yn . It turns out that the variables (Rn )n also allow us to control the asymptotics of the transfer matrices. Let Tω,n = Tω,n · · · Tω,2 Tω,1 so that Xn+1 = Tω,n X1 . We write Yn (θ ) when the recursion (3.4) is started from   cos θ # # Y1 = θ := , and similarly for Rn (θ ). sin θ Lemma 3.2 For any pair of initial angles θ1 = θ2 and initial norm Y1  = 1, there exists constants c(θ1 , θ2 ), C(θ1 , θ2 ) such that c(θ1 , θ2 ) max{Rn (θ1 ), Rn (θ2 )} ≤ Tω,n−1  ≤ C(θ1 , θ2 ) max{Rn (θ1 ), Rn (θ2 )}, for all n ≥ 1. Proof From the relation Tω,n−1 X1 = n Yn , we obtain Tω,n−1 2 ≥ n Yn 2 ≥

sin2 k Yn 2 . 4

(3.11)

The upper bound is more delicate and follows from a general result on unimodular matrices that we defer to Lemma A.1 in Appendix A.1.   Starting from this point, we specialize to the random vanishing case. Recall that (ωn )n≥0 is an i.i.d. sequence of bounded centered random variables defined on a

46

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probability space (!, F, P) with an absolutely continuous distribution and denote by E the expected value with respect to P so that E(ωn ) = 0 for all n ≥ 0. We also assume that E(ωn2 ) = 1. All the quantities defined above depend on ω, λ and E but these parameters will be omitted from most of the notations whenever no confusion is possible.

4 Super-Critical Case: Absolutely Continuous Spectrum This is based on a criterion of Last and Simon [35] that relates spectral properties to transfer matrices behavior. Let Tn (E) denote the product of transfer matrices associated to a bounded Schrödinger operator H on l 2 ([0, ∞)) and an energy E. Theorem 4.1 ([35, Teorem 1.3]) Suppose that 

b

Tn (E)4 dE < ∞.

lim inf n

(4.1)

a

Then, (a, b) ⊂ σ (H ) and the spectral measure is purely absolutely continuous on (a, b). The criterion is valid for any power larger than 2. There is nothing special about the power 4 except that it makes the computations easier. In the following, we write Tω,n (E), Yn (E) and Rn (E) when we want to emphasise the dependence on the energy E. We write Vω,n = λan ωn for n ≥ 0. Proof of Theorem 2.1, Part 1 (Super-Critical Case) Let θ0 be any initial angle and let [a, b] ⊂ (−2, 2). According to Lemma 3.2, it is enough to show $

b

lim inf E n

a

% < ∞.

Rn4 (E)dE

(4.2)

Then, by Fatou’s lemma, $



E lim inf n

a

b

% Rn4 (E)dE

$ ≤ lim inf E n

a

b

% Rn4 (E)dE

< ∞,

(4.3)

which implies that (4.1) holds almost surely. Squaring (3.7), we obtain 4 Rn+1 (E)

  Vω,n ¯ sin(2θn ) + Aω,n (E) Rn4 (E), = 1−2 sin k

(4.4)

where Aω,n (E) collects all the terms of higher order in Vω,n . An inspection at those terms shows that there exists c = c(a, b) ∈ (0, ∞) such that |Aω,n (E)| ≤ cn−2α for all E ∈ [a, b].

Anderson Model in a Decaying Potential

47

Observing that Rn is Fn−1 -measurable and bounded, we have % $     Vω,n   4 sin(2θ¯n ) + Aω,n (E)Fn−1 Rn4 (E). (4.5) (E)Fn−1 = E 1 − 2 E Rn+1 sin k Now, as θ¯n is Fn−1 -measurable and Vω,n is independent of Fn−1 and centered,       E Vω,n sin(2θ¯n )Fn−1 = sin(2θ¯n )E Vω,n = 0.

(4.6)

Collecting all the above observations, we conclude that    c  4 E Rn+1 (E)Fn−1 ≤ 1 + 2α Rn4 (E), n

(4.7)

for all E ∈ [a, b] and all n ≥ 1. Integrating with respect to P and iterating, we obtain  n    "  c c  4 1 + 2α , (E) ≤ 1 + 2α E Rn4 (E) ≤ E Rn+1 n j

(4.8)

j =1

for all E ∈ [a, b] and all n ≥ 1. As



j −2α < ∞ for α > 12 , the product above is

j

bounded uniformly in n and E ∈ [a, b]. This finishes the proof.

 

5 Critical Case: Transition from Singular Continuous to Point Spectrum 5.1 General Scheme To prove the absence of a.c. spectrum, we use the following criterion of Last and Simon. With the notations of the beginning of Sect. 4: Theorem 5.1 ([35, Theorem 1.2]) Suppose lim Tn (E) = ∞ for a.e. E ∈ n→∞

[a, b]. Then, μac ([a, b]) = 0, where μac is the absolutely continuous part of the spectral measure for H . We will prove that, for α = 12 and all λ ∈ R and all E ∈ (−2, 2) corresponding to values of k different from π4 , π2 and 3π 4 , one has   log Tω,n (E) λ2 , β = β(E, λ) := lim &n = −2α n→∞ 2(4 − E 2 ) j =1 j

P − a.s.

(5.1)

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O. Bourget et al.

which in particular implies that the norms diverge so that Theorem 5.1 can be applied. This will be done by working the fine asymptotics of (Rn )n . Next, we will argue that, for all λ and E as above, there exists a random initial direction ϑˆ 0 such that log Tω,n (E)ϑˆ 0  = −β, n→∞ log n lim

P − a.s.

(5.2)

This guaranties that the solution of the eigenvalue equation with initial conditions x0 = cos ϑ0 and x1 = sin ϑ0 satisfies

lim

 2 log xn2 + xn−1 log n

n→∞

= −β,

P − a.s.

(5.3)

Summarising, for almost every pair (E, ω) there is a unique decaying solution √ with rate of decay n−β . If β > 12 , it is in l 2 (N). This holds if and only if |λ| > 4 − E 2 . This √ condition is always met if |λ| ≥√2. If |λ| < 2, the condition fails once |λ| ≤ 4 − E 2 or, equivalently, if |E| ≤ 4 − λ2 . In this case, there is no l 2 solution. The result follows from the general theory of rank 1 perturbations (see, for instance, Section II in Simon’s lecture notes [37, Section II]).

5.2 Asymptotics of Transfer Matrices The following computation is valid in the critical and sub-critical region i.e. α ≤ 12 . Recall the relation E = 2 cos k. Proposition 5.2 Let 0 < α ≤ 12 . Let Rn (E, θ0 ) denote the recurrence (3.7) corresponding to an energy E, an initial direction θ0 and R1 = 1. Then, for all initial direction θ0 and E ∈ (−2, 2) corresponding to values of k different from π4 , π 3π 2 and 4 , we have log Rn (E, θ0 ) λ2 &n = , −2α n→∞ 8 sin2 k j =1 j lim

P − a.s.

(5.4)

Proof We take a0 = 1 and an = n−α to simplify the presentation and write again Vω,n = λn−α ωn for n ≥ 0. Remember the recursion, Rn2

=

n−1 " j =1

' 2 Vω,j Vω,j 2 ¯ ¯ cos (θj ) . 1− sin(2θj ) + sin k sin2 k



Using the Taylor expansion log(1 + ε) = ε −

ε2 2

+ O(ε3 ), we obtain

(5.5)

Anderson Model in a Decaying Potential

log

n−1 "



j =1

=

n−1  j =1

=



2 Vω,j Vω,j sin(2θ¯j ) + cos2 (θ¯j ) 1− sin k sin2 k

2 Vω,j Vω,j − sin(2θ¯j ) + sin k sin2 k

 n−1 E[V 2 ]  ω,j j =1

49

4 sin2 k

−

' (5.6)

'   1 2 2 ¯ 3 ¯ cos (θj ) − sin (2θj ) + O(Vω,j ) 2

2 − E[V 2 ] Vω,j Vω,j ω,j sin(2θ¯j ) + sin k 4 sin2 k

2 − E[V 2 ] Vω,j ω,j

(5.8)



 1 1 ¯j + cos 4θ¯j cos 2 θ 2 4 sin2 k '   2 E[Vω,j ] 1 1 3 + cos 2θ¯j + cos 4θ¯j + O(Vω,j ) . 2 4 sin2 k

+

(5.7)

(5.9)

(5.10)

The main contribution comes from the first term above. The & tree next terms n−1 −2α are martingales with respect to Fn and are shown to be o using j =1 j martingale arguments (Lemma 5.3 and Remark 5.4). The fifth term is more delicate and will be treated by ad-hoc methods in Appendix A.2. Finally, to estimate the 3 contribution of the error terms, we use the bound | log(1 + ε) − ε + ε2 | ≤ cε3 for 3 ) is of order j −3α since the some constant c, to see that the remaining term O(Vω,j  & n−1 −2α .   random variables are bounded and summing over j we obtain o j =1 j The following lemma is the key to estimate the martingale terms. It corresponds to [31, Lemma 8.4] but this shorter proof is taken from [8]. Lemma 5.3 Let (Xj )j be i.i.d. random variables with E[Xn ] = 0, let Fn = σ (X1 , · · · , Xn ) and let Yn ∈ Fn−1 for n ≥ 1. Let γ > 0 and define Mn =

n  Xj Yj j =1

jγ

and

sn =

n  1 . j 2γ

(5.11)

j =1

Assume |Xn |, |Yn | ≤ 1. Then, (Mn )n is an (Fn )n -martingale and (i) For γ ≤

1 2

and all ε > 0, − 1+ε 2

lim sn

n→∞

Mn = 0,

P − a.s.

(5.12)

(ii) For γ > 12 , (Mn )n converges P-almost surely to a finite (random) limit M∞ and, for all κ < γ − 12 , we have

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O. Bourget et al.

lim nκ (M∞ − Mn ) = 0,

P − a.s.

n→∞

(5.13)

Proof The reader can consult the book [18] for the general properties of martingales used below. The sequence (Mn )n is a martingale thanks to our hypothesis on (Xn )n and (Yn )n : indeed, since Mn , Yn ∈ Fn and Xn+1 is independent of Fn and centered, we have $  % Xn+1 Yn+1  E[Mn+1 |Fn ] = E + Mn Fn (5.14) (n + 1)γ = E[Xn+1 ]

Yn+1 + Mn = Mn . (n + 1)γ

(5.15)

Let γ ≤ 12 . We use Azuma’s inequality [4]: let (Mn )n be a martingale such that |Mn − Mn−1 | ≤ cn for all n ≥ 1. Then, 

t2 P [|Mn − M0 | ≥ t] ≤ 2 exp − &n 2 j =1 cj2

' .

(5.16)

1+ε

In our case, M0 = 0, cj = 2j −γ , and taking t = sn 2 for 0 < ε < ε, we obtain $

1+ε 2

P |Mn | ≥ sn

%



 ≤ 2 exp −Cnε ,

(5.17)

for some C > 0. The claim (i) then follows from Borel-Cantelli’s lemma. Now, let γ > 12 . Noticing that, for k < l, E[Xk Yk Xl Yl ] = E[E[Xk Yk Xl Yl |Fl−1 ]] = E[Xk Yk Yl E[Xl |Fl−1 ]] = 0, (5.18) we have sup E[Mn2 ] = sup n

n

n E[X 2 Y 2 ]  j j j =1

j 2γ

≤

 1 < ∞. j 2γ

(5.19)

j ≥1

Hence, (Mn )n is an L2 -martingale and converges, i.e., there exists a random variable M∞ such that limn→∞ Mn = M∞ , P-a.s.. Finally, applying Azuma’s inequality to the martingale (Mn+k − Mn )k≥0 , we obtain

  1 P nκ |Mn+k − Mn | ≥ 1 ≤ 2 exp −cn2(γ − 2 −κ) ,

(5.20)

for all k ≥ 0. Choosing κ < γ − 12 , the last claim follows from Fatou’s lemma, the convergence of (Mn )n and Borel-Cantelli.  

Anderson Model in a Decaying Potential

51

Remark 5.4 The martingale property for the second, third and fourth terms in decomposition (5.10) follows from Lemma 5.3, recalling that θ¯j ∈ Fn for all j ≤ n + 1. &  n −2α by All the martingale terms in Proposition 5.2 are seen to be o j =1 j taking γ = α in Lemma 5.3 for the first one and β = 2α for the others. The lemma actually guaranties that they are much smaller. There is still room in Azuma’s inequality to allow the support of the random variables (Xn )n to grow with n. Unbounded random variables could be handled under some moment assumptions with some extra effort.

5.3 Control of Generalized Eigenfunctions The next proposition provides l 2 eigenfunctions in the proper region. For an angle θ , we write # θ = (cos θ, sin θ ). Proposition 5.5 Let α = 12 and k = π4 , 2π 4 , exists an initial angle ϑ0 = ϑ0 (ω) such that

3π 4 .

Then, for P-almost every ω, there

log Tω,n (E)# ϑ0  λ2 &n = − , −2α n→∞ 8 sin2 k j =1 j lim

P − a.s.

(5.21)

The proof is given in details in Appendix A.1. We now provide a non-asymptotic lower bound on eigenfunctions that will be the key to the proof of absence of dynamical localization. The next lemma actually provides a lower bound on any non-trivial generalized solution of Hω,λ x = Ex for α = 12 , λ > 0, uniformly in E ranging over compact intervals of (−2, 2). The proof is taken from [8] where it was developed in the context of the discrete Dirac model. Lemma 5.6 Let α = 12 and fix λ > 0. For E ∈ (−2, 2) define xω,E = (xω,E,n )n as the solution of Hω,λ x = Ex with a possibly random initial condition # ϑ0 . Then, for each compact interval I ⊂ (−2, 2), there exists a deterministic constant κ = κ(I ) > 0 such that, for P-almost every ω, there exists cω = cω (I ) > 0 such that 

|xω,E,n |2 + |xω,E,n−1 |2 ≥ cω n−κ ,

∀ n ∈ Z.

(5.22)

Proof We prove the bound for all n ≥ 1,  case being similar. We can  the opposite xω,E,n = Tω,n−1# ϑ0 . This implies in reconstruct xω,E through the recurrence xω,E,n−1 particular that  |xω,E,n |2 + |xω,E,n−1 |2 ≥ Tω,n−1 −1 .

(5.23)

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Hence, using Lemma 3.2 with some ϑ1 = ϑ2 ,     (5.24) P Xω,E,n  ≤ n−κ ≤ P Tω,n−1 (E) ≥ nκ   ≤ n−2κ E Tω,n−1 (E)2 (5.25)     ≤ C1 (ϑ1 , ϑ2 )n−2κ E Rn2 (E, ϑ1 ) + E Rn2 (E, ϑ1 ) , (5.26) for some C1 (ϑ1 , ϑ2 ) > 0. Keeping in mind the recursion (3.8), the argument of the proof of Part 1 of Theorem 2.1 given in Sect. 4 can be reproduced and yields E



Rn2 (E, ϑ1 )



 n  " b 1+ ≤ C2 nb−2κ , ≤ j

(5.27)

j =1

for some constants b = b(I ) and C2 = C2 (b). The estimate for Rn2 (E, ϑ2 ) is of course similar. Taking b − 2κ < −1, the result follows by Borel-Cantelli.  

5.4 Absence of Dynamical Localization The following result shows that no dynamical localization can arise in the critical case. This was already stated in Theorem 2.2 from [24]. The simple argument below uses our lower bound on eigenfunctions from Lemma 5.6 and is once again taken from [8]. Proposition 5.7 Let α = 12 and λ > 0. Let I be a compact interval contained in the interior of σpp (Hω,λ ). Then, there exists p0 > 0 such that, for P-almost every ω,  2   lim sup |X|p/2 e−itHω,λ ψ  = ∞,

(5.28)

t→∞

for all p > p0 and ψ ∈ RanPI (Hω,λ ). Proof Let cω and κ be as in Lemma 5.6. Let (ψl )l be a basis of RanPI (Hω,λ ) consisting of eigenfunctions of the operator Hω,λ , with corresponding eigenvalues (El )l . Define the truncated position operator XN = Xχ[−N,N ] . Then, taking p > κ − 1,  2    |n|p |ψl (n)|2 |XN |p/2 ψl  = |n|≤N

(5.29)

Anderson Model in a Decaying Potential

≥

1 2

53



(|n| − 1)p max{|ψl (n)|2 , |ψl (n − 1)|2 } (5.30)

|n|≤N −1

cω ≥ 2



(|n| − 1)p |n|−κ ≥ cω N p−κ+1 ,

(5.31)

|n|≤N −1

for some cω > 0 and for all l. Let ψ ∈ RanPI (Hω,λ ) and write ψ = Hence,  2    αl α l  e−it (El −El  ) ψl  , |XN |p/2 ψl . |XN |p/2 e−itHω,λ ψ  =

&

l al ψl .

(5.32)

l,l 

After a careful application of the dominated convergence theorem to exchange sums and integrals, we obtain 1 T →∞ T



T

lim

0

  2 2      |al |2 |XN |p/2 ψl  ≥ cω N p−κ+1 (5.33) . |XN |p/2 e−itHω,λ ψ  dt = l

Hence, there exists an diverging (random) sequence (TN )N such that 1 TN



TN

0

 2 c   |XN |p/2 e−itHω,λ ψ  dt ≥ ω N p−κ+1 , 2

(5.34)

for all N ≥ 1. From here, we can find a diverging (random) sequence (tN )N such that 2 c    |XN |p/2 e−itN Hω,λ ψ  ≥ ω N p−κ+1 , 4 for all N ≥ 1. This finishes the proof.

(5.35)  

6 Sub-critical Case: Dynamical Localization Theorem 2.3 was proved in [36] using the Kunz-Souillard method. We briefly outline the approach used in [7] to prove dynamical localization for the analogue continuum model and which becomes particularly simple when adapted to the discrete case (see [8] for the related discrete Dirac model). We use the fractional moment method [1, 2]. The key to this approach is to estimate the correlator by the fractional moments of the Green’s function in boxes. Let Hω,L denote the restriction of the operator Hω,λ to [−L, L], let Rω,L (E) = (Hω,L − E)−1 and let Gω,L (m, n) = δm , Rω,L (E)δn .

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A suitable adaptation of the arguments of [2, Section 7] to the decaying potential case shows that      −1 (6.1) lim inf E |Gω,L (m, n)|s dE, E Qω (m, n; I )2 ≤ C|λ|−1 am L→∞

I

for s ∈ (0, 1) small enough, for some C = C(s) > 0 and all m, n ∈ N. It is then possible to show that there exists a constant C = C(s, I ) such that     −2s #m −s , E |Gω,L (m, n)|s ≤ C|λ|−2s am E Tω,[m,n] (E)X

(6.2)

for all E ∈ I and all m, n ∈ N, where Tω,[m,n] (E) = Tω,n · · · Tω,m and #m = Xm −1 Xm with Xm = Tω,m X0 . The inverse fractional moments of transfer X matrices are shown to decrease exponentially in [10] in the ergodic case. The proof uses the positivity of the Lyapunov exponent as an input. The decaying potential case requires a finer analysis [7, 8] and uses the asymptotics of transfer matrices given in Proposition 5.2 as a starting point. More precisely, it is possible to show that, for s ∈ (0, 1) small enough, there exists constants C = C(m, s, I ) > 0 and c = c(s, I ) > 0 such that   #m −s ≤ Ce−cn1−2α , E Tω,[m,n] (E)X

(6.3)

for all E ∈ I and all n ∈ N. Theorem 2.3 then follows from a concatenation of the above estimates. The proof of the upper bound in Proposition 2.4 in standard (see [11, Theorem 9.22]). The lower bound can be proved along the lines of Lemma 5.6. Proposition 2.5 can then be proved as Proposition 5.7 using Proposition 2.4 instead of Lemma 5.6. Acknowledgments O. Bourget is partially supported by Fondecyt grant 1161732. G. Moreno Flores is partially supported by Fondecyt grant 1171257, Núcleo Milenio ‘Modelos Estocásticos de Sistemas Complejos y Desordenados’ and MATH Amsud ‘Random Structures and Processes in Statistical Mechanics’. A. Taarabt is partially supported by Fondecyt grant 11190084.

Appendix A: Some Technical Lemmas A.1 Two Results on Unimodular Matrices We say that a matrix is unimodular if it has determinant equal to 1. The following lemma is used to compare the asymptotics of the transfer matrix with those of the sequence (Rn )n . Lemma A.1 Let A be an unimodular matrix and let θˆ = (cos θ, sin θ ). Then, for all pair of angles |θ1 − θ2 | ≤ π2 ,

Anderson Model in a Decaying Potential

A ≤ sin



|θ1 −θ2 | 2

55

−1

max{Aθˆ1 , Aθˆ2 }.

(A.1)

Proof First, there exists angles θ0 and σ0 such that A∗ σˆ 0 = Aθˆ0 . Then, for any angle θ , | cos(θ − θ0 )| = |θˆ , θˆ0 | =

1 1 Aθˆ  |θˆ , A∗ σˆ 0 | = |Aθˆ , σˆ 0 | ≤ . A A A

This way, for any pair of angles, we obtain A max{| cos(θ1 − θ0 )|, | cos(θ2 − θ0 )|} ≤ max{Aθˆ1 , Aθˆ2 }. To conclude, just note that for all |γ | ≤ π/2, the minimum of the function x → max{| cos(x)|, | cos(x + γ )|} is attained at π/2 − γ /2 and is equal to | cos(π/2 − γ /2)| = | sin(γ /2)|.   The following lemma is used to find eigenfunctions with the proper decay. Lemma A.2 For a unimodular matrix with A > 1, define ϑ = ϑ(A) as the ˆ = A−1 . We also define r(A) = unique angle ϑ ∈ (− π2 , π2 ] such that Aϑ T T A(1, 0) /A(0, 1) . Let (An )n be a sequence of unimodular matrices with An  > 1 and write ϑn = ϑ(A) and rn = r(An ). Assume that (i) lim An  = ∞, n→∞

An+1 A−1 n  = 0. n→∞ An  An+1 

(ii) lim Then,

1. (ϑn )n has a limit ϑ∞ ∈ (−π/2, π/2) if and only if (rn )n has a limit r∞ ∈ [0, ∞). If ϑn → ±π/2, then rn → ∞ but, if rn → ∞, we can only conclude that |ϑn | → π/2. 2. Suppose (ϑn )n has a limit ϑ∞ = 0, π2 . Then, lim

n→∞

log An ϑˆ ∞  = −1 log An 

if and only if

lim sup n

log |rn − r∞ | ≤ −2. log An 

(A.2)

Proof Recall ϑn denotes the unique angle in (−π/2, π/2] such that An ϑˆ n  = An −1 . Let ϑn ∈ (ϑn − π/2, ϑn + π/2] be the unique angle such that An ϑˆ n  = An  and let ϑn⊥ be either ϑn − π/2 or ϑn + π/2 and such that ϑn lies between ϑn and ϑn⊥ .     1 0 Let v0 = , w0 = , vn = An v0 and wn = An w0 . Then, 0 1

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ϑˆ n = cos ϑn v0 + sin ϑn w0 ,

(A.3)

and applying An on both sides, ±

1 ϑˆ n = An ϑˆ n = cos ϑn vn + sin ϑn wn " 0, An 

(A.4)

as An  → ∞. Hence, rn = r(An ) =

vn  " | tan ϑn |. wn 

(A.5)

This shows that, if ϑn → ϑ∞ , then rn → | tan ϑ∞ | ∈ [0, ∞]. On the other hand, if rn → r∞ ∈ [0, ∞], then |ϑn | → arctan(ρ∞ ), with the convention arctan(∞) = π/2. If ρ∞ = ∞, we only get that |ϑn | → π/2. Otherwise, it is enough to prove that ϑn − ϑn−1 → 0 to show the convergence of ϑn . For this, we use the decomposition ϑˆ = cos(ϑ − ϑn )ϑˆ n + sin(ϑ − ϑn )ϑˆ n⊥ .

(A.6)

Applied to ϑn+1 , this yields ϑˆ n+1 = cos(ϑn+1 − ϑn )ϑˆ n + sin(ϑn+1 − ϑn )ϑˆ n⊥ ,

(A.7)

An ϑˆ n+1 = ±An −1 cos(ϑn+1 − ϑn )ϑˆ n + sin(ϑn+1 − ϑn )An ϑˆ n⊥

(A.8)

and

" sin(ϑn+1 − ϑn )An ϑˆ n⊥ . Hence, An ϑˆ n⊥ | sin(ϑn+1 − ϑn )| " An ϑˆ n+1  ˆ ≤ An A−1 n+1 An+1 ϑn+1  =

(A.9) A−1 n+1 An  An+1 

.

In the last step, we used that the An ’s are unimodular to switch from An A−1 n+1  to −1 ⊥ An+1 An . By condition (ii), it is enough to show that An ϑˆ n   An . For this, decompose ϑn⊥ = αn ϑˆ n + βn ϑˆ n ,

(A.10)

for some coefficients such that α 2 + β 2 = 1. Note that, by construction, |αn | > |βn | and hence |αn | ≥ 1/2. Therefore,

Anderson Model in a Decaying Potential

57

An ϑˆ n⊥ 2 = α 2 An 2 + β 2 An −2 ≥

1 An 2 . 4

This finishes the proof Part 1. To prove Part 2, assume ϑn → ϑ∞ ∈ (0, π/2) and apply (A.6) to ϑ = ϑ∞ to obtain An ϑ∞ = ±An −1 cos(ϑ∞ − ϑn )ϑˆ n + sin(ϑ∞ − ϑn )An ϑˆ n⊥ .

(A.11)

Recalling that An  → ∞, ϑn − ϑ∞ → 0 and An ϑˆ n⊥  # An ,   log | sin(ϑ∞ − ϑn )| log An ϑ∞  " max −1, +1 . log An  log An 

(A.12)

Hence, the left condition in (A.2) is satisfied if and only if lim sup n

log | sin(ϑ∞ − ϑn )| ≤ −2. log An 

(A.13)

But, disregarding multiplicative constants which will disappear in the limit, sin(ϑ∞ − ϑn ) # tan(ϑ∞ − ϑn ) # tan ϑ∞ − tan ϑn = r∞ − rn .

(A.14)  

We apply this with An = Tω,n to prove Proposition 5.5. Define (1) xn+1 (1) xn

!

  1 = Tω,n 0

and

!   (2) xn+1 0 = Tω,n (2) 1 xn 1

(A.15)

and let Rn(i) , θn(i) , n ≥ 1, i = 1, 2 be the corresponding Prüfer radii and phases. We (1) (2) let rn = Rn /Rn and ϑn be as in Lemma A.2. Then it follows from (3.5) and some elementary trigonometry that (1) (2) (2) xn − xn(1) xn+1 = Rn(1) Rn(2) sin k sin(θn(2) − θn(1) ), xn+1

(A.16)

(i) (i) where θ¯n = nk − θn . On the other hand,

   10 (1) (2) xn+1 xn(2) − xn(1) xn+1 = det Tω,n = 1. 01

(A.17)

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This, together with the convergence (i)

Rn = β, n→∞ log n lim

i = 1, 2,

(A.18)

gives log | sin(θn − θn )| log | sin(θ¯n − θ¯n )| = lim = −2β. (A.19) n→∞ n→∞ log n log n (2)

(1)

(2)

(1)

lim

Remember the decomposition (5.10). We have to estimate the difference of the (1) (2) (2) (1) expansions for log Rn and log Rn . By (A.19), | sin(θ¯n − θ¯n )|  n−β+ , for any  > 0. Hence, there exists random sequences (mn )n ⊂ N and (n )n ⊂ R such that θ¯n(1) − θ¯n(2) = mn π + n and |n |  n−β+ . Therefore, sin(2θ¯n(2) ) = sin(2θ¯n(1) + 2n ) " sin(2θ¯n(1) ) + 2 cos(2θ¯n(1) )n .

(A.20)

This shows that

and

   1  (1) (2)  Vj sin(2θ¯j ) − sin(2θ¯j )   j − 2 −2β+ ,     2 (1) (2)  Vj cos2 (θ¯j ) − cos2 (θ¯j )   j −1−2β+ ,

by a similar argument. Hence, log rn = −

n  Vω,j  j =1

sin k

n   (1) (2) sin(2θ¯j ) − sin(2θ¯j ) + Aj

(A.21)

j =1

where the first sum is a convergent martingale by Lemma 5.3 with γ = 12 + 2β −  and the second one is absolutely convergent as Aj = O(j −1−2β+ ). This shows that rn → r∞ ∈ (0, ∞) almost surely which implies that ϑn has a limit ϑ∞ = 0, π/2 by the first part of Lemma A.2. The equivalence (A.2) in our context corresponds to lim

n→∞

log Rn (ϑ∞ ) = −β log n

if and only if

lim sup n

log |rn − r∞ | ≤ −2β. (A.22) log n

Let us denote by log rn = Mn + Sn the decomposition (A.21) and log r∞ = M∞ + S∞ where M∞ and S∞ are the almost sure limits of Mn and Sn respectively. Then,     |r∞ − rn | = eM∞ +S∞ 1 − eMn −M∞ +Sn −S∞  " eM∞ +S∞ |Mn − M∞ + Sn − S∞ |  eM∞ +S∞ n−2β+2 ,

(A.23) (A.24)

Anderson Model in a Decaying Potential

by the last statement of Lemma 5.3 with γ = Proposition 5.5.

59 1 2

+ 2β − . This finishes the proof of

A.2 Analysis of the Prüfer Phases We begin with a simple observation: from (3.6),    ρ   |Vω,a |   n+1  i(θn+1 −θn ) ≤ 1, − 1 =  − 1 ≤ e ρn | sin k|

(A.25)

for n large enough. Hence, for n large enough, |θn+1 − θn | < π/2 and |θn+1 − θn | ≤

 π π   | sin(θn+1 − θn )| ≤ ei(θn+1 −θn ) − 1  n−α . 2 2

(A.26)

This can be written in the equivalent form |θ¯n+1 − θ¯n − k| ≤ c0 n−α ,

(A.27)

for some c0 > 0, which will be more suitable for our purposes. The next lemma provides the control of the Prüfer phases needed to complete the proof of Proposition 5.2. Lemma A.3 Let 0 < α ≤ 12 . Let E ∈ (−2, 2) corresponding to values of k different from π4 , π2 and 3π 4 . Then,  2 ] 1 cos 2θ¯ + E[V j j =1 ω,j 2 &n −2α j =1 j

&n lim sup n→∞

1 4

cos 4θ¯j

 = 0.

(A.28)

Proof The key to the proof is [31, Lemma 8.5] which states: suppose that y ∈ R is not in π Z. Then, there exists a sequence of integers ql → ∞ such that     ql ql        θj − θ0 − jy  , cos θj  ≤ 1 +   j =1 j =1

(A.29)

for all (θj )j ≥0 ⊂ R. We will treat the term with cos(4θ¯j ) as the other one is similar. We will take y = 4k above. Let n be large enough so that it can be written as n = n0 + Kql with −2 n0 ≥ ql2 and 4c0 n−α 0 ≤ ql (where c0 is the constant from (A.27)). Then,

60

O. Bourget et al.    K q    n l       −2α  −2α  ¯j ) =  ¯ j cos(4 θ (n + mq + r) cos(4 θ(n + mq + r))  (A.30) 0 l 0 l      j =n0 +1 m=0 r=1 ≤

K  m=0

+

q  l     (n0 + mql )−2α  cos(4θ¯ (n0 + mql + r))  

(A.31)

r=1

ql  K      (n0 + mql + r)−2α − (n0 + mql )−2α 

(A.32)

m=0 r=1

=: A + B.

(A.33)

By (A.29), A≤

K 

(n0 + mql )

−2α

1+4

m=0

ql 

! ¯ 0 + mql + r) − θ¯ (n0 + mql ) − kr| . (A.34) |θ(n

r=1

Now, by (A.27),

4

ql 

¯ 0 + mql + r) − θ¯ (n0 + mql ) − kr| ≤ c0 |θ(n

r=1

ql  r  (n0 + mql + r)−α (A.35) r=1 s=1

≤ c0 (n0 + mql )−α

ql 

r ≤ c0 ql2 n−α 0 ≤ 1.

(A.36)

r=1

Thus,

A≤2

K 

(n0 + mql )−2α ≤ 2ql−2α

m=0

K 

(n0 ql−1 + m)−2α ≤ c1 ql−2α K 1−2α ≤ c1 ql−1 n1−2α ,

m=0

for some finite c1 > 0. To estimate B, we use that     (n0 + mql + r)−2α − (n0 + mql )−2α  ≤ c2 (n0 + mql )−2α−1 r,

(A.37)

for some finite c2 > 0 which shows that B ≤ c2

ql K K    −1 −2α (n0 + mql )−2α−1 r ≤ c2 ql2 n−1 (1 + n−1 0 0 mql ) (n0 + mql ) m=0 r=1

≤ c2

K  m=0

(n0 + mql )−2α ,

(A.38)

m=0

(A.39)

Anderson Model in a Decaying Potential

61

where we used ql2 n−1 0 ≤ 1. This last sum can be estimated as above. Summarizing,    ql     cos θj  ≤ c3 ql−1 n1−2α ,  j =1  for some finite c3 > 0. This finishes the proof.

(A.40)  

References 1. M. Aizenman, S. Molchanov, Localization at large disorder and at extreme energies: an elementary derivations, Comm. Math. Phy. 157, 245–278 (1993) 2. M. Aizenman, S. Warzel, Random Operators: Disordered effects on Quantum Spectra and Dynamics. Graduate Studies in Mathematics, vol 168 (American Mathematical Society, Providence, 2016) 3. M. Aizenman, R. Sims, S. Warzel, Stability of the absolutely continuous spectrum of random Schrödinger operators on tree graphs. Probab. Theory Related Fields 136, 363–394 (2006) 4. K. Azuma, Weighted sums of certain dependent random variables. Tôhoku Math. J. 19(3), 357–367 (1967) 5. J. Bourgain, On random Schrödinger operators on Z2 . Discret Contin. Dyn. Syst. 8, 1–15 (2002) 6. J. Bourgain, Random lattice schrödinger operators with decaying potential: some higher dimensional phenomena, in Geometric Aspects of Functional Analysis. Lectures Notes in Mathematics, vol. 1807 (Springer, Berlin, 2003), pp. 70–98 7. O. Bourget, G. Moreno, A. Taarabt, Dynamical localization for the one-dimensional continuum Anderson model in a decaying random potential. Ann. Henri Poincaré, 21(8), (2020) 8. O. Bourget, G. Moreno, A. Taarabt, One-dimensional discrete dirac operators in a decaying random potential I: Spectrum and dynamics. Math. Phys. Anal. Geom. 23(20), (2020). 9. R. Carmona, Exponential localization in one dimensional disordered systems. Duke Math. J. 49, 191–213 (1982) 10. R. Carmona, A. Klein, F. Martinelli, Anderson localization for bernoulli and other singular potentials. Commun. Math. Phys. 108, 41–66 (1987) 11. H.L. Cycon, R.G. Froese, W. Kirsch, B. Simon, Schrödinger Operators with Applications to Quantum Mechanics and Global Geometry. Texts and Monographs in Physics, Springer Study Edition (Springer, Berlin, 1987) 12. D. Damanik, A. Gorodetski, An extension of the Kunz-Souillard approach to localization in one dimension and applications to almost-periodic Schrödinger operators. Adv. Math. 297, 149–173 (2016) 13. D. Damanik, G. Stolz, A continuum version of the Kunz–Souillard approach to localization in one dimension. J. Für die reine und Angewandte Math. (Crelles J.) 660, 99–130 (2011) 14. R. Del Rio, S. Jitomirskaya, Y. Last, B. Simon, What is localization? Phys. Rev. Lett. 75, 117–119 (1995) 15. R. Del Rio, S. Jitomirskaya, Y. Last, B. Simon, Operators with singular continuous spectrum IV: Hausdorff dimensions, rank one perturbations and localization. J. Anal. Math. 69, 153–200 (1996) 16. F. Delyon, Appearance of a purely singular continuous spectrum in a class of random Schrödinger operators. J. Statist. Phys. 40, 621–630 (1985) 17. F. Delyon, B. Simon, B. Souillard, From power pure point to continuous spectrum in disordered systems. Ann. Henri Poincaré, 42(6), 283–309 (1985)

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18. R. Durrett, Probability: Theory and Examples. Cambridge Series in Statistical and Probabilistic Mathematics, 4th edn. (Cambridge University Press, New York, 2010) 19. A. Figotin, F. Germinet, A. Klein, P. Müller, Persistence of Anderson localization in Schrödinger operators with decaying random potentials. Ark. Mat. 45. 15–30 (2007) 20. R. Froese, D. Hasler, W. Spitzer, Absolutely continuous spectrum for the Anderson model on a tree: a geometric proof of Klein’s theorem. Comm. Math. Phys. 269, 239–257 (2007) 21. L. Ge, X. Zhao, Exponential dynamical localization in expectation foe one dimensional Anderson model. J. Spect. Theory 22. F. Germinet, A. Klein, Bootstrap multiscale analysis and localization in random media. Commun. Math. Phys. 222, 415–448 (2001) 23. F. Germinet, A. Taarabt, Spectral properties of dynamical localization for Schrödinger operators. Rev. Math. Phys. 25(9), 1350016 (2013) 24. F. Germinet, A. Kiselev, S. Tcheremchantsev, Transfer matrices and transport for Schrödinger operators. Ann. Inst. Fourier 54, 787–830 (2004) 25. F. Germinet, A. Klein, J. Schenker, Dynamical delocalization in random Landau Hamiltonians. Ann. Math. 166, 215–244 (2007) 26. I. Goldsheid, S. Molchanov, L. Pastur, A pure point spectrum of the stochastic one-dimensional Schrödinger equation. Funct. Anal. Appl. 11, 1–10 (1977) 27. V. Jak˘si´c, Y. Last, Spectral structure of Anderson type Hamiltonians. Inven. Math. 141, 561– 577 (2000) 28. S. Jitomirskaya, X. Zhu, Large deviations of the Lyapunov exponent and localization for the 1D Anderson model. Comm. Phys. Math. 370, 311–324 (2019) 29. A. Klein, Extended states in the anderson model on the bethe lattice. Adv. Math. 133, 163–184 (1998) 30. W. Kirsch, M. Krishna, J. Obermeit, Anderson model with decaying randomness: mobility edge. Math. Z. 235, 421–433 (2000) 31. A. Kisselev, Y. Last, B. Simon, Modified Prüfer and EFGP transforms and the spectral analysis of one-dimensional schrödinger operators. Comm. Math. Phys. 194, 1–45 (1998) 32. S. Kotani, N. Ushiroya, One-dimensional Schrödinger operators with random decaying potentials. Comm. Math. Phys. 115, 247–266 (1988) 33. M. Krishna, Anderson model with decaying randomness: existence of extended states. Proc. Indian Acad. Sci. (Math. Sci.) 100, 285–294 (1990) 34. H. Kunz, B. Souillard, Sur le spectre des opérateurs aux différences finies aléatoires. Comm. Math. Phys. 78, 201–246 (1980) 35. Y. Last, B. Simon, Eigenfunctions, transfer matrices, and absolutely continuous spectrum of one-dimensional Schrödinger operators. Invent. Math. 135, 329–367 (1999) 36. B. Simon, Some Jacobi matrices with decaying potential and dense point spectrum. Comm. Math. Phys. 87, 253–258 (1982) 37. B. Simon, Spectral Analysis of Rank One Perturbations and Applications. CRM Lectures Notes, vol. 8 (American Mathematical Society, Providence, 1995), pp. 109–149

On Non-selfadjoint Operators with Finite Discrete Spectrum Olivier Bourget, Diomba Sambou, and Amal Taarabt

1 Introduction The spectral theory of non-selfadjoint perturbations of selfadjoint operators has made significant advances in the last decade. For an historical panorama on the relationships between non-selfadjoint operators and quantum mechanics, we refer the reader to e.g. [1] and references therein. The distributional properties of the discrete spectrum have been one of the main issues considered in this field, among the many results obtained so far. In the present note, we keep going on with the spectral analysis of compact nonselfadjoint perturbations of the discrete Schrödinger operator. In [3], we have proved that in dimension 1 and under adequate regularity conditions, the discrete spectrum of the perturbed operator remains finite. We have also exhibited some Limiting Absorption Principle, thus completing some previous results obtained by [4, 5] in the framework of Jacobi matrices. Presently, we extend these results to non-selfadjoint perturbations of some fibered version of the discrete one dimensional Schrödinger operator. Following the methodology developed in [3], the paper is organized as follows. The model and the results are introduced in Section 2. Theorem 2.1 is proved in Section 3.1, through an analysis far from the thresholds involving complex scaling arguments. In Section 3.2, an analysis of the resonances located in a neighbourhood of the thresholds (Theorem 2.2) allows to conclude with the proof of Theorem 2.3.

O. Bourget () · D. Sambou · A. Taarabt Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Santiago, Chile e-mail: [email protected]; [email protected]; [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 P. Miranda et al. (eds.), Spectral Theory and Mathematical Physics, Latin American Mathematics Series, https://doi.org/10.1007/978-3-030-55556-6_4

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Notations and Basic Concepts Throughout this paper, we adopt the notations of [3]. Z, Z+ and N denote respectively the sets of integral numbers, non-negative and positive integral numbers. For γ ≥ 0, we define the weighted Hilbert spaces 2 ±γ (Z)

   := x ∈ CZ : e±γ |n| |x(n)|2 < ∞ . n∈Z

In particular, 2 (Z) = 20 (Z) and one has the inclusions 2γ (Z) ⊂ 2 (Z) ⊂ 2−γ (Z). For γ > 0, one defines the multiplication operators Wγ : 2γ (Z) → 2 (Z) by 



Wγ x (n) := e(γ /2)|n| x(n), and W−γ : 2 (Z) → 2−γ (Z) by W−γ x (n) := e−(γ /2)|n| x(n). We denote by (δn )n∈Z the canonical orthonormal basis of 2 (Z). The discrete Fourier transform F : 2 (Z) → L2 (T), where T := R/2π Z, is defined for any x ∈ 2 (Z) and f ∈ L2 (T) by 1  −inϑ (Fx)(ϑ) := √ e x(n), 2π n ∈ Z

−1  1 F f (n) := √ 2π

 T

einϑ f (ϑ)dϑ.

(1.1) The operator F is unitary. For any bounded (resp. selfadjoint) operator X acting on 2 (Z), we define the bounded (resp. selfadjoint) operator X # acting on L2 (T) by # := FXF −1 . X

(1.2)

More generally, if H is an auxiliary Hilbert space, for any bounded (resp. selfadjoint) operator Y acting on 2 (Z) ⊗ H , we define the bounded (resp. selfadjoint) # acting on L2 (T) ⊗ H by operator Y # := (F ⊗ I )Y (F −1 ⊗ I ). Y

(1.3)

If H is a separable Hilbert space, B(H ) and GL(H ) denotes the algebras of bounded linear operators and boundedly invertible linear operators acting on H . S∞ (H ) stands for the ideal of compact operators. For any operator H ∈ B(H ), we denote its spectrum by σ (H ), its resolvent set by ρ(H ), the set of its eigenvalues by Ep (H ). We also define its point spectrum as the closure of the set of its eigenvalues and write it σpp (H ) = Ep (H ). Since this article deals with nonselfadjoint (bounded) operators, it is also convenient to clarify the different notions of spectra we use. Let T be a closed linear operator acting on a Hilbert space H , and z be an isolated point of σ (T ). If γ is a small contour positively oriented containing z as the only point of σ (T ), the Riesz projection Pz associated to z is defined by 1 Pz := 2iπ

(

(T − ζ )−1 dζ. γ

The algebraic multiplicity of z is then defined by

On Non-selfadjoint Operators with Finite Discrete Spectrum

65

m(z) := rank(Pz ),

(1.4)

and when it is finite, the point z is called a discrete eigenvalue of the operator T .

 Note that one has the inequality m (z) ≥ dim Ker(T − z) , which is the geometric multiplicity of z. The equality holds if T is normal (see e.g. [10]). So, one defines the discrete spectrum of T as   σdisc (T ) := z ∈ σ (T ) : z is a discrete eigenvalue of T .

(1.5)

A closed linear operator is said to be of Fredholm if it has a closed range and both its kernel and cokernel are finite-dimensional. We define the essential spectrum of T as   σess (T ) := z ∈ C : T − z is not a Fredholm operator .

(1.6)

For ! ⊆ C an open domain and B a Banach space, Hol(!, B) denotes the set of holomorphic functions from ! with valuesin B. For two subsets 1 and 2 of R,  we denote as a subset of C, 1 + i2 := z ∈ C : Re(z) ∈ 1 and  Im(z)∗ ∈ 2 . For R > 0 and ζ0 ∈ C, we set DR (ζ0 ) := z ∈ C : |z − ζ0 | < R and DR (ζ0 ) := DR (ζ0 ) \ {ζ0 }. By 0 < |η| 0. Let A be a selfadjoint operator defined on H . An operator B ∈ B(H ) belongs to the class AR (A) iθA −iθA , defined for θ ∈ R, has an extension lying in if the

map θ → e Be Hol DR (0), B(H ) . In this case, we write B ∈ AR (A). The collection of bounded operators for which a complex scaling w.r.t. A can be performed is simply denoted by: A(A) := AR (A). R>0

The main properties of the classes A(A) are listed in [3, Section 6], and we will frequently refer to them (see Sect. 4). Given the operator H0 introduced previously, we define the auxiliary selfadjoint operator A0 acting on 2 (Z, H ) by: A0 := A0 ⊗ I,

(2.5)

#0 is the unique selfadjoint extension of #0 F and the operator A where A0 := F −1 A #0 := sin ϑ(−i∂ϑ ) + (−i∂ϑ ) sin ϑ. the symmetric operator defined on C ∞ (T) by A The operators A0 and A0 are respectively the conjugate operator of H0 and L0 in the sense of the Mourre theory. Remark 2.1 Examples of operators which belong to A(A0 ): (a) any operator of the form |ψ ϕ| ⊗ B where ϕ and ψ are analytic vectors for A0 and B ∈ B(H ) (see Sect. 4.1). (b) any V ∈ B( 2 (Z, H )) which satisfies Assumption 2.1 below (see Corollary 4.1).

 If the perturbation V ∈ S∞ 2 (Z, H ) is compact, it follows from the Weyl criterion on the invariance of the essential spectrum under compact perturbations

On Non-selfadjoint Operators with Finite Discrete Spectrum

67

and from+ [7, Theorem 2.1, p. 373], that one has the disjoint union σ (HV ) = σess (HV ) σdisc (HV ), where σess (HV ) = [0, 4]. Furthermore, the only possible limit points of σdisc (HV ) are contained in σess (HV ). Under additional regularity conditions on V , the next theorems give more information about the distribution of σdisc (HV ) near σess (HV ), and that of Ep (HV ) inside σess (HV ).

 Theorem 2.1 Let V ∈ S∞ 2 (Z, H ) ∩ AR (A0 ) for some R > 0. Then: 1. The only possible limit points of σdisc (HV ) belong to the spectral thresholds {0, 4}. 2. There exists a discrete subset D ⊂ (0, 4) whose only possible limit points belong to {0, 4} and for which the following holds: given any relatively compact interval 0 , 0 ⊂ (0, 4)\D, there exist δ0 > 0 such that for any vectors ϕ and ψ analytic w.r.t. A0 , sup

|ϕ, (z − HV )−1 ψ | < ∞,

sup

|ϕ, (z − HV )−1 ψ | < ∞.

z∈0 +i(0,δ0 ) z∈0 −i(0,δ0 )

Remark 2.2 • For any subset  such that  ⊂ (0, 4),  ∩ D is finite. • If HV is selfadjoint, D coincides with the set of its embedded eigenvalues. If not, we expect that the embedded eigenvalues belong to D. We refer to Sect. 3.1 for a proof of Theorem 2.1. Remark 2.3 Examples of compact operators acting on

2 (Z, H

):

(a) any operator of the form K1 ⊗ K2 where K1 ∈ S∞ ( and K2 ∈ S∞ (H ). (b) any operator satisfying the conditions discussed in Remark 4.1. 2 (Z))

2.2 Resonances The reader may have noted that Theorem 2.1 does not give any information about the distribution of Ep (HV ) around the spectral thresholds {0, 4}. This is a nontrivial problem; under Assumptions 2.1 and 2.2 below, we give an answer by means of resonances techniques and characteristic values methods (see e.g. [2]). First, we fix some notations and definitions. W ∈ B( 2 (Z, H )) can be represented as a matrix with entries

An operator  w(n, m) (n,m)∈Z2 ⊂ B(H ), where for any φ = (φ(n))n∈Z ∈ 2 (Z, H ), (W φ)(n) =

 m∈Z

w(n, m)φ(m),

n ∈ Z.

(2.6)

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Remark 2.4 (a) If H = Cd for d ≥ 1, then w(n, m) ∈ Md (C). If d = 1 (i.e. H = C), w(n, m) (n,m)∈Z2 coincides with the matrix representation of W in the canonical orthonormal basis of 2 (Z). (b) Let (ej )j ≥0 be an orthonormal basis of H . For each (n, m) ∈ Z2 , the operator w(n, m) admits also a matrix representation in this basis (with complex entries), which will be denoted:

 wj k (n, m) j,k≥0 ,

wj k (n, m) := ej , w(n, m)ek H .

It is also convenient to decompose W = (n, m) ∈ Z2 fixed, Wn,m is defined by:

& (n,m)∈Z2

(Wn,m φ)(k) = w(n, m)φ(m)δkn ,

(2.7)

Wn,m where for each

k ∈ Z,

for any φ ∈ 2 (Z, H ). Abusing notations, we also represent the operators W and (Wn,m )(n,m)∈Z2 as operators acting on 2 (Z) ⊗ H : Wn,m = |δn δm | ⊗ w(n, m),  W = |δn δm | ⊗ w(n, m).

(2.8) (2.9)

(n,m)∈Z2

We refer to Remark 4.1 for sufficient conditions concerning the compactness of the operator W . Let us introduce our main assumptions on the (non-selfadjoint) perturbation V ∈ B( 2 (Z, H )): Assumption 2.1 There exist (%1 , %2 , W ) ∈ B(H ) × B(H ) × B( 2 (Z, H )) and γ > 0 such that: V = (I ⊗ %1 )W (I ⊗ %2 ) and sup (n,m)∈Z2

w(n, m)H eγ (|n|+|m|) < ∞,

where (w(n, m)) denotes the matrix representation of W according to (2.6). Assumption 2.2 There exist (%1 , %2 , W ) ∈ B(H ) × S∞ (H ) × B( 2 (Z, H )) such that: V = (I ⊗ %1 )W (I ⊗ %2 ) and %2 is of finite rank. We note that if there exist (%1 , %2 , W ) ∈ B(H ) × B(H ) × B( 2 (Z, H )) such that V = (I ⊗ %1 )W (I ⊗ %2 ), then making use of the representations (2.6), (2.8) and (2.9), we can rewrite V as: V =

 (n,m)∈Z2

Vn,m

On Non-selfadjoint Operators with Finite Discrete Spectrum

69

where Vn,m = |δn δm | ⊗ v(n, m) and v(n, m) = %1 w(n, m)%2 for (n, m) ∈ Z2 . In other words, (v(n, m))(n,m)∈Z2 = (%1 w(n, m)%2 )(n,m)∈Z2 provides a matrix representation of V in the sense of (2.6). Remark 2.5 (a) If dim(H ) < ∞, the finite rank condition set on %2 in Assumption 2.2 holds trivially. In particular, if %j = I , j = 1, 2, then V = W . (b) If dim(H ) = ∞ and Assumption 2.2 holds, the compact operator %2 plays the role of a regularization in the component H of the space 2 (Z) ⊗ H , which is crucial to define the resonances. (c) A perturbation V which satisfies Assumptions 2.1 and 2.2 is compact (see Lemma 3.2). Under Assumptions 2.1 and 2.2, the resonances of the operator HV near the spectral thresholds {0, 4} can be defined as poles of the meromorphic extension of the resolvent (HV − z)−1 in some Banach weighted spaces. Moreover, they are parametrized respectively near 0 and 4 by z0 (λ) := λ2

and

z4 (λ) := 4 − λ2 ,

(2.10)

and are defined in some two-sheets Riemann surfaces. In particular, the discrete and the embedded eigenvalues of HV near {0, 4} are resonances. One refers to Definitions 3.1, 3.2 and Sect. 3.2 for more details. Note that these definitions are actually established in a slightly more general framework. Remark 2.6 (a) The resonances can be defined by meromorphic extension of the weighted resolvent (see Sect. 2.2) or by complex dilation (see Sect. 2.1). When the resonances are discrete eigenvalues, these two definitions coincide. If not, this is still an open issue. We refer to e.g. [8], for the case of perturbations of the Schrödinger operator in a semi-classical regime. (b) Under Assumptions 2.1 and 2.2, the resonances are defined in pointed neighborhoods of the thresholds {0, 4}. Our method does not allow us to conclude about the nature of the thresholds as resonances. (c) The finite rank restriction set on %2 may look restrictive: it is actually enough for our purpose, i.e. to exhibit finite discrete spectrum. A study of the accumulation of eigenvalues (or resonances) at the thresholds {0, 4} without this restriction would be welcome. Our second main result is the following: Theorem 2.2 Let Assumptions 2.1 and 2.2 hold. Let ε0 > 0 be as defined in Proposition 3.6. Then, for some 0 ∈ (0, ε0 ], HV has no resonance in the subsets {zμ (λ); λ ∈ D∗ (0)}, μ ∈ {0, 4}. 0

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Fig. 1 Resonances zμ (λ) in variable λ, near 0

According to Theorem 2.2, the operator HV has no resonance in a punctured neighborhood of 0 and 4 in the two-sheets Riemann surfaces where they are defined, see Fig. 1 below for a graphic illustration. The proof of Theorem 2.2 is postponed to Sect. 3.2. We will also show that: Theorem 2.3 Let Assumptions 2.1 and 2.2 hold. Then: • σess (HV ) = [0, 4] and σdisc (HV ) is finite. • There exists a finite set D ⊂ (0, 4) for which the following holds: given any relatively compact interval 0 , 0 ⊂ (0, 4) \ D , there exists δ0 > 0 such that for any vectors ϕ and ψ in 2γ (Z; H ) " 2γ (Z) ⊗ H , sup

|ϕ, (z − HV )−1 ψ | < ∞

sup

|ϕ, (z − HV )−1 ψ | < ∞.

z∈0 +i(0,δ0 ) z∈0 −i(0,δ0 )

If in addition, the perturbation is selfadjoint, we get: Corollary 2.1 Assume that V is selfadjoint and satisfies Assumptions 2.1 and 2.2. Then: • σess (HV ) = [0, 4] and σdisc (HV ) is finite. • There is at most a finite number of eigenvalues embedded in [0, 4]. Each eigenvalue embedded in (0, 4) has finite multiplicity. • The singular continuous spectrum σsc (HV ) = ∅ and the following LAP holds: given any relatively compact interval 0 , 0 ⊂ (0, 4) \ Ep (HV ), there exists

On Non-selfadjoint Operators with Finite Discrete Spectrum

δ0 > 0 such that for any vectors ϕ and ψ in

2 (Z; H γ

71

)"

sup

|ϕ, (z − HV )−1 ψ | < ∞

sup

|ϕ, (z − HV )−1 ψ | < ∞.

z∈0 +i(0,δ0 ) z∈0 −i(0,δ0 )

2 (Z) ⊗ H γ

,

Remark 2.7 Setting H = C in Theorems 2.1, 2.2, 2.3 and Corollary 2.1, we recover Theorems 2.2, 2.3, 2.4, 2.5 and Corollary 2.1 of [3].

3 Proofs of the Main Results 3.1 Complex Scaling In this section, we take advantage of the complex scaling techniques developed in [3] to study σ (HV ) for compact perturbations V ∈ A(A0 ). Since the operators HV * and H V are unitarily equivalent, we focus our attention on the analysis on the latter. #V ∈ AR (# Note that HV ∈ AR (A0 ) for some R > 0 if and only if H A0 ).

3.1.1

Complex Scaling for H0

)0 . For We describe the complex scaling process for the unperturbed operator H complementary references, see e.g. [9, 11] and [12]. First, we observe that f = T ◦ cos, where T : C → C, T (z) := 2(1 − z). The map T is bijective and maps [−1, 1] onto [0, 4]. The points T (−1) = 4 and )0 . T (1) = 0 are the thresholds of H0 and H We also have that for any θ ∈ R, #

#

eiθ A0 = eiθ A0 ⊗ I. It follows from [3, Section 3.1], that for any θ ∈ R, #

#

#

#

#0 )) ⊗ I, )0 e−iθ A0 = eiθ A0 L #0 e−iθ A0 ⊗ I = (Gθ (L eiθ A0 H

(3.1)

where for θ ∈ R, the function Gθ is defined on [0, 4] by Gθ := T ◦ Fθ ◦ T −1 with Fθ (λ) :=

λ − th(2θ ) , 1 − λth(2θ )

λ ∈ [−1, 1].

(3.2)

  Remark 3.1 In (3.2), the denominator does not vanish since th(2θ )λ < 1 for λ ∈ [−1, 1].

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In order to perform our complex scaling argument, we need to precise the meaning of formula (3.1) and (3.2), for possibly complex values of the deformation parameter θ . To this end, we observe that: Proposition 3.1 Let D := D1 (0) denote the open unit disk of the complex plane C. Then: (a) For any λ ∈ [−1, 1], the map θ → Fθ (λ) is holomorphic in D π4 (0). (b) For θ ∈ C such that |θ | < π8 , the map λ → Fθ (λ) is a homographic transformation with Fθ−1 = F−θ . In particular, for θ ∈ R, Fθ (D) = D and Fθ ([−1, 1]) = [−1, 1]. (c) For θ ∈ C such that 0 < |θ | < π8 , the unique fixed points of Fθ are ±1. (d) For θ1 , θ2 ∈ C with |θ1 |, |θ2 | < π8 , we have that: Fθ1 ◦ Fθ2 = Fθ1 +θ2 . From (3.1) and Proposition 3.1, it follows that: Proposition 3.2 The bounded operator valued-function

 # ) −iθ # # # −iθ A #0 A0 θ → eiθ A0 H = eiθ A0 L ⊗ I ∈ B L2 (T, H ) , 0e 0e 

admits an analytic extension from − π8 , π8 to D π8 (0), with extension given for #0 ) ⊗ I , where Gθ (L #0 ) is the θ ∈ D π8 (0) by the operator-valued map θ → Gθ (L multiplication operator by the function Gθ ◦ f = T ◦ Fθ ◦ cos = Gθ ◦ T ◦ cos. In )0 (θ ). the sequel, this extension is denoted H #0 ∈ A π (# Paraphrasing Proposition 3.2, we have that H A0 ) and for any θ ∈ 8 D π8 (0), )0 (θ ) = Gθ (L #0 ) ⊗ I. H Combining the continuous functional calculus, Proposition 3.2 and unitary equivalence properties, we get for θ ∈ D π8 (0), )0 )) = Gθ (σ (H )0 )) = Gθ (σ (H0 )) = Gθ ([0, 4]). )0 (θ )) = σ (Gθ (H σ (H

 )0 (θ ) is a smooth parametrized curve described by Thus, for θ ∈ D π8 (0), σ H   

)0 (θ ) = T ◦ Fθ (λ) = Gθ ◦ T (λ) : λ ∈ [−1, 1] . σ H Quoting [3, Proposition 3.3], the following properties hold: )0 (θ ))θ∈D π (0) be the family of bounded operators defined in Proposition 3.3 Let (H 8 Proposition 3.2. Then, it holds:

 )0 (θ1 ) = (a) For θ1 , θ2 ∈ D π8 (0) such that Im(θ1 ) = Im(θ2 ), one has σ H



 )0 (θ2 ) . That is, the curve σ H )0 (θ ) does not depend on the choice of Re(θ ). σ H

On Non-selfadjoint Operators with Finite Discrete Spectrum

73

* Fig. 2 Spectral structure of the operator H V (θ) for θ ∈ D π8 (0) and Im(θ) ≥ 0

 )0 (θ ) lies in C± . (b) For ± Im(θ ) > 0, θ ∈ D π8 (0), the curve σ H

 )0 (θ ) is an arc of a circle (c) Let θ ∈ D π8 (0). If Im(θ ) = 0, the curve σ H

 )0 (θ ) = [0, 4]. containing the points 0 and 4. If Im(θ ) = 0, σ H We refer to Fig. 2 above for a graphic illustration.

3.1.2

Complex Scaling for HV

# and the corresponding Now, we proceed to the complex scaling of the perturbation V # * ) # # perturbed operator HV := H0 + V . For V ∈ AR (A0 ), R > 0, and for all θ ∈ DR (0), A0 V A0 . # e−iθ # #(θ ) := eiθ # we write: V Quoting [11, Lemma 5, Section XIII.5], we recall that:

 # ∈ S∞ L2 (T, H ) ∩ AR (# Lemma 3.1 Let V A0 ), for some R > 0. Then, for all # θ ∈ DR (0), V (θ ) is compact. # # Following Proposition  3.2, we can define for V ∈ AR (A0 ) and all θ ∈ D2R  (0)

with 2R  := min R, π8 , )0 (θ ) + V #(θ ). * H V (θ ) := H * By construction, (H V (θ ))θ∈D2R  (0) is a holomorphic family of bounded operators * * A0 ) and the map θ → H on D2R  (0). Also, the operator H V belongs to A2R  (# V (θ ) #0 * −iθ A #0 iθ A actually coincides with the holomorphic extension of the map θ → e HV e   from the interval (−2R , 2R ) to D2R  (0). In addition, we have that: # ∈ A R (A #0 ), R > 0. Then, for any θ  ∈ R such that |θ  | < R  , Proposition 3.4 Let V we have #0 *  −iθ A #0  iθ A * , H HV (θ ) e V (θ + θ ) = e

for all θ ∈ DR  (0).

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Proof Fix θ  ∈ R with |θ  | < R  and observe that DR  (0) ⊂ DR (0) ∩ DR (−θ  ). The maps #

#

−iθ A0 * θ −→ eiθ A0 H V (θ )e

and

 * θ −→ H V (θ + θ ),

are bounded operator-valued and holomorphic on DR  (0). Moreover, they coincide on R ∩ DR  (0) = (−R  , R  ). Hence, they also coincide on DR  (0). The next result is the key to the proof of Theorem 2.1. One refers to Fig. 2 for a graphic illustration. # ∈ S∞ (L2 (T)) ∩ AR (A #0 ), and let R  > 0 such Proposition 3.5 Let R > 0 and V that 2R  = min(R, π8 ). Then, for any θ ∈ DR  (0), we have * (a) σ (H V (θ )) depends only on Im(θ ). * )0 (θ )) = σ (H )0 (θ )) and (b) It holds: σess (H V (θ )) = σess (H * * σ (H V (θ )) = σdisc (H V (θ ))

,

)0 (θ )), σess (H

* )0 (θ )). where the possible limit points of σdisc (H V (θ )) belong to σess (H Proof Statement (a) is a consequence of the unitary equivalence established in Proposition 3.4. Statement (b) follows from Lemma 3.1, the Weyl criterion on the invariance of the essential spectrum and [7, Theorem 2.1, p. 373].

3.1.3

Proof of Theorem 2.1

The proof of Statements 1 and 2 follows now from a straightforward adaptation to our setting of the contents of [3, Section 3.3] and [3, Section 3.4] respectively.

3.2 Resonances In this section, we follow the notations introduced in Sect. √ 2. We also adopt the following principal determination of the complex square root:     · : C\(−∞, 0] −→ z ∈ C : Im(z) ≥ 0 , and we set C+ := z ∈ C : Im(z) > 0 . Let J be the selfadjoint unitary operator defined on 2 (Z, H ) by J := J ⊗ I

with

(J ϕ)(n) := (−1)|n| ϕ(n).

The operator J commutes with any multiplication operator acting on V ∈ B( 2 (Z, H )), we write: VJ := JV J−1 .

(3.3) 2 (Z).

Given

(3.4)

On Non-selfadjoint Operators with Finite Discrete Spectrum

75

The next discussion will be performed on the basis of Assumption 2.2 and a generalization of Assumption 2.1: Assumption 3.1 There exist (1 , 2 ) ∈ B( 2 (Z)) × B( 2 (Z)), (%1 , %2 , W ) ∈ B(H ) × B(H ) × B( 2 (Z, H )) and γ > 0 such that: V = (1 ⊗ %1 )W (2 ⊗ %2 ) and • sup(n,m)∈Z2 w(n, m)H eγ (|n|+|m|) < ∞, • j , j = 1, 2, commute with the operators W−γ and J , where (w(n, m)) denotes the matrix representation of W according to (2.6). Under Assumption 3.1, J and J−1 commute with the operators j W−γ ⊗ %j , j = 1, 2.

3.2.1

Definition of the Resonances

We define the resonances of the operator HV near the spectral thresholds {0, 4}. Notice that there is a simple way to reduce the analysis near the threshold 4 to another one near the threshold 0 (see (3.7)). But some preliminaries are required. Recall that W±γ denote the multiplication γ operators on 2 (Z) by the functions; n −→ e± 2 |n| . Our first result deals with the compactness of the perturbation V :

 Lemma 3.2 Let Assumptions 2.2 and 3.1 hold. There exists W ∈ B 2 (Z, H ) such that W = (W−γ ⊗ I )W (W−γ ⊗ I ). In particular, one has V = (1 W−γ ⊗ %1 )W (W−γ 2 ⊗ %2 ). Moreover, V is compact. γ

Proof Let W−γ denote the multiplication operator by the function n −→ e− 2 |n| on 2 (Z, H ). Thus W −γ = W−γ ⊗ I and one has W = (W−γ ⊗ I )W (W−γ ⊗ I )



γ γ with W := e 2 |n| w(n, m)e 2 |m| (n,m)∈Z2 ,

γ γ & i.e. (W φ)(n) = m∈Z e 2 |n| w(n, m)e 2 |m| φ(m) for any φ ∈ 2 (Z, H ). Therefore, to get the first claim of the lemma, it suffices to show that W is a bounded operator as follows. Similarly to (2.9), W can be represented canonically by W = γ γ & 2 |n| w(n, m)e 2 |m| ). Then, one obtains from Assumption 3.1 that n,m |δn δm | ⊗ (e

W  ≤

 n,m

γ

γ

|δn δm |e 2 |n| w(n, m)e 2 |m| H 



γ

e− 2 |n|

2

< ∞,

n

and the claim follows. The compactness of V follows from that of W−γ 2 ⊗ %2 , since the operator W−γ ⊗ %2 is compact. As an immediate consequence of Lemma 3.2 , we deduce that:

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 Corollary 3.1 Let Assumptions 2.2 and 3.1 hold. There exists WJ ∈ B 2 (Z, H ) such that VJ = (1 W−γ ⊗ %1 )WJ (W−γ 2 ⊗ %2 ). In particular, VJ is compact. For further use, one recalls the following result established in [3]: Lemma 3.3 ([3, Lemma 4.1]) Set z(λ) := λ2 . Then, there exists 0 < ε0 ≤

 enough such that the operator-valued function, with values in S∞ 2 (Z) ,

γ 8

small

−1 λ → W−γ L0 − z(λ) W−γ admits a holomorphic extension from Dε∗0 (0) ∩ C+ to Dε∗0 (0). The next result is crucial for our analysis: Lemma 3.4 Let Assumptions 2.2 and 3.1 hold. Set z(λ) := λ2 . There exists 0 < ε0 ≤ γ8 small enough such that both,

 • the operator-valued function, with values in S∞ 2 (Z, H )

−1 λ → (W−γ ⊗ %2 ) H0 − z(λ) (W−γ ⊗ I ), • and the operator-valued function, with values in B



2 (Z, H

)



−1 λ → (W−γ ⊗ I ) H0 − z(λ) (W−γ ⊗ I ), admit a holomorphic extension from Dε∗0 (0) ∩ C+ to Dε∗0 (0). Proof For λ ∈ Dε∗0 (0) ∩ C+ , we have that:

−1 −1

(W−γ ⊗ %2 ) H0 − z(λ) (W−γ ⊗ I ) = W−γ L0 − z(λ) W−γ ⊗ %2



−1 −1 (W−γ ⊗ I ) H0 − z(λ) (W−γ ⊗ I ) = W−γ L0 − z(λ) W−γ ⊗ I. Mind that the operator %2 is compact. The conclusion follows from Lemma 3.3. We draw a first consequence: 2 Lemma 3.5 Let Assumptions and 3.1

2.2  hold. Set z(λ) := λ . The operator-valued 2 functions, with values in S∞ (Z, H ) ,

−1

 • λ −→ TV z(λ) := (1 ⊗ %1 )W (W−γ 2 ⊗ %2 ) H0 − z(λ) (W−γ ⊗ I ),



−1 • λ −→ T−VJ z(λ) := −(1 ⊗ %1 )WJ (W−γ 2 ⊗ %2 ) H0 − z(λ) (W−γ ⊗ I ),

admit holomorphic extensions from Dε∗0 (0) ∩ C+ to Dε∗0 (0). Proof For λ ∈ Dε∗0 (0) ∩ C+ , we write



−1 TV z(λ) = (1 ⊗ %1 )W (2 ⊗ %2 )(W−γ ⊗ I ) H0 − z(λ) (W−γ ⊗ I ),

On Non-selfadjoint Operators with Finite Discrete Spectrum

77



−1 T−VJ z(λ) = −(1 ⊗ %1 )WJ (2 ⊗ %2 )(W−γ ⊗ I ) H0 − z(λ) (W−γ ⊗ I ).



 We conclude for the map TV z(·) (resp. T−VJ z(·) ), by combining Lemma 3.2 (resp. Corollary 3.1) with the second statement of Lemma 3.4.

 Let V ∈ {V , −VJ }. Due to the resolvent identity (HV −z)−1 I + V(H0 − z)−1 = (H0 − z)−1 , we have that (W−γ ⊗ I )(HV − z)−1 (W−γ ⊗ I ) = (W−γ ⊗ I )(H0 − z)−1 (W−γ ⊗ I ) −1  · I + (Wγ ⊗ I )V(H0 − z)−1 (W−γ ⊗ I ) . If Assumptions 2.2 and 3.1 hold, we can combine Lemma 3.2 (resp. Corollary

3.1) with Lemma 3.4 to deduce that the operator-valued function λ −→ (Wγ ⊗I )V H0 −

 −1 z(λ) (W−γ ⊗ I ) is holomorphic in Dε∗0 (0) with values in S∞ 2 (Z, H ) . By the analytic Fredholm extension theorem, the operator-valued function

−1 −1

λ −→ I + (Wγ ⊗ I )V H0 − z(λ) (W−γ ⊗ I ) admits a meromorphic extension from Dε∗0 (0) ∩ C+ to Dε∗0 (0). Since the map λ → (W−γ ⊗ I )(H0 − z(λ))−1 (W−γ ⊗ I ) is also holomorphic in Dε∗0 (0) (see again Lemma 3.4), we have just proved that the map λ → (W−γ ⊗ I )(HV − z(λ))−1 (W−γ ⊗ I ) extends meromorphically from Dε∗0 (0) ∩ C+ to Dε∗0 (0). Keeping in mind that 2 ±γ (Z, H

)"

2 ±γ (Z) ⊗ H

,

(3.5)

we summarize it as: Proposition 3.6 Let Assumptions 2.2 and 3.1 hold. Set z(λ) := λ2 and let V ∈ {V , −VJ }. There exists 0 < ε0 ≤ γ8 small enough such that the operator-valued   function, with values in B 2γ (Z, H ), 2−γ (Z, H ) ,

−1 λ −→ HV − z(λ) admits a meromorphic extension from Dε∗0 (0) ∩ C+ to Dε∗0 (0). This extension will be denoted RV (z). One can now define the resonances of the operator HV near the spectral thresholds {0, 4}. Notice that in the next definitions, the quantity I ndC (·) is defined in the appendix by (4.4). Definition 3.1 The resonances of the operator HV near 0 are the points z(λ), which are the poles of the meromorphic extension RV (z) as introduced in Proposition 3.6.

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For λ ∈ Dε∗0 (0), set z0 (λ) := λ2 . Given λ1 ∈ Dε∗0 (0), let z1 = z0 (λ1 ). By Proposition 3.7, z1 is a resonance of HV near 0 if and only if λ1 is a characteristic value of I + TV z0 (·) . The multiplicity of the resonance z1 is defined as the multiplicity of the characteristic value λ1 , namely



 mult(z1 ) := I ndC I + TV z0 (·) ,

(3.6)

where C is a positively oriented circle centered at λ1 and chosen sufficiently small so that λ1 is the only characteristic value enclosed in C. The resonances of the operator HV near 4 are defined via a reduction, which is based on a simple relation between the two thresholds {0, 4}. Indeed, since J L0 J −1 = −L0 + 4, then one has JH0 J−1 = −H0 + 4, which implies that J(HV − z)J−1 = −H0 + VJ + 4 − z. So,

−1 J(W−γ ⊗I )(HV −z)−1 (W−γ ⊗I )J−1 = −(W−γ ⊗I ) H−VJ −(4−z) (W−γ ⊗I ). (3.7) Let us set u := 4 − z. Since u is close to 0 for z near 4, we can use identity (3.7) to define the resonances of the operator HV near 4 as the points z = 4 − u where u is a resonance of the operator H−VJ near 0. Precisely, one has: Definition 3.2 The resonances of the operator HV near 4 are the points z(λ) = 4 − u(λ) where u(λ) are the poles of the meromorphic extension R−VJ (u) as introduced in Proposition 3.6. For λ ∈ Dε∗0 (0), set z4 (λ) := 4 − λ2 . Given λ1 ∈ Dε∗0 (0), let z1 = z4 (λ1 ). By Proposition 3.8, z1 is a resonance of HV near 4 if and only if λ1 is a characteristic

 value of I + T−VJ (4 − z4 ·) . The multiplicity of the resonance z1 is defined as the multiplicity of the characteristic value λ1 , namely



 mult(z1 ) := I ndC I + T−VJ 4 − z4 (·) ,

(3.8)

where C is a positively oriented circle centered at λ1 and chosen sufficiently small so that λ1 is the only characteristic value enclosed in C. Remark 3.2 The resonances zμ (λ) near the spectral thresholds μ ∈ {0, 4} are defined respectively in some two-sheets Riemann surfaces Mμ . The discrete eigenvalues of the operator HV near μ are resonances. Furthermore, the algebraic multiplicity (1.4) of a discrete eigenvalue coincides with its multiplicity as a resonance near μ respectively defined by (3.6) and (3.8). This can be shown for instance as in [3].

On Non-selfadjoint Operators with Finite Discrete Spectrum

3.2.2

79

Proof of Theorem 2.2

As introduced previously, we first reformulate our characterization of the resonances near 0 and 4 in terms of characteristic values. We presume that Assumptions 2.2 and 3.1 hold throughout this section. Proposition 3.7 For λ1 ∈ Dε∗0 (0), the following assertions are equivalent: (a) (b) (c) (d)

z0 (λ1 ) = λ21 ∈ M0 is a resonance of HV , z1 = z0 (λ1 ) is a pole of RV (z),  −1 is an eigenvalue of TV z0 (λ1 ) ,

 λ1 is a characteristic value of I + TV z0 (·) .

Thanks to (3.6), the multiplicity of the resonance z0 (λ1 ) coincides with that of the characteristic value λ1 . Proof The equivalence between (a) and (b) is just Definition 3.1. The equivalence between (b) and (c) is a consequence of the identity

 I + (1 ⊗ %1 )W (W−γ 2 ⊗ %2 )(H0 − z)−1 (W−γ ⊗ I )

 · I − (1 ⊗ %1 )W (W−γ 2 ⊗ %2 )(HV − z)−1 (W−γ ⊗ I ) = I, which follows itself from the resolvent equation. Finally, the equivalence between (c) and (d) follows from Definition 4.2 and (4.4). Similarly, one can prove that: Proposition 3.8 For λ1 ∈ Dε∗0 (0), the following assertions are equivalent: (a) (b) (c) (d)

z4 (λ1 ) = 4 − λ21 ∈ M4 is a resonance of HV , 4 − z4 (λ1 ) = u(λ1 ) := λ21 is a pole of R−VJ (u), −1 is an eigenvalue of T−VJ (4 − z4 (λ1 )),  λ1 is a characteristic value of I + T−VJ (4 − z4 (·) .

Thanks to (3.8), the multiplicity of the resonance z4 (λ1 ) coincides with that of the characteristic value λ1 .

 The next goal consists in splitting the sandwiched resolvent TV z(λ) , z(λ) = λ2 , for V ∈ {V , −VJ }, into a sum of a singular part at λ = 0 and a holomorphic part in the full open disk Dε0 (0). By Assumption 3.1, one has

−1 −1

(W−γ 2 ⊗ %2 ) H0 − z(λ) (W−γ ⊗ I ) = 2 W−γ L0 − z(λ) W−γ ⊗ %2 . (3.9)

From [3], we know that the summation kernel of the operator W−γ L0 −

 γ −1 γ z(λ) W−γ is given by e− 2 |n| R0 z(λ), n − m e− 2 |m| , with

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λ 2

  i|n−m|2 arcsin λ2 i e − 1 i = √ + √ λ 4 − λ2 λ 4 − λ2

 ie R0 z(λ), n − m = √ λ 4 − λ2 i = + α(λ) + β(λ), 2λ i|n−m|2 arcsin

(3.10)

where the functions α and β are defined by  α(λ) := i

1 1 − √ 2 2λ λ 4−λ

 and

  λ i ei|n−m|2 arcsin 2 − 1 β(λ) := . √ λ 4 − λ2

Note that α and β can be extended to holomorphic functions in Dε0 (0). By (3.10), for λ ∈ Dε∗0 (0) 

γ γ   ie− 2 |n| e− 2 |m| x(m)



−1 W−γ L0 − z(λ) W−γ x (n) = + A(λ) x (n), 2λ m∈Z (3.11) where the operator A(λ) is defined by

 

 γ γ γ γ A(λ) x (n) := e− 2 |n| α(λ)e− 2 |m| x(m) + e− 2 |n| β(λ)e− 2 |m| x(m). m∈Z

m∈Z γ

Introduce the rank-one operator & : 2 (Z) −→ C by & := e− 2 |·| | so that its

 γ adjoint &∗ : C −→ 2 (Z) be given by &∗ (η) (n) := ηe− 2 |n| . By putting this together with (3.11) one gets 

  

−1 i &∗ & x (n)

+ A(λ) x (n). W−γ L0 − z(λ) W−γ x (n) = 2λ

(3.12)

For simplification, we set W :=

 W −WJ

if

V = V,

if

V = −VJ .

Thus, we deduce from (3.12) and the above computations that: Proposition 3.9 Let λ ∈ Dε∗0 (0) and V ∈ {V , −VJ }. Then,

 i (1 ⊗ %1 )W(2 ⊗ %2 )M∗ M + (1 ⊗ %1 )W(2 ⊗ %2 )(A(λ) ⊗ I ), TV z(λ) = 2λ where M := & ⊗ I and z(λ) = λ2 . Furthermore, λ →

(2 ⊗ %2 )(A(λ) ⊗ I ) is holomorphic in the open disk Dε0 (0) with values in S∞ 2 (Z, H ) .

On Non-selfadjoint Operators with Finite Discrete Spectrum

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The third and last step consists in applying Proposition 4.3 as follows: Let μ ∈ {0, 4}. Then, from Propositions 3.7 and 3.8 together with Proposition 3.9, it follows that zμ (λ) is a resonance of HV near μ if and only if λ is a characteristic value of the corresponding operator

 i I + TV z(λ) = I + (1 ⊗ %1 )W(2 ⊗ %2 )M∗ M + (1 ⊗ %1 )W(2 ⊗ %2 )(A(λ) ⊗ I ), 2λ

with z(λ) = λ2 . Mind that (I ⊗ %2 )M∗ M = &∗ & ⊗ %2 is finite rank. Since (1 ⊗

%1 )W(2 ⊗%2 )(A(λ)⊗I ) is holomorphic in Dε0 (0) with values in S∞ 2 (Z, H ) while i(1 ⊗ %1 )W(2 ⊗ %2 )M∗ M is finite-rank, then Theorem 2.2 follows

 by applying Proposition 4.3 with D = Dε0 (0), Z = {0}, and F = I + TV z(·) . We refer to [3, Section 4.4] for complementary details.

3.2.3

Proof of Theorem 2.3

This proof is a straightforward adaptation of the contents of [3, Section 4.5] to our setting. The first statement follows from Theorem 2.2 and the first statement of Theorem 2.1. Keeping in mind Corollary 4.2, the second part follows from Proposition 3.6, Theorem 2.2 and the second statement of Theorem 2.1.

4 Appendix 4.1 The A(A0 ) Class One refers to [10, Chapter III] for general considerations on bounded operatorvalued analytic maps. The next lemma provides examples of analytic vectors for A0 . Lemma 4.1 For any n ∈ Z and any vector u ∈ H , δn ⊗ u is an analytic vector for A0 . Proof For any θ ∈ D 1 (0), it follows from [3, Lemma 6.2] that the power series 2

∞ ∞ ∞       |θ |k  |θ |k  |θ |k  Ak (δn ⊗ u) = (Ak ⊗ I )δn ⊗ u = Ak δn u 0 0 0 k! k! k! k=0

k=0

converges. This concludes the proof.

k=0

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In the continuity of Remark 2.1 (i), one has the following result: Proposition 4.1 Let W ∈ B( 2 (Z, H )), (n, m) ∈ Z2 and Wn,m be the operator defined by (2.8). Then, Wn,m ∈ A 1 (A0 ). The holomorphic extension map is given 2 by ¯

D 1 (0)  θ −→ |eiθA0 δn ei θ A0 δm | ⊗ w(n, m). 2

In particular, any finite linear combination of such Wn,m belongs to A 1 (A0 ). 2

Proof This is an immediate consequence of [3, Lemma 6.2]. Now, we aim at showing that if a bounded operator V satisfies Assumption 2.1, then it belongs to ARγ (A0 ) for some Rγ > 0 (see Corollary 4.1). Referring to [3, Section 3.1] for the details, we recall that for any ψ ∈ L2 (T),  # (eiθ A0 ψ)(α) = ψ(ϕθ (α)) J (ϕθ )(α) , α ∈ T where



• (ϕθ )θ∈R is the flow solution of the system:

(4.1)

 ∂θ ϕθ (α) = 2 sin ϕθ (α) ,

ϕ0 (α) = idT (α) = α for each α ∈ T, • J (ϕθ )(α) denotes the Jacobian of the transformation α → ϕθ (α).

Explicitly, one has:  ϕθ (α) = ± arccos

−th(2θ ) + cos α 1 − th(2θ ) cos α



for ±α ∈ T. By using (4.1) and the fact that ϕθ1 ◦ ϕθ2 = ϕθ1 +θ2 for all (θ1 , θ2 ) ∈ R2 , one gets for all θ ∈ R #

#

#0 e−iθ A0 ψ)(α) = f (ϕθ (α))ψ(α). (eiθ A0 L Paraphrasing [3, Lemma 6.3], we have that: Lemma 4.2 There exists 0 < R0 < 1/2 such that for any 0 < R < R0 , sup n∈Z

  −|n| sup (θ,α)∈DR (0)×T | Im(ϕθ (α))| sup eiθA0 δn e

! < ∞.

(4.2)

θ∈DR (0)

Proposition 4.2 Consider W ∈ B( 2 (Z, H )) with its representation (2.9). Assume there exists γ > 0 such that sup (n,m)∈Z2

w(n, m)H eγ (|n|+|m|) < ∞.

On Non-selfadjoint Operators with Finite Discrete Spectrum

83

Then, there exists Rγ > 0 such that W belongs to ARγ (A0 ). The extension map is given by DRγ (0)  θ −→ eiθA0 W e−iθA0 =



¯

|eiθA0 δn ei θ A0 δm | ⊗ w(n, m).

n,m

Proof Mind that W =



Wn,m =

n,m



|δn δm | ⊗ w(n, m).

n,m

Now as in [3, Proposition 6.6], with the help of Lemma 4.2 one can pick 0 < Rγ <  iθA  γ 1  0 δ  ≤ Ce 2 |n| , for some constant C > 0 n 2 small enough such that supθ∈DRγ (0) e (independent of n). It follows that 

 iθA  e 0 Wn,m e−iθA0 

sup

n,m θ∈DRγ (0)

=



sup

  iθA (e 0 ⊗ I )(|δn δm | ⊗ w(n, m))(e−iθA0 ⊗ I )

n,m θ∈DRγ (0)

=



sup

 iθA  i θ¯ A  e 0 δn e 0 δm w(n, m)

n,m θ∈DRγ (0)





γ

e 2 (|n|+|m|) w(n, m) < ∞.

n,m

By arguing as in [3, Proposition 6.6], one gets the claim. Corollary 4.1 Let Assumption 2.1 holds for V ∈ B( 2 (Z, H )). Then, V belongs to ARγ (A0 ) where Rγ > 0 has been defined in Proposition 4.2. The extension map is given by DRγ (0)  θ −→ eiθA0 V e−iθA0 =



¯

|eiθA0 δn ei θ A0 δm | ⊗ %1 w(n, m)%2 .

n,m

Proof Mind that V = (I ⊗ %1 )W (I ⊗ %2 ) =



|δn δm | ⊗ %1 w(n, m)%2 .

n,m

There are at least two ways to conclude. Given % ∈ B(H ), the operator I ⊗ % commute with the unitary group (e−iθA0 )θ∈R . It follows that I ⊗% ∈ AR (A0 ) for any R > 0, with constant extension map:

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DR (0)  θ −→ eiθA0 (I ⊗ %)e−iθA0 = I ⊗ %. Due to Proposition 4.2, W ∈ ARγ (A0 ). We deduce that V = (I ⊗ %1 )W (I ⊗ %2 ) ∈ ARγ (A0 ) since ARγ (A0 ) is stable under product. The conclusion follows. Alternatively, we observe that V can also be represented according to (2.9): V =



Vn,m =

n,m



|δn δm | ⊗ v(n, m),

n,m

where v(n, m) = %1 w(n, m)%2 . It follows that sup

v(n, m) eγ (|n|+|m|) ≤ %1 %2 

(n,m)∈Z2

sup

w(n, m) eγ (|n|+|m|) < ∞.

(n,m)∈Z2

The conclusion follows after applying Proposition 4.2 directly to the operator V . Corollary 4.2 Let (ϕ, u) ∈ 2 (Z) × H . Suppose that supn∈Z eγ |n| |ϕ, δn | < ∞ for some γ > 0. Then, ϕ ⊗ u is an analytic vector for A0 and the power series ∞  |θ |k k=0

k!

Ak0 (ϕ ⊗ u)

converges for any |θ | < Rγ , where Rγ > 0 is defined in Proposition 4.2. & Proof Let L = |ϕ ϕ| ⊗ |u u|. We observe that L = n,m Ln,m where Ln,m = |δn δm | ⊗ l(n, m) and l(n, m) = ϕ, δn ϕ, δm |u u|. In particular, sup

l(n, m) eγ (|n|+|m|) < ∞.

(n,m)∈Z2

Due to Proposition 4.2, L belongs to ARγ (A0 ). The conclusion follows from O’Connor lemma, see [11] p.196 and [3, Proposition 6.4, Remark 6.1]. One concludes this section by the following general observations: Remark 4.1 Consider W ∈ B( 2 (Z, H )) with its representation (2.9).

 (a) If w(n, m) ∈ S∞ (H ) for some (n, m) ∈ Z2 , then Wn,m ∈ S∞ 2 (Z, H ) . & (b) If w(n, m) ∈ S∞ (H ) for all (n, m) ∈ Z2 and if n,m w(n, m) < 

2 ∞, then W ∈ S∞ (Z, H ) . Indeed, for all (n, m) ∈ Z2 , Wn,m  = δn δm w(n, m) = w(n, m), so W is the limit of an absolutely convergent series of compact operators.

On Non-selfadjoint Operators with Finite Discrete Spectrum

85

4.2 Characteristic Values We recall some tools we need on characteristic values of finite meromorphic operator-valued functions. For more details on the subject, one refers to [6] and the book [7, Section 4]. The content of this section follows [7, Section 4]. Let H be Hilbert space as above. Definition 4.1 Let U be a neighborhood of a fixed point w ∈ C, and F : U \ {w} −→ B(H ) be a holomorphic operator-valued function. The function F is said to be finite meromorphic at w if its Laurent expansion at w has the form F (z) = & +∞ n n=m (z − w) An , m > −∞, where (if m < 0) the operators Am , . . . , A−1 are of finite rank. Moreover, if A0 is a Fredholm operator, then the function F is said to be Fredholm at w. In that case, the Fredholm index of A0 is called the Fredholm index of F at w. One has the following proposition: Proposition 4.3 ([7, Proposition 4.1.4]) Let D ⊆ C be a connected open set, Z ⊆ D be a closed and discrete subset of D, and F : D −→ B(H ) be a holomorphic operator-valued function in D\Z. Assume that: F is finite meromorphic on D (i.e. it is finite meromorphic near each point of Z), F is Fredholm at each point of D, there exists w0 ∈ D\Z such that F (w0 ) is invertible. Then, there exists a closed and discrete subset Z  of D such that: Z ⊆ Z  , F (z) is invertible for each z ∈ D\Z  , F −1 : D\Z  −→ GL(H ) is finite meromorphic and Fredholm at each point of D. In the setting of Proposition 4.3, one defines the characteristic values of F and their multiplicities: Definition 4.2 The points of Z  where the function F or F −1 is not holomorphic are called the characteristic values of F . The multiplicity of a characteristic value w0 is defined by 1 Tr mult(w0 ) := 2iπ

( |w−w0 |=ρ

F  (z)F (z)−1 dz,

(4.3)

  where ρ > 0 is chosen small enough so that w ∈ C : |w − w0 | ≤ ρ ∩ Z  = {w0 }. According to Definition 4.2, if the function F is holomorphic in D, then the characteristic values of F are just the complex numbers w where the operator F (w) is not invertible. Then, results of [6] and [7, Section 4] imply that mult(w) is an integer. Let ! ⊆ D be a connected domain with boundary ∂! not intersecting Z  . The sum of the multiplicities of the characteristic values of the function F lying in ! is called the index of F with respect to the contour ∂! and is defined by I nd∂! F :=

1 Tr 2iπ

( ∂!

F  (z)F (z)−1 dz =

1 Tr 2iπ

( ∂!

F (z)−1 F  (z)dz.

(4.4)

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Acknowledgments O. Bourget is supported by the Chilean Fondecyt Grant 1161732. D. Sambou is supported by the Chilean Fondecyt Grant 3170411. A. Taarabt is supported by Fondecyt grant 11190084.

References 1. F. Bagarello, J-P. Gazeau, F.H. Szafraniec, M. Znojil, Non-selfadjoint Operators in Quantum Physics: Mathematical Aspects (Wiley, 2015) 2. J.-F. Bony, V. Bruneau, G. Raikov, Counting function of characteristic values and magnetic resonances. Commun. PDE. 39, 274–305 (2014) 3. O. Bourget, D. Sambou, A. Taarabt, On the Spectral Properties of Non-selfadjoint Discrete Schrödinger Operators. J. Math. Pures Appl. 141, 1–49 (2020) 4. I. Egorova, L. Golinskii, On limit sets for the discrete spectrum of complex Jacobi matrices. Mat. Sborn. 196, 43–70 (2005) 5. I. Egorova, L. Golinskii, On the location of the discrete spectrum for complex Jacobi matrices. Proc. AMS. 133, 3635–3641 (2005) 6. I. Gohberg, E.I. Sigal, An operator generalization of the logarithmic residue theorem and Rouché’s theorem. Mat. Sb. (N.S.) 84(126), 607–629 (1971) 7. I. Gohberg, S. Goldberg, M.A. Kaashoek, Classes of Linear Operators. Operator Theory, Advances and Applications, vol. 49 (Birkhäuser, 1990) 8. B. Helffer, A. Martinez, Comparaison entre les diverses notions de résonances. Helv. Phys. Acta 60, 992–1003 (1987) 9. P. Hislop, Sigal, Introduction to Spectral Theory with Applications to Schrödinger Operators (Springer, New York, 1996) 10. T. Kato, Perturbation Theory for Lineal Operators (Springer, Berlin, 1995). Reprint of the 1980 edition. MR 1335452 1 11. M. Reed, B. Simon, Analysis of Operators IV. Methods of Modern Mathematical Physics (Academic Press, 1979) 12. I.M. Sigal, Complex transformation method and resonances in one-body quantum systems. Ann. Hen. Poincaré A 41, 103–114 (1984)

Pseudo-Differential Perturbations of the Landau Hamiltonian Esteban Cárdenas

1 Introduction and Known Results Given b > 0, we consider on L2 (R2 ) the so called Landau Hamiltonian, i.e. the self-adjoint operator 2  2  b b ∂ ∂ + y + −i − x , H0 = −i ∂x 2 ∂y 2

(x, y) ∈ R2

(1)

initially defined on C0∞ (R2 ) and then closed. As is well known, H0 represents the energy operator for a quantum particle moving in two dimensions, under the action of a uniform transversal magnetic field of strength b. Its spectrum is given by the collection of the Landau levels σ (H0 ) =

∞

{%q } ,

%q := b (2q + 1) ,

q ∈ Z+ = {0, 1, 2, . . .}

(2)

q=0

where each %q is an eigenvalue of infinite multiplicity, hence making σ (H0 ) purely essential. Suppose that you are given a linear operator V for which H0 + V remains selfadjoint and the essential spectrum remains invariant, that is σess (H0 + V ) = σess (H0 ) .

(3)

E. Cárdenas () Department of Mathematics, The University of Texas at Austin, Austin, TX, USA e-mail: [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 P. Miranda et al. (eds.), Spectral Theory and Mathematical Physics, Latin American Mathematics Series, https://doi.org/10.1007/978-3-030-55556-6_5

87

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E. Cárdenas

In such a situation, the Landau levels %q will be the only possible accumulation points of σd (H0 + V ) and one may seek to answer the following question: If σd (H0 + V ) accumulates at %q , what is the rate of convergence? In this section, we develop a noncomprehensive account of the results that answer this question when V plays the role of a perturbation that is either electric (multiplication operator), magnetic (first order differential operator) or metric (second order differential operator). For the purpose of the exposition, we refrain from referring to details regarding the verification of the self-adjointness of H0 + V and the invariance of the essential spectrum, namely, (3). The interested reader can consult the references to be cited, but this should always be understood to hold under the hypothesis considered.

1.1 Notation Given a self-adjoint operator T in L2 (R2 ) with purely discrete spectrum in (a, b) ⊂ R, we denote by N(T ; a, b) the number of eigenvalues of T lying inside (a, b), counted with their multiplicities. Under the condition σess (T ) = σess (H0 ) we introduce, for q ∈ Z+ , the counting function Nq+ (T , λ) := N(T ; %q + λ, %q + b) ,

λ ∈ (0, b) .

(4)

λ ∈ (0, b) .

(5)

Analogously, given q ∈ Z+ we define Nq− (T , λ) := N(T ; %q − b, %q − λ) ,

If, in addition, for every q ∈ Z+ σd (T ) accumulates at %q only from above, we denote by (λ+ q,k (T ))k the eigenvalues of T , lying in the interval (%q , %q+1 ) counted accordingly to their multiplicities and ordered decreasingly. On the other hand, if for every q ∈ Z+ σd (T ) accumulates at %q only from below, (λ− q,k (T ))k denotes the eigenvalues of T lying in (%q−1 , %q ) counted accordingly to their multiplicities and ordered increasingly, with the convention %−1 = −∞.

1.2 Electric Perturbations Throughout this subsection we assume that there exists a bounded measurable function V : R2 → R which we naturally identify with our latter perturbation, playing the role of an electric potential. We consider operators given by the operator sum H := H0 + V .

(6)

DO Perturbations of the Landau Hamiltonian

89

To state the results, given any measurable function f : Rn → R we introduce its volume functions as vol± (f, λ) := (2π )−n |{x ∈ Rn | ± f (x) > λ}| ,

λ ∈ (0, 1) ,

(7)

where | · | stands for Lebesgue measure in Rn . It is easy to check that, for fixed f , these are non-increasing functions of λ. In what follows, the following regularity requirement will be needed: (C). A non-increasing function φ : (0, 1) → [0, ∞) is said to satisfy C if there exist positive constants λ0 , c1 and c2 such that φ is differentiable on (0, λ0 ) and c1 φ(λ) ≤ −λ φ  (λ) ≤ c2 φ(λ) ,

λ ∈ (0, λ0 ) .

(8)

The first result that we present analyzes the case in which the perturbation V has, in a suitable sense, power-like decay. Theorem 1.1 ([10], Theorem 2.6) Let H be given by (6), where V ∈ C 1 (R2 , R) is such that there exist positive constants c1 , c2 and α satisfying |V (x)| ≤ c1 x −α

and

|∇V (x)| ≤ c2 x −α−1

(9)

for all x ∈ R2 . Further, assume that both vol± (V , ·) satisfy C. Then, if the condition lim inf λ2/α vol+ (V , λ) > 0

(10)

λ↓0

holds, we have that for all q ∈ Z+ Nq+ (H, λ) = b vol+ (V , λ) (1 + o(1)) ,

λ ↓ 0.

(11)

Similarly, if lim infλ↓0 λ2/α vol− (V , λ) > 0 we then have for all q ∈ Z+ Nq− (H, λ) = b vol+ (V , λ) (1 + o(1)) ,

λ ↓ 0.

(12)

Remark 1 Equations (9) and (10) imply vol+ (V , λ) # λ−2/α as λ ↓ 0 and similarly for vol− (V , λ). Remark 2 In view of [3] Sect. 2, Case 2.B, we can give a simpler (but more restrictive) criteria for the last theorem to hold. Indeed, note that whenever V ∈ C 1 (R2 , R) satisfies the estimates c0 x −α ≤ V (x) ,

|x| ≥ R0 ,

(13)

c1 V (x) ≤ −x, ∇V (x) ≤ c2 V (x) ,

|x| ≥ R1 ,

(14)

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vol+ (V , ·) satisfies C together with (10). An analogous criteria for vol− (V , ·) can also be given. The next two results deal with potentials that decay faster than polynomials. The first one treats perturbations that have exponential decay, while the second those of compact support. We highlight that we will require V to have a definite sign. Theorem 1.2 ([11], Theorem 2.1) Let H be given by (6) with V measurable, bounded and non-negative. In addition, assume that there exist μ > 0 and β > 0 such that ln V (x) = −μ . |x|→∞ |x|2β lim

(15)

Then, for every q ∈ Z+ : (i) If 0 < β < 1 we have Nq+ (H, λ) =

b | ln λ|1/β (1 + o(1)) , 2μ1/β

λ ↓ 0.

(16)

1 | ln λ| (1 + o(1)) , 1 + 2μ/b

λ ↓ 0.

(17)

(ii) If β = 1 we have Nq+ (H, λ) = (iii) If β > 1 we have Nq+ (H, λ) =

| ln λ| β (1 + o(1)) , β − 1 ln| lnλ|

λ ↓ 0.

(18)

Theorem 1.3 ([11], Theorem 2.2) Let H be given by (6) with V measurable, bounded and non-negative. Further, assume that V is compactly supported and that there exists C > 0 such that V ≥ C on an open disk. Then, for every q ∈ Z+ we have that Nq+ (H, λ) =

| ln λ| (1 + o(1)) , ln| lnλ|

λ ↓ 0.

(19)

For results on the remainder terms of Theorems 1 and 3, see [6] and [5], respectively. See also [9] for a related result for power-like decaying electric potentials.

DO Perturbations of the Landau Hamiltonian

91

1.3 Magnetic Perturbations We now present results that treat perturbations that are differential operators of first order, physically playing the role of a perturbation to the magnetic field. More precisely, introduce the vector field A0 (x, y) = b20 (−y, x), for (x, y) ∈ R2 , so that (1) may be written as H0 = (−i ∇ − A0 )2 with magnetic field strength b0 > 0. Then, given the vector field a = (a1 , a2 ) : R2 → R2 we consider perturbed operators given by H := (−i ∇ − A)2 ,

A := A0 + a .

(20)

Since physics is independent of the chosen gauge, results are naturally given in terms of the magnetic field associated with the perturbative magnetic vector potential, i.e. in terms of b (x) := ∂1 a2 (x) − ∂2 a1 (x) ,

x ∈ R2 .

(21)

The next Theorem is a simplification of a more general result which deals with eigenvalue asymptotics of H0 near the Landau levels, while being simultaneously perturbed by an electric field, a magnetic field and a metric perturbation. Perturbations to the magnetic field have, in a suitable sense, power-like decay at infinity. Theorem 1.4 ([6], Theorem 11.3.17) Let H be given by (20) with b ∈ C ∞ (R2 , R). Fix q ∈ Z+ and let C, m, and ε be positive constants. Consider the conditions (i) For all |x| ≥ C and α ∈ Z2+ |D α b(x)| ≤ C 1+|α| x −|α|−m .

(22)

(ii)q,± For all |x| ≥ C we have |b0 − b(x)| ≥ ε together with ± (2q + 1)b(x) ≥ −εx −m (⇒ (2q + 1)|∇b(x)| ≥ εx −m−1 .

(23)

If (i) and (ii)q,+ hold true, then the following asymptotic formula holds as λ ↓ 0 Nq+ (H, λ)

= (2π )

−1



b0 + b(x) dx + O(ln λ) = O(λ−2/m ) .

(24)

(2q+1)b>λ

If (i) and (ii)q,− hold true, then the analogous formula holds as λ ↓ 0 Nq− (H, λ)

= (2π )

−1

 −(2q+1)b>λ

b0 + b(x) dx + O(ln λ) = O(λ−2/m ) .

(25)

92

E. Cárdenas

In each case, suppose in addition that for |x| ≥ C one has ±(2q +1)b(x) ≥ εx −m . Then, Nq± (H, λ) # λ−2/m , the improved asymptotics hold Nq± (H, λ)

= (2π )

−1

 ±(2q+1)b>λ

b0 + b(x) dx + O(1)

(26)

and %q ∓ 0 is not an accumulation point of the discrete spectrum. Eigenvalue asymptotics for magnetic perturbations that have power-like decay at infinity has also been studied in [13]. There, conditions (22) and (23) are dropped, but it is proven that if b satisfies |b(x)| = O(|x|−β ),

|∂xki b(x)| = O(|x|−β−δ ),

|x| → ∞ ,

where β > 2 and δ ∈ (0, β), together with vol+ (b, ·) satisfying (10) and lim lim sup ε↓0

λ↓0

k∈N (27)

vol+ (b, λ(1 − ε)) =1 vol+ (b, λ)

(28)

then, the main asymptotic term is given by Nq+ (H, λ) = b0 vol+ (b, λ) (1 + o(1)),

λ↓0

(29)

and similarly for Nq− (H, λ). The case in which the Landau Hamiltonian is perturbed by a magnetic field of compact support is treated in [12], see Theorem 6.2 and Theorem 6.3. Due to the length of the hypothesis, we omit to write these results.

1.4 Metric Perturbations Throughout this subsection, m(x) = {mj,k (x)}2j,k=1 denotes a 2 × 2 Hermitian, positive-definite matrix satisfying mj,k ∈ C ∞ (R2 ) ,

sup |D α mj,k (x)| < ∞ , x∈R2

j, k = 1, 2 , α ∈ Z2+ .

(30)

Let us set 'j := −i

∂ − Aj ∂xj

(31)

DO Perturbations of the Landau Hamiltonian

93

with (A1 , A2 ) := b2 (−x2 , x1 ), so that the Landau Hamiltonian with magnetic field strength b > 0 may be written as H0 = '21 + '22 . We introduce its metric perturbations as H± :=

2 j,k=1

'j (δj,k ± mj,k )'k = H0 ± W ,

(32)

&2 with W = j,k=1 'j mj,k 'k ≥ 0. Under these conditions, σd (H+ ) may only accumulate at the Landau levels from above. If additionally, supx∈R2 |m(x)| < ∞, the same statement holds for H− but having instead accumulation at the Landau levels from below. We refer to [8] for a justification of these facts. The first result we present deals with compactly supported metric perturbations, whereas the second deals with metric perturbations with suitable exponential decay. For x ∈ R2 we denote by m< (x) ≤ m> (x) the two eigenvalues of m(x). Theorem 1.5 ([8], Theorem 2.1) Let H± be given by (32) and assume that the support of m is compact but m< does not vanish identically. Then, for all q ∈ Z+

 ln ± (λ± q,k (H± ) − %q ) = −k ln k + O(k),

k → ∞.

(33)

Theorem 1.6 ([8] Theorem 2.2) Let H± be given by (32) and assume that there exist constants β > 0 and γ such that ln m< (x) = ln m> (x) + O(ln |x|) = −γ |x|2β + O(ln |x|) ,

|x| → ∞

(34)

Denote μ = γ (2/b)β . Then, for fixed q ∈ Z+ (i) If β ∈ (0, 1), there exist constants fj = fj (b, μ), j ∈ N, with f1 = μ such that  

fj k 1−j (1−β) + O(ln k) , k → ∞ . ln ±(λ± q,k (H± )−%q ) = − 1≤j ≤(1−β)−1

(35) (ii) If β = 1,

 ln ± (λ± k,q (H± ) − %q ) = − ln(1 + μ) k + O(ln k) ,

k → ∞.

(36)

(iii) If β ∈ (1, ∞), there exist constants gj = gj (β, μ), j ∈ N, such that ln



 ± (λ± q,k (H± ) − %q ) =

 −

   β − 1 − ln(μβ) β −1 k ln k + k β β (37)

−

 1≤j ≤(1−β1 )−1

gj k

1−j (1−β1 )

+ O(ln k),

k → ∞.

94

E. Cárdenas

For the sake of comparison with other asymptotic expressions, we note that we can deduce from (33) the less precise version Nq± (H± , λ) =

| ln λ| (1 + o(1)) , ln| lnλ|

λ ↓ 0.

(38)

Similarly, we deduce from (35), (36) and (37) the following less precise asymptotics: ⎧ −1/β | ln λ|1/β (1 + o(1)) , β ∈ (0, 1) ⎪ ⎪ ⎨μ Nq± (H± , λ) =

1

1+μ ⎪ ⎪ ⎩ β

| ln λ| (1 + o(1)) ,

| ln λ| β−1 ln | ln λ|

(1 + o(1)) ,

β=1

λ ↓ 0.

(39)

β ∈ (1, ∞)

We conclude this section by adding that the case in which the metric perturbation has power-like decay has also been studied. See [6] Theorem 11.3.17 and [8] Theorem 2.3.

2 Pseudo-Differential Perturbations In the last section we gave a noncomprehensive account on the results about the behaviour of eigenvalues near the essential spectrum of the Landau Hamiltoninan, perturbed by certain multiplication and differential operators. In addition, there has been interest in the mathematical physics community in Schrödinger operators with non-local potentials, i.e. operators of the form  V ψ(x) =

K(x, y)ψ(y)dy

(40)

where K is an appropriate integral kernel, see for instance [1, 4] and [7]. Since these are all examples of pseudo-differential operators (DOs), one may then ask if there is a unified approach towards the subject, that is, to determine the spectral asymptotics of H0 + V near the Landau levels whenever V is a suitable DO. In this section, we summarize some results found in [2], which gives a first step in this direction.

2.1 Classes of Admissible Symbols Let N and n be natural numbers and denote by S(RN ) the Schwartz class of functions. Then, for a ∈ S(R2n ), Opw (a) denotes the operator with integral kernel

DO Perturbations of the Landau Hamiltonian −n

 x+y , ξ ei(x−y)ξ dξ , a 2 Rn 



K(x, y) = (2π )

95

(x, y) ∈ R2n

(41)

initially defined in S(Rn ). For a given pair u, v ∈ S(Rn ) we introduce its Wigner transform W (u, v) as −n



W (u, v)(x, ξ ) = (2π )

iyξ

Rn

e

    y y u x− v x+ dy , 2 2

(x, ξ ) ∈ R2n . (42)

Since W (u, v) ∈ S(R2n ), we have that for every a ∈ S(R2n ) and u, v ∈ S(Rn ) Opw (a) u, v L2 (Rn ) = a, W (v, u) L2 (R2n ) .

(43)

Therefore, through (43), we may define Opw (a) as a continuous linear mapping from S(Rn ) to S  (Rn ), for any a ∈ S  (R2n ). We denote by Γw (R2n ) the class of functions a : R2n → C for which the norm  aw := sup

sup (x,ξ )∈R2n

β |Dxα Dξ

 a(x, ξ )| : |α|, |β| ≤ [n/2] + 1

(44)

is finite. We name it the class of Weyl symbols. Whenever a ∈ Γw (R2n ), Opw (a) can be given sense as a bounded operator in L2 (Rn ), as the next proposition shows Proposition 1 If a ∈ Γw (R2n ), then Opw (a) extends to a bounded operator in L2 (Rn ). Moreover, there exists a constant c0 > 0 such that  w  Op (a) ≤ c0 aw

(45)

for every a ∈ Γw (R2n ). We now turn to introduce another useful class of symbols. Let us set Gn (x) = for x ∈ R2n and define for a ∈ S(R2n )

2 π −n e−|x|

Opaw (a) = Opw (a ∗ Gn ) .

(46)

Then, for every u, v ∈ S(Rn ) we have the expression Opaw (a)u, v L2 (Rn ) = a, Gn ∗ W (v, u) L2 (R2n )

(47)

which allows us to define Opaw (a) as a continuous linear mapping from S(Rn ) to S  (Rn ), for any a ∈ S  (R2n ). The interest in this quantization procedure lays on the representation Opaw (a) = (2π )−n

 R2n

a(x, ξ )Px,ξ dxdξ ,

a ∈ S(R2n ) ,

(48)

96

E. Cárdenas

to be understood in the weak sense, where Px,ξ are explicit rank-one orthogonal projections given by Px,ξ = ·, φx,ξ L2 (Rn ) φx,ξ ,

φx,ξ (y) = π −n/4 eiyξ e−|x−y|

2 /2

.

(49)

It allows us to define Opaw (a) as an operator in L2 (Rn ), in a way in which interesting properties of a are naturally transferred to properties of Opaw (a). In particular, let Sp (L2 (Rn )) denote the p-th Schatten-von Neumann class of operators in L2 (Rn ) and ·p its associated operator norm. Proposition 2 ([9], Lemma 2.5) (i) If a ∈ L∞ (R2n ), then Opaw (a) extends to a bounded operator in L2 (Rn ) satisfying  aw  Op (a) ≤ a

L∞ (R2n ) .

(50)

(ii) If a ∈ Lp (R2n ) for p ∈ [1, ∞), then Opaw (a) extends to an operator in the class Sp (L2 (Rn )) satisfying  aw p Op (a) ≤ (2π )−n ap p 2n . p L (R )

(51)

We call Γaw (R2n ) := L1 (R2n ) + L∞ (R2n ) the class of Anti-wick symbols. In particular, we have the important monotonicity property a ≥ 0 (⇒ Opaw (a) ≥ 0

(52)

where the left hand side is understood as an almost everywhere inequality. It should be stressed that not every DO with Weyl symbol admits an anti-Wick symbol, and that this class should be understood as a restriction of the first. One last class of functions will be defined. For ρ ∈ (0, 1] and m ∈ R we introduce the Hörmander-Shubin class as   Sρm (RN ) := a ∈ C ∞ (RN ) : |D α a(x)| ≤ Cα x m−ρ|α| , x ∈ RN , α ∈ Z+ . (53) In particular, Sρm (R2n ) ⊂ Γw (R2n ) whenever m ≤ 0. For the rest of the section, v denotes a real-valued symbol in the class Γw (R4 ) and we consider self-adjoint operators given by Hv := H0 + Opw (v) .

(54)

DO Perturbations of the Landau Hamiltonian

97

2.2 Unitary Equivalences To find the accumulation rate of eigenvalues near the Landau levels, we will reduce the asymptotics of Hv to that of a Toeplitz-type operator. For q ∈ Z+ , let pq denote the orthogonal projection onto the eigenspace associated to %q and define Tq (v) := pq Opw (v)pq

(55)

as an operator in pq L2 (R2 ). One of the main results of this subsection is to characterize Tq (v) as being unitarily equivalent to a pseudo-differential operator Opw (vq ) acting in L2 (R), where vq is an explicit effective symbol. First, we recall the well known unitary equivalence between H0 and the onedimensional harmonic oscillator h=−

d2 + x2 , dx 2

x ∈ R,

(56)

essentially self-adjoint on C0∞ (R) with respect to L2 (R). Namely, introduce the unitary transformation on L2 (R2 ) √ 

 b ( U u) (x, y) = exp i φ(x, y, x , y  ) u(x , y  ) dx  dy  , 2π R2

u ∈ L2 (R2 ) (57)

where φ(x, y, x , y  ) =

√ b xy + b (xy  − yx  ) − x  y  , 2

(x, y, x , y  ) ∈ R4 .

(58)

Next, let κ : R2 → R2 denote the symplectomorphism given by κ(x, ξ ) =

! √ √ 1 b b 1 (ξ + y), − (η + x) . √ (x − η), √ (ξ − y), 2 2 b b

(59)

for x = (x, y), ξ = (ξ, η) ∈ R2 . Then, Proposition 3 ([2], Proposition 3.2) If I denotes the identity operator on L2 (R), we have U ∗ H0 U = b h ⊗ I .

(60)

Moreover, for every v ∈ Γw (R4 ) one has that U ∗ Opw (v) U = Opw (v ◦ κ).

(61)

98

E. Cárdenas

From this representation, we can obtain a criteria to verify the H0 -compactness of Opw (v). Corollary 1 ([2], Proposition 3.3) Let v ∈ Sρ0 (R4 ), ρ ∈ (0, 1], be such that lim

x 2 +y 2 +ξ 2 +η2 →∞

(v ◦ κ)(x, y, ξ, η) = 0. x2 + ξ 2

(62)

Then, Opw (v) is bounded and relatively compact with respect to H0 , in L2 (R2 ). Remark 3 Any real-valued symbol v satisfying the conditions listed above will give rise to an operator V = Opw (v) for which Hv is self-adjoint and, thanks to Weyl’s criteria, verifies (3). Our results are mostly proved using convenient orthonormal basis for the spaces pq L2 (R2 ) and L2 (R) which we now introduce. First, on S(R2 ) introduce the mutually adjoint operators   b a = −2i ∂z¯ + z , 4

  b a ∗ = −2i ∂z − z¯ 4

where for (x, y) ∈ R2 we have denoted z = (x + iy), ∂z = ∂z¯ = 12 (∂x + i∂y ). Next, for k, q ∈ Z+ define  ϕk,q (x, y) := (a ∗ )q zk e−b|x|

2 /4

,

(63) 1 2 (∂x

− i∂y ) and

  ϕk,q L2 (R2 ) . ϕq,k :=  ϕk,q / 

(64)

Similarly, on S(R) introduce the mutually adjoint operators  α = −i

d +x dx

 ,

α ∗ = −i



d −x dx

 (65)

and for k ∈ Z+ let k (x) := (−i)q (α ∗ )q e−x 2 /2 , ψ

k /ψ k L2 (R) . ψk := ψ

(66)

It is a well known fact that {ψk }k∈Z+ is an orthonormal basis of L2 (R), being the eigenfunctions of h. On the other hand, for fixed q ∈ Z+ , {ϕk,q }k∈Z+ is an orthonormal basis of pq L2 (R2 ), the eigenspace of H0 associated to %q . The next lemma relates these two basis via the unitary transformation U defined in (57). Lemma 1 ([2], Lemma 3.5) For every k, q ∈ Z+ we have that U ∗ ϕk,q = iq−k ψk ⊗ ψq . For q ∈ Z+ consider the unitary transformation

(67)

DO Perturbations of the Landau Hamiltonian

99

Uq : L2 (R) → pq L2 (R2 )

(68)

defined by sending ψk into i k ϕk,q , for every k ∈ Z+ . Further, for every j, k ∈ Z+ write j,k := W (ψj , ψk )

(69)

for the Wigner transform of the pair ψj , ψk . Then, for q ∈ Z+ and v ∈ Γw (R4 ) define the effective symbol  vq (y, η) :=

R2

(v ◦ κ) (x, y; ξ, η) q,q (x, ξ ) dxdξ ,

(y, η) ∈ R2 .

(70)

Corollary 2 ([2], Corollary 3.7) If v ∈ Γw (R4 ) and q ∈ Z+ , then Uq∗ Tq (v) Uq = Opw (vq ) .

(71)

Tq (v) being defined in (55). We note that one may find an explicit expression for the Wigner transform j,k in terms of special functions. As a particular case, we have q (x, ξ ) := q,q (x, ξ ) =

(−1)q 2 2 Lq (2(x 2 + ξ 2 )) e−(x +ξ ) , π

q

d q −x ) are the Laguerre polynomials. where Lq (x) = (q!)−1 ex dx q (x e

(x, ξ ) ∈ R2 (72)

2.3 Spectral Properties of DOs with Radial Weyl Symbols The results on this subsection can be easily generalized to higher dimensions (see [2], Section 4), but we refrain to do so in order to retain notational simplicity. We say that a function v : R4 → C is radial provided there exists a function Rv : R2+ → C such that v (x ; ξ ) = v (x1 , x2 ; ξ1 , ξ2 ) = Rv ( x12 + ξ12 , x22 + ξ22 ) ,

x, ξ ∈ R2 .

(73)

Introduce on L2 (R2+ ) the orthonormal basis {L(q,k) }(q,k)∈Z2 given by +

L(q,k) (t1 , t2 ) = Lq (t1 )Lk (t2 ) e−t1 /2 e−t2 /2 ,

(t1 , t2 ) ∈ R2+ .

(74)

The following proposition states that DOs with radial symbols diagonalize in the same basis as the harmonic oscillator h.

100

E. Cárdenas

Proposition 4 ([2], Proposition 4.1) (i) If v ∈ Γw (R4 ) is radial, then Opw (v) has eigenfunctions {ψq ⊗ ψk }(q,k)∈Z2 + whose associated eigenvalues are μw (q,k) (v) :=

(−1)q+k 4

 R2+

Rv (t1 , t2 ) L(q,k) (t1 , t2 ) dt1 dt2 .

(75)

(ii) If v ∈ Γaw (R4 ) is radial, then Opaw (v) has eigenfunctions {ψq ⊗ ψk }(q,k)∈Z2 + whose associated eigenvalues are μaw (q,k) (v) :=

1 q! k!



Rv (2t1 , 2t2 ) t1 t2k e−t1 e−t2 dt1 dt2 . q

R2+

(76)

Corollary 3 ([2], Corollary 4.5) Let v ∈ Γw (R4 ) be a Weyl symbol for which v ◦ κ is radial. Then, Hv has eigenfunctions {ϕq,k }(q,k)∈Z2 whose associated eigenvalues + are %q + μw (q,k) (v ◦ κ) ,

q, k ∈ Z+ .

(77)

The above corollary allows us to show that an important difference arises when considering asymptotics of H0 perturbed by multiplication operators or by pseudo-differential operators. Namely, Theorem 3 from Section 1 shows that the rate of convergence to %q cannot be arbitrarily fast when the perturbation is a multiplication operator. However, the next proposition shows that this rate can be quite arbitrary, when sufficiently fast, in the case of a DO perturbation. Proposition 5 ([2], Proposition 5.1) (i) Let (c1,q )q∈Z+ and (c2,k )k∈Z+ be two sequences of positive numbers satisfying, for every m ∈ Z+ , limq→∞ q m c1,q = 0 and limk→∞ k m c1,k = 0. Then there exists a real valued symbol v ∈ S(R4 ) such that v ◦ κ is radial, Opw (v) ≥ 0 and λ+ k,q (Hv ) − %q = c1,q c2,k ,

k, q ∈ Z+ .

(78)

(i) For any given a sequence (mq )q∈Z+ ⊆ Z+ ∪ {∞} there exists a real valued symbol v ∈ S(R4 ) such that v ◦ κ is radial, Opw (v) ≥ 0 and lim Nq+ (Hv , λ) = mq , λ↓0

q ∈ Z+ .

(79)

DO Perturbations of the Landau Hamiltonian

101

2.4 Asymptotics: Slowly Decaying Symbols Let v be a given symbol with suitable power-like decay. To provide an asymptotic formula for Nq± (Hv , λ) one may employ techniques used in [10] to reduce the analysis to that of the Toeplitz-type operator Tq (v) introduced in (55), see [2] Proposition 5.9. Then, using Corollary 2, the Weyl inequalities and the mini-max principle, it is possible to reduce the analysis to that of the pseudo-differential operator with effective symbol vq introduced in (70), see [2] Corollary 5.10. Finally, the asymptotics of the latter may be deduced from [3] Theorem 1.3. One obtains −γ

Theorem 2.1 ([2], Theorem 5.4) Let v ∈ Sρ (R4 ) with ρ ∈ (0, 1) and γ > 0 be real-valued. Fix q ∈ Z+ and assume that both vol± (vq , ·) satisfy C, defined in Sect. 1.2. Then, if lim infλ↓0 λ2/γ vol+ (vq , λ) > 0, we have the asymptotic formula Nq+ (Hv , λ) = vol+ (vq , λ)(1 + o(1)) ,

λ ↓ 0.

(80)

λ ↓ 0.

(81)

Similarly, if lim infλ↓0 λ2/γ vol− (vq , λ) > 0 we then have Nq− (Hv , λ) = vol− (vq , λ)(1 + o(1)) , −γ

Remark 4 The condition v ∈ Sρ (R4 ) is a simplified version of a more general assumption for which the above result still holds, see [2].

2.5 Asymptotics: Fast Decaying Symbols To deal with symbols v that are, in a suitable sense, fast decaying, we ask our symbols to belong to a smaller class in order to reduce the problem to the analysis of Tq (v), as done in the previous case. Most notably, in addition to self-adjointess and relative compactness, we assume that Opw (v) has a definite sign and the following condition holds. (Hq,r ) Opw (vq ) admits an anti-Wick symbol v˜q ∈ Γaw (R2√ ) for which √ there exists r ∈ N and 0 ≤ ζ ∈ L∞ (R2 ) such that for ωq (x, y) := v˜q (− b y, − b x) we have  ωq = Lr

  − ζ 2b

(82)

Lr being the Laguerre polynomials defined at the end of Sect. 2.2. The set of symbols v satisfying Hq,r is non-trivial and examples may be constructed in the following way. Pick ζ ≥ 0 in L∞ (R2 ) and define ω := Lr (−Δ/2b)ζ , together with u(x, ˜ y) := ω(−b1/2 y, b1/2 x) and u := u˜ ∗ G1 . Then, the symbol

102

E. Cárdenas

v = 2π(q,q ⊗ u) ◦ κ −1

(83)

satisfies vq = u and so admits the anti-Wick symbol u˜ fulfilling (82). Under the above assumptions, it will then be possible to reduce the asymptotic analysis of Hv near %q to that of the Toeplitz-type operator Tq (v) using the (generalized) Birman-Schwinger principle and the Weyl inequalities. Then, using again Corollary 2 we reduce the problem to the analysis of the pseudo-differential operator with effective symbol vq (see [2] Proposition 5.5). The main consequence of assumption Hq,r comes however at this stage. Namely, it allows us to pass (up to unitary equivalence) from the non-local operator Opw (vq ) to the local multiplier ζ . More precisely consider for r ∈ Z+ the unitary transformation Ur : p0 L2 (R2 ) → pr L2 (R2 )

(84)

defined by sending, for every k ∈ Z, ϕk,0 into ϕk,r . Then, it can be deduced from the results of Sect. 2.2 that Proposition 6 ([2], Corollaries 3.8, 3.9) Let q ∈ Z+ and v ∈ Γw (R4 ). Assume that Opw (vq ) satisfies Hq,r for some r ∈ N. Then Opw (vq ) = Opaw (v˜q ) = U0∗ p0 ωq p0 U0 = (Ur U0 )∗ pr ζpr (Ur U0 )

(85)

U0 being defined in (68). Finally, techniques used in [5] and [8] may again be employed to determine the asymptotics of pr ζpr when ζ is of compact support or of exponential decay, respectively. To state the upcoming result, we introduce the logarithmic capacity of a compact set K ⊂ R2 , C(K), defined by  I(K) :=

inf

μ∈M(K) K×K

ln |x − y|−1 dμ(x)dμ(y) ,

C(K) := e−I (K)

(86)

M(K) being the collection of probability measures on K. Theorem 2.2 ([2], Theorem 5.2) Assume that Opw (v) is bounded, self-adjoint, non-negative and Opw (v)H0−1 is compact. For fixed q, r ∈ Z+ assume that Hq,r holds true. Then, if ζ ∈ C(R2 ) and if there exists a bounded domain ! ⊂ R2 with Lipschitz boundary ∂! such that supp(ζ ) = ! and ζ > 0 on !, we have ln



 ±(λ± k,q (H±v )−%q )

   b 2 k +o(k) , C(!) = −k ln k + 1+ln 2

k → ∞. (87)

Theorem 2.3 ([2], Theorem 5.3) Assume that Opw (v) is bounded, self-adjoint, non-negative and Opw (v)H0−1 is compact. For fixed q, r ∈ Z+ assume that Hq,r holds true. Assume that there exists β > 0 and γ > 0 such that

DO Perturbations of the Landau Hamiltonian

103

ln ζ (x) = −γ |x|2β + O(ln |x|) ,

|x| → ∞ ,

(88)

uniformly with respect to x/|x| ∈ S1 . Denote μ = γ (2/b)β . Then (i) If β ∈ (0, 1), there exist constants fj = fj (b, μ), j ∈ N, with f1 = μ such that  

(H )−% ) = − fj k 1−j (1−β) + O(ln k), k → ∞. ln ±(λ± ±v q q,k 1≤j ≤(1−β)−1

(89) (ii) If β = 1,

 ln ± (λ± k,q (H±v ) − %q ) = − ln(1 + μ) k + O(ln k),

k → ∞.

(90)

(iii) If β ∈ (1, ∞), there exist constants gj = gj (β, μ), j ∈ N, such that

 ln ± (λ± q,k (H±v ) − %q ) = −



   β − 1 − ln(μβ) β −1 k ln k + k β β (91)

−



gj k

1−j (1− β1 )

+ O(ln k),

k → ∞.

1≤j ≤(1− β1 )−1

It remains an open question to understand when can we solve for ζ ≥ 0 in terms of ωq , so as to make our assumptions less restrictive. In other words, suppose you are given the Weyl symbol vq with anti-Wick symbol v˜q . Under what conditions can we find r ∈ N and 0 ≤ ζ ∈ L∞ (R2 ) such that (82) holds?

References 1. A.M. Berthier, P. Collet, Existence and completeness of the wave operators in scattering theory with momentum-dependent potentials. J. Funct. Anal. 26, 1–15 (1977) 2. E. Cárdenas, G. Raikov, I. Tejeda, Spectral properties of Landau Hamiltonians with non-local potentials. arxiv.org/abs/1901.04370. To appear in Asymptotic Analysis 3. M. Dauge, D. Robert, in Weyl’s Formula for a Class of Pseudodifferential Operators with Negative Order on L2 (Rn ). Pseudodifferential operators (Oberwolfach, 1986), pp. 91–122. Lecture Notes in Math., vol. 1256 (Springer, Berlin, 1987) 4. A.L. Figotin, L.A. Pastur, Schrödinger operator with a nonlocal potential whose absolutely continuous and point spectra coexist. Commun. Math. Phys. 130, 357–380 (1990) 5. N. Filonov, A. Pushnitski, Spectral asymptotics of Pauli operators and orthogonal polynomials in complex domains. Commun. Math. Phys. 264, 759–772 (2006) 6. V. Ivrii, in Microlocal Analysis and Precise Spectral Asymptotics. Springer Monographs in Mathematics (Springer, Berlin, 1998)

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7. A. Jensen, Some remarks on eigenfunction expansions for Schrödinger operators with nonlocal potentials. Math. Scand. 41, 347–357 (1977) 8. T. Lungenstrass, G. Raikov, Local spectral asymptotics for metric perturbations of the Landau Hamiltonian. Anal. PDE 8, 1237–1262 (2015) 9. A. Pushnitski, G. Raikov, C. Villegas-Blas, Asymptotic density of eigenvalue clusters for the perturbed Landau Hamiltonian. Commun. Math. Phys. 320, 425–453 (2013) 10. G. Raikov, Eigenvalue asymptotics for the Schrödinger operator. I. Behaviour near the essential spectrum tips. Commun. Partial Differ. Equ. 15, 407–434 (1990) 11. G. Raikov, S. Warzel, Quasi-classical versus non-classical spectral asymptotics for magnetic Schrödinger operators with decreasing electric potentials. Rev. Math. Phys. 14, 1051–1072 (2002) 12. G. Rozenblum, G. Tashchiyan On the spectral properties of the perturbed Landau Hamiltonian. Commun. Partial Differ. Equ. 33, 1048–1081 (2008) 13. G. Rozenblum, G. Tashchiyan, in On the Spectral Properties of the Landau Hamiltonian Perturbed by a Moderately Decaying Magnetic Field. Spectral and Scattering Theory for Quantum Magnetic Systems, pp. 169–186. Contemp. Math., vol. 500 (Amer. Math. Soc., Providence, RI, 2009)

Semiclassical Surface Wave Tomography of Isotropic Media Maarten V. de Hoop and Alexei Iantchenko

1 Surface Wave Tomography The theoretical existence of elastic surface waves, that is, propagating wave solutions which decay exponentially away from the boundary of a homogeneous elastic half-space, was first discovered in 1885 by Rayleigh [17]. In 1911 Love [14] was the first to argue that surface-wave dispersion is responsible for the oscillatory character of the main shock of an earthquake tremor, following the “primary” and “secondary” arrivals. Rayleigh and Love waves can be identified with Earth’s free oscillations n Sl and n Tl with n , l/4 assuming spherical symmetry. We study the linear elastic wave equation in the half space IR3− = IR2x ×(−∞, 0]z , with coordinates (x, z), x = (x1 , x2 ) ∈ IR2 , z ∈ IR− = (−∞, 0]. ∂ 2u σ (u) , = div 2 ρ ∂t where u is displacement vector, σ (u) is the stress tensor given by Hooke’s law σ (u) = Cε(u). Here, ε(u) is strain tensor, C is the fourth-order stiffness tensor with components cij kl , ρ is the density of mass. In [4] we give the semiclassical description of surface waves in a general elastic medium stratified near the boundary in full generality. This semiclassical framework was first formulated by Colin

M. V. de Hoop Computational and Applied Mathematics and Earth Science, Rice University, Houston, TX, USA e-mail: [email protected] A. Iantchenko () Department of Materials Science and Applied Mathematics, Faculty of Technology and Society, Malmö University, Malmö, Sweden e-mail: [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 P. Miranda et al. (eds.), Spectral Theory and Mathematical Physics, Latin American Mathematics Series, https://doi.org/10.1007/978-3-030-55556-6_6

105

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de Verdière [1] to describe acoustic surface waves, who inspired our work. The medium stratification near the boundary is modeled by scaling the equation in the vertical direction with semiclassical parameter . Effective Hamiltonians of surface waves correspond with eigenvalues of ordinary differential operators, which, to leading order, define their phase velocities. Using these Hamiltonians, we obtain pseudodifferential surface wave equations. We carry out the semiclassical construction of parametrices associated with these equations. More than a decade ago, it was discovered [7, 18] that cross correlation of ambient noise yields Green’s function for surface waves. This enabled the possibility to extend the applicability of surface-wave tomography not only to any area where seismic sensors can be placed, but also to short-path measurements and frequencies at which the data are most sensitive to shallow depths. Since the pioneering work on inference from the dispersion of surface waves half a century ago [9, 12], the surface wave tomography based on dispersion of waveforms from earthquake data has played an important role in studies of the structure of the earth’s crust and upper mantle on both regional scales (such as Western Europe [16] and Scandinavia [3]) and global scale [13]. Crustal studies based on ambient noise tomography are typically conducted in the period band of 5−40 s, but shorter period surface waves (∼ 1 s, using station spacing of ∼ 20 km or less) have been used to investigate shallow crustal or even near surface shear-wave speed variations [20]. Building on the semiclassical description of surface waves [4], we solve the inverse problems for Love [5] and Rayleigh [6] waves and prove a uniqueness result for a smooth S-wave speed exhibiting multiple minima and maxima in depth from the semiclassical spectrum (spectrum in the high-frequency limit [2]). The semiclassical spectrum itself is related to the Green’s function via the trace formula (normal mode summation) which we derived in [4]. The reconstruction method is based on ideas from [2] and uses the inversion of Abel-type integral transforms for each well and a gluing argument. However, [2] does not account for the Neumann boundary condition at the free surface. To directly apply [2] a reflection principle could be invoked to yield results for Love waves; however, this principle does not apply for Rayleigh waves. The remedy is an explicit WKB construction near the boundary. Moreover, in contrast to the whole line problem [2], the presence of the boundary implies the rigidity of the solution.

2 Semiclassical Analysis of Surface Waves 2.1 Semiclassical Analysis and Hamiltonians We consider solutions u = (u1 , u2 , u3 ) of (1), generated by interior (point) sources and satisfying the Neumann boundary condition at ∂IR3− = {z = 0},

Semiclassical Surface Wave Tomography

107

∂t2 ui + Mil ul = 0,

(1)

u(t = 0, x, z) = 0, ∂t u(t = 0, x, z) = h(x, z), ci3kl ∂k ul (t, x, z = 0) = 0, ρ

(2)

where &2 & c 3l (x,z) ∂ cij kl (x,z) ∂ ∂ ∂ ci33l (x,z) ∂ Mil = − ∂z − 2j =1 ∂x∂ j ijρ(x,z) j,k=1 ρ(x,z) ∂z − ρ(x,z) ∂x ∂x ∂z  j k    & &2 &2 (x,z) ∂ ∂ ∂ ci3kl (x,z) ∂ ∂ cij kl (x,z) ∂ − 2k=1 ci3kl − − k=1 ∂z ρ(x,z) j,k=1 ∂xj ρ(x,z) ρ(x,z) ∂z ∂xk ∂xk ∂xk . Assumption 1 We assume that media are isotropic and smooth, cij kl (x, z) = λ(x, z)δij δkl+μ(x, z)(δik δj l+δil δj k ) ∈ C ∞ (IR3− ), ρ(x, z) ∈ C ∞ (IR3− ), where λ, μ ∈ C ∞ (IR3− ) are the two Lamé moduli and ρ, λ, μ obey: z λ(x, z) ˆ = λ(x, ), ρ(x, z) 

μ(x, z) z = μ(x, ˆ ),  ∈ (0, 0 ], ρ(x, z) 

stratification near the boundary at scale  comparable to wave length; μ(x, ˆ Z) = μ(x, ˆ ZI ) = μˆ I (x),

ˆ ˆ λ(x, Z) = λ(x, ZI ) = λˆ I (x),

for Z ≤ ZI < 0, (3) for fixed x, media are homogeneous in vertical direction beneath depth ZI < 0.

General assumptions needed for semiclassical description of elastic surface waves can be found in [4]. The assumption that the media are is needed in  isotropic z order to solve the inverse problem. We write u(t, x, z) = v t, x, and Z = z . To  prove existence of the surface waves we need the following Assumption 2 We assume that the shear modulus close to the free surface is smaller than its values in the interior: inf μ(x, ˆ Z) < μ(x, ˆ ZI ). Z≤0

Then (1) can be rewritten as  2 ∂t2 v + H (x, Dx ; )v ∼ 0, where 1 H (x, Dx ; )v(x) = (2π )2



 H

 x+y , ξ ;  eiξ,x−y / v(y)dydξ 2

is a semiclassical pseudodifferential operator (see [8]) with symbol H (x, ξ ; ) = H0 (x, ξ ) + H1 (x, ξ ),

(4)

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 H0 =

 0 H0L (x, ξ ) , 0 H0R (x, ξ )

(5)

where H0L (x, ξ )ϕ1 = −

∂ ∂ μˆ ϕ1 + μˆ |ξ |2 ϕ1 ∂Z ∂Z

(6)

supplemented with Neumann boundary condition ∂ ϕ1 (0) = 0, ∂Z for Love waves, and  ⎞  ⎛   ∂ϕ2 ∂ ∂ ∂ 2ϕ ˆ ˆ + ( λ + 2 μ)|ξ ˆ | ( μ ˆ ) − i|ξ | ( μϕ ˆ ) + λ ϕ − 3 3 2 ϕ2 ∂Z  ∂Z ∂Z  ∂Z   ⎠ H0R (x, ξ ) =⎝ ˆ 2 ) + μˆ ∂ ϕ2 + μ|ξ ϕ3 ˆ ∂ϕ3 − i|ξ | ∂ (λϕ ˆ |2 ϕ3 − ∂ (λˆ + 2μ) ∂Z

∂Z

∂Z

∂Z

(7)

supplemented with Neumann boundary condition ∂ϕ3 (0) = 0, ∂Z ∂ϕ2 i|ξ |μϕ ˆ 3 (0) + μˆ (0) = 0, ∂Z

ˆ iλˆ |ξ |ϕ2 (0) + (λˆ + 2μ)

(8) (9)

for Rayleigh waves. We use eigenvalues and eigenfunctions of H0 (x, ξ ) to construct approximate solutions to (4).

2.2 Pseudodifferential Surface “Wave” Equation and Mode Summation We assume that H0 (x, ξ ) has M + 1 eigenvalues Λα (x, ξ ) with eigenfunctions Φα,0 (Z, x, ξ ). Then H Φα,0 = Λα Φα,0 + O(). We solve H ◦ Φα, = Φα, ◦Λα, + O( ∞ ), ◦ the composition of symbols, Φα, ∼ Φα,0 + Φα,1 + . . . , Λα, ∼ Λα,0 + . . . We define 1 Jα, (Z, x, ξ ) = √ Φα,0 (Z, x, ξ ).  (The factor √1 makes the operator Jα, (Z, x, Dx ) microlocally unitary) Now, the approximate solution to (4) with initial values

Semiclassical Surface Wave Tomography

h(x, Z) =

M 

109

Jα, (Z, x, Dx )Wα, (x, Z)

α=0

is constructed in the form v (t, x, Z) =

M 

Jα, (Z, x, Dx )Wα, (t, x, Z), repre-

α=0

senting surface waves. Let Wα, be the solution (up to leading order) to [ 2 ∂t2 + Λα (x, Dx )]Wα, (t, x, Z) = 0, α = 0, 1, . . . , M,

(10)

Wα, (0, x, Z) = 0, ∂t Wα, (0, x, Z) = Jα, (Z, x, Dx )Wα (x, Z). Let Gα,±,0 be Green’s functions for one-way wave equations associated with (10). The approximate Green’s function for surface waves (microlocalized in x), up to leading order, is given by G0 (Z, x, t, Z  , ξ ; ) =   M  i i Jα, (Z, x, ξ ) Gα,+,0 (x, t, ξ, ) − Gα,−,0 (x, t, ξ, ) 2 2 α=0

(x, ξ )Jα, (Z  , x, ξ ) ×Λ−1/2 α

2.3 Trace Formulas and Data On the diagonal Z = Z  we take the semiclassical Fourier transform in t, use stationary phase and get  ∂ t G0 (Z, x, ω, Z, ξ ; ) ≡ ∼



∞

−∞ M 

∂t G0 (Z, x, t, Z, ξ ; )e−

itω 

dt

2 Φα,0 (Z, x, ξ )δ(ω2 − Λα (x, ξ ))Λ1/2 α (x, ξ ).

α=0

Integrating in Z and using the normalization we get  IR−

 ∂ t G0 (Z, x, ω, Z, ξ ; )dZ ∼

M 

δ(ω2 − Λα (x, ξ ))Λ1/2 α (x, ξ )

(11)

α=0

=

M 1  δ(y − λα (h)), 2h2 α=0

(12)

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where y = h2 ω2 , λα = h2 Λα and h = |ξ |−1 plays the role of “Planck’s constant”. The geometrical behavior of Green’s function G0 is exactly what seismologists can get from ambient noises [7, 18, 20, 21]. Then, by the above formula, we can extract the eigenvalues Λα , α = 0, 1, 2, · · · , M, and use them to recover the profile of μ. ˆ

2.4 Computation of Eigenvalues and Eigenfunctions From here on we suppress the dependence on x in the notations (so x ∈ IR2 is fixed). Due to the decoupling of the Hamiltonian (5) in Love and Rayleigh operators, we can either study scalar Love spectral problem or matrix-valued Rayleigh problem. As the Love Hamiltonian H0L can in (our) smooth case be transformed into a well-studied standard Schrödinger operator, we will mostly focus on the Rayleigh problem. Using Assumption (3), that beneath certain depth ZI the coefficients in (7) − are constant (in Z) we introduce the so called “free” Jost solutions fP−,I , fS,I , for R H0 , (see [15] for the classical introduction) fP−,I

− fS,I

!

1

=

e−iZqP , qP :=

− |ξ1| qP 1 |ξ | qS

=

1

! e−iZqS , qS :=



5

ω2 μˆ I

ω2 λˆ I +2μˆ I

− |ξ |2 ,

− |ξ |2 ,

satisfying equation H0R f = ω2 f

(13)

√ only for Z < ZI . Here, z denotes the principal branch of the square root which is positive for z > 0. By introducing a minimal Riemann surface where the quasimomenta qp (ω), qS (ω) are single-valued holomorphic functions, we get − that at the eigenvalues Λα = ωα2 < |ξ |2 μI , we have fP−,I , fS,I ∈ L2 (IR− ) as  Im qP (ωα ) = Im i |ξ | − 2

ωα2

λˆ I + 2μˆ I

 12 > 0,

1  ωα2 2 2 Im qS (ωα ) = Im i |ξ | − > 0. μˆ I

Now, we define the Jost solutions fP− , fS− , as solutions of equation (13) for all − Z < 0 and coinciding with fP−,I , fS,I , below ZI :   

− − (Z) fP (Z) fS− (Z) = fP−,I (Z) fS,I

for

Z < ZI .

(14)

Semiclassical Surface Wave Tomography

111

As these solutions form a linear independent system of L2 (IR− ) solutions to (13), we get an eigenfunction of H0R with eigenvalue Λα as linear combination of fP− (Z), fS− (Z) satisfying boundary conditions (8), (9). This implies that the eigenvalues Λα = ωα2 are the real zeros of the Rayleigh determinant (see [19]) 

 a(fP− ) a(fS− ) ΔR = det , b(fP− ) b(fS− ) where the boundary operators a, b are given by 3 ˆ |ϕ2 (0) + (λˆ + 2μ) a(f ) := iλ|ξ ˆ ∂ϕ ∂Z (0), ∂ϕ2 b(f ) := i|ξ |μϕ ˆ 3 (0) + μˆ ∂Z (0), f = (ϕ2 , ϕ3 )T .

Therefore, the direct problem of computing of eigenvalues is reduced to construction of the Jost solutions by finding solutions of (13) satisfying conditions (14) and to computing zeros of the Rayleigh determinant: ΔR (ω) = 0. Using the eigenfunctions and eigenvalues we can compute the trace (11), (12) by normal mode summation and by mollifying the delta-function. Moreover, from the trace formula (21) we can get the classical actions as explained in Sect. 3.

3 Bohr-Sommerfeld Rules 3.1 Semiclassical Spectrum and Main Result Here, we discuss converting the trace to “new” data, classical actions. We consider the matrix-valued Rayleigh operator H0R (ξ ) given by (7) subject to the Neumann boundary conditions (8), (9). We introduce h = |ξ |−1 as another semiclassical R = h2 H R (ξ ), that is, parameter (the first one was ) and put H0,h 0  R H0,h

ϕ2 ϕ3



⎛

    ⎞ ∂ ∂ ∂ 2 −h2 ∂Z μˆ ∂ϕ − ih ∂Z ˆ 2 (μϕ ˆ 3 ) + λˆ ∂Z ϕ3 + (λˆ + 2μ)ϕ ∂Z ⎠,     =⎝ ∂ ∂ ˆ ∂ 3 (λˆ + 2μ) ˆ ∂ϕ − ih + μϕ ˆ −h2 ∂Z ( λϕ ) + μ ˆ ϕ 2 2 3 ∂Z ∂Z ∂Z

which has eigenvalues λα (h) = h2 Λα . We write E0 = μ(0). ˆ The spectrum of H0,h is divided in two parts, R R R ) = σd (H0,h ) ∪ σess (H0,h ), σ (H0,h

where the discrete spectrum σd (H0,h ) consists of a finite number of eigenvalues below the branching point μˆ I , that is, λ0 (h) < E0 < λ1 (h) < λ2 (h) < . . . < λM (h) < μˆ I ,

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and the essential spectrum σess (H0,h ) = [μˆ I , ∞) (see [4]). (The essential spectrum is not absolutely continuous for the Rayleigh wave operator). The lowest eigenvalue λ0 (h) lies below μ(0) ˆ for h sufficiently small. Its existence and uniqueness under certain conditions (which are satisfied here) are explained in Theorem 4.3 in [4]. No such phenomenon occurs in the Love case. The number of eigenvalues, M increases as h decreases. We study how to reconstruct the profile of μˆ using the semiclassical spectrum as in [2]. Definition 1 For a given E with E0 < E ≤ μ(Z ˆ I ) and a positive real number N , a sequence μα (h), α = 0, 1, 2, . . . is a semiclassical spectrum of H0,h mod o(hN ) in ] − ∞, E[ if, for all λα (h) < E, λα (h) = μα (h) + o(hN ) uniformly on every compact subset K of ] − ∞, E[ . Our main result is (see [6]) Theorem 3.1 Under Assumptions 3–8, the function μˆ can be uniquely recovered R modulo o(h5/2 ) below μ from the semiclassical spectrum of H0,h ˆI. A similar result for Love waves was proven in [5].

3.2 Bohr-Sommerfeld Rules for Wells Separated from the Boundary and Boundary Half Well For the scalar Love operator, H0L , the Bohr-Sommerfeld rules are well known [2]. For the semiclassical wells separated from the boundary, Z = 0, thanks to Remark 1, we get the Bohr-Sommerfeld rules for the semiclassical spectrum of the matrix-valued Rayleigh operator H0R by applying the semiclassical diagonalization procedure (see [6] for details) Theorem 3.2 (Diagonalization) There exists a unitary pseudodifferential operator U and diagonal R #0,h H (x, ξ ) =



0 H0,h,1 (x, ξ ) 0 H0,h,2 (x, ξ )



such that R R #0,h U ∗ H0,h (x, ξ )U = H (x, ξ ) + O(h∞ ) in L(L2 , L2 ).

Here H0,h,i (x, ξ ), i = 1, 2, are pseudodifferential operators with symbols

(15)

Semiclassical Surface Wave Tomography

113

σ W (H0,h,1 (x, ξ )) = (λˆ + 2μ)(1 ˆ + ζ 2 ) + h2 α2 + . . . , σ W (H0,h,2 (x, ξ )) = μ(1 ˆ + ζ 2 ) + h 2 δ2 + . . . ,

(16)

where '   )2 (2λ ˆ λˆ  + 2μˆ  1 + 3 μ) ˆ 4( μ ˆ 1  α2 = + 2 , − λˆ + μˆ  + 4 2 ζ +1 (λˆ + μ) ˆ 2 '  1 3  μˆ  4λˆ (μˆ  )2 + 2 μˆ − δ2 = . 4 ζ +1 2 (λˆ + μ) ˆ 2

(17)

The h-order terms are absent. To reconstruct μˆ we will only use δ2 . For wells separated from the boundary, the analysis is purely based on the diagonalized system and, hence, follows the corresponding analysis for Love waves. That is, we consider operator H0,h,2 (cf. (15)). Note that the Weyl symbol of the Love operator H0L differs from σ W (H0,h,2 ) by the second term in the right-hand side of (17). We label the critical values of μ(Z) ˆ as E1 < . . . < EM < μˆ I and the corresponding critical points by Z1 , · · · , ZM . We denote Z0 = 0, E0 = μ(Z ˆ 0) and EM+1 = μˆ I . A well of order k is a connected component of {Z ∈]ZI , 0[ : μ(Z) ˆ < Ek } that is not connected to the boundary at Z = 0. We refer to the connected component connected to the boundary as a half well of order k. We denote Jk =]Ek−1 , Ek [, k = 1, 2, 3, · · · and let Nk (≤ k) be the number of wells of order k. The set {Z ∈ ˆ < Ek } consists of Nk wells and one half well (ZI , 0) : μ(Z)  k (E), (∪Nk W k (E)) ∪ W  k (E) ⊂ [ZI , 0[ . Wjk (E), j = 1, 2, · · · , Nk , and W j =1 j  k (E) is connected to the boundary at Z = 0. The half well W The semiclassical spectrum mod o(h5/2 ) in Jk is the union of Nk + 1 spectra: k k k ∪N j =1 Σj (h) ∪ Σ (h).

(18)

Here, Σjk (h) is the semiclassical spectrum associated with well Wjk , and the k. k (h) is the semiclassical spectrum associated with half well W spectrum Σ Remark 1 Separation of semiclassical spectra associated with different wells and independence of such spectra from the boundary condition come from the fact that the associated eigenfunctions are O(h∞ ) outside the wells and is related to the exponentially small “tunneling” effects [10, 11].

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We have Bohr-Sommerfeld rules for separated wells, Σjk (h) = {μα (h) : Ek−1 < μα (h) < Ek and S k,j (μα (h)) = 2π hα}, where S k,j = S k,j (E) : ]Ek−1 , Ek [→ IR admits the asymptotics in h k,j

k,j

S k,j (E) = S0 (E) + hπ + h2 S2 (E) + · · · k (h) = {να (h) : Ek−1 < να (h) < Ek and  S k (να (h)) = 2π hα}, and Σ k k where  S = S (E) : ]Ek−1 , Ek [→ IR admits the asymptotics in h 1 1  S k (E) = S˜0k (E) + h S1k (E) + h2 S2k (E) + · · · , 2 2 for the boundary half well. For the Love operator, H0L , the Bohr-Sommerfeld rules for a well separated from the boundary are also well known [2]. As the Weyl symbol of H0,h,2 differs from the Love one by the second term in the right-hand side of (17), k,j we get a new action S2 which is calculated in [6]. For the boundary half well the k actions  Sl , l = 0, 1, 2 were explicitly calculated only for Love operator H0L in [5]. For the matrix-valued Rayleigh operator, H0,h,2 , near the boundary we cannot use the diagonalization procedure as in Theorem 3.2 and therefore computing of lower order actions is technically challenging. However, only leading order term (counting function) is used in the reconstruction procedure. Therefore, in [6] we calculated explicitly only S˜lk , l = 0, 1.

4 Inverse Problems 4.1 Assumptions on the Material Parameters The statements are formulated for the Rayleigh operator H0R and can be naturally adapted (simplified) for the Love operator H0L . 4.1.1

General Restrictions on Lamé Moduli

Assumption 3 Poisson’s ratio ν, with λˆ =

2ν ˆ 1−2ν μ,

of the elastic solid is constant.

For a Poisson solid, ν = 1/4. However, we only assume that ν is known. We may thus express λˆ in terms of μ. ˆ We strengthen condition (3) and Assumption 2. ˆ ˆ I ). Moreover, Assumption 4 For all Z ≤ ZI , μ(Z) ˆ = μ(Z ˆ I ) and λ(Z) = λ(Z ˆ < μˆ I = sup μ(Z) ˆ = μ(Z ˆ I ), 0 < μ(0) ˆ = inf μ(Z) Z≤0

Z≤0

(19)

Semiclassical Surface Wave Tomography

115

ˆ + 2ˆ λ μ

μ ˆ

f+ (E)

Wjk f− (E)

Ek E Ek−1

Z → −∞

Fig. 1 Illustration of a well of order k (Nk = 1) and associated f±

for all Z ∈ [ZI , 0] we have λˆ (Z) + 2μ(Z) ˆ ≥ μ(Z ˆ I ).

(20)

The assumption that μˆ attains its minimum at the boundary and its maximum in the “deep zone” (Z  ≤ ZI , cf. (19)) is realistic in seismology.  ˆ CS = μˆ correspond with longitudinal (primary), shear Here, CP = λˆ + 2μ, wavespeed, respectively. Bound (20) is needed in order to separate the P − and S− semiclassical spectra (Fig. 1) and allows us to focus only on the S spectrum and shear modulus, μ. ˆ

4.1.2

Multiple Semiclassical Wells

Here, we formulate the sufficient conditions on the profile μˆ which allow reconstruction and assures the uniqueness. Assumption 6 can be generalized (see [2]). Assumption 5 There is a Z ∗ < 0 such that μˆ  (Z ∗ ) = 0, μˆ  (Z ∗ ) < 0 and μˆ  (Z) < 0 for Z ∈ ]Z ∗ , 0[ . Assumption 6 The function μ(Z) ˆ has non-degenerate critical values at a finite set {Z1 , Z2 , · · · , ZM } in ]ZI , 0[ and all critical points are non-degenerate extrema. None of the critical values of μ(Z) ˆ are equal, that is, μ(Z ˆ j ) = μ(Z ˆ k ) if j = k.

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M. V. de Hoop and A. Iantchenko

Assumption 7 The function μˆ has a generic symmetry defect: If there exists X± satisfying μ(X ˆ − ) = μ(X ˆ + ) < E, and for all N ∈ IN, μˆ (N ) (X− ) = N (N ) (−1) μˆ (X+ ), then μˆ is globally even with respect to 12 (X+ +X− ) in the interval {Z : μ(Z) ˆ < E}. We postpone one more condition, Assumption 8 on weak transversality of classical periods, to Sect. 4.2, as it needs additional definitions.

4.2 Concentration of Eigenvalues and Separation of Clusters R below μ(0) In [4], it was proved that there exists a unique eigenvalue of H0,h ˆ for small h. This eigenvalue cannot be related to any well. Therefore, for Rayleigh waves, we first separate out this fundamental mode to continue our presentation. 2 We find E0 = μ(0) ˆ from λ1 (h) using that λ1 (h) = μ(0) ˆ + O(h 3 ). Here, λ1 (h) is the first eigenvalue for the Love operator or the second eigenvalue for the Rayleigh operator. Then we use the property of clustering of the semiclassical spectrum around (above) minima of the multiple semiclassical wells, which we formulate unformally as follows

Theorem 4.1 The eigenvalues λα (h), α = 1, 2, . . . , are concentrated at minima of multiple semiclassical wells as h → 0. Exact formulation and the proof can be found in [10], and a simpler presentation is given in [11], Theorem 4.2.1. By this result we can read off the lowest local minimum E1 > E0 and the higher local minima. In order to localize the intervals Jk =]Ek−1 , Ek [, k = 1, 2, 3, · · · , and the wells Wjk (E), j = 1, . . . , Nk , we need to find also the local maxima. Then we use (see [2, Lemma 11.1]) that the local minima Ek of μˆ also show up  as singularities of the counting function or A(E) = μ(Z)(1+ζ 2 )≤E |dZdζ |, so that ˆ A(E) is smooth at [Ek − , Ek ],  > 0. The other singularities of the counting function are local maxima. The other way around, we detect all jumps of the counting functions. The minima are singularities which are at the bottoms of the clusters, the maxima are the other singularities, as they do not show up in eigenvalue concentration. This gives us the right intervals Jk =]Ek−1 , Ek [, k = 1, 2, 3, · · · . In addition, from the singularity at Ek one can extract the value of μˆ  (Zk ), k = 1, 2, . . .. As each interval Jk is the union of Nk + 1 semiclassical spectra (18) modulo 5 k,j k,j o(h 3 ), we need to extract actions {S0 , S2 } from each cluster. We apply the trace formula as distributions on each Jk  α∈ZZ

δ(E − μα (h)) =

Nk   j =1 m∈ZZ

k Zm,j (E) +

 m∈ZZ

k Zm,N (E) + o(1) k +1

(21)

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with k Zm,j (E) =

k,j 1 −1 k,j (−1)m eimS0 (E)h Tjk (E)(1 + imhS2 (E)), 2π h

k (E) = Zm,N k +1

k,N +1

j = 1, · · · , Nk ,

k,N +1

1 imS1 k (E) imS1 k (E)h−1 e 2π h e ·(TNkk +1 (E) + h(S1k,Nk +1 ) (E))(1 + imhS2k,Nk +1 (E)),

k. where the last term is associated with the boundary half well W Here Tjk (E) = (S0 ) (E), j = 1, . . . , Nk , are classical periods, T k (E) = (S˜ k ) (E) is half-period. We invoke k,j

Nk +1

0

Assumption 8 (Weak Transversality) For any k = 1, 2, · · · and any j with 1 ≤ j < l ≤ Nk + 1, the classical periods (half-period if j = Nk + 1) Tjk (E) and Tlk (E) are weakly transverse in Jk , that is, there exists an integer N such that the N th derivative (Tjk (E) − Tlk (E))(N ) does not vanish. By the weak transversality assumption, it follows that set Bk = {E ∈ Jk : ∃j = l,

Tjk (E) = Tlk (E)},

is a discrete subset of Jk . We let the distributions Dh (E) =



δ(E − μα (h)) be α∈ZZ Jk ∩ (−∞, μ(0)) ˆ =∅

given on the interval J = Jk modulo o(1) using (21). Since for any k, for the Rayleigh operator, we can ignore the lowest eigenvalue λ0 . These distributions are determined mod o(1) by the semiclassical spectra mod o(h5/2 ). We denote by Zh the finite sum defined by the right-hand side of (21) restricted to m = 1, Zhk (E) =

N k +1

k Z1,j (E),

j =1

while suppressing k in the notation. Assuming that we already have recovered μ(0), ˆ we obtain  S0k (E). By analyzing the microsupport of Dh and Zh [2, Lemmas 12.2 and 12.3], we find Lemma 1 Under the weak transversality assumption, the sets B and the distributions Zhk mod o(1) are determined by the distributions Dh mod o(1). Lemma 2 Assuming that the S j ’s are smooth and the aj ’s do not vanish, there is a unique splitting of Zh as a sum

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Zh (E) =

Nk +1 1  j (aj (E) + hbj (E))eiS (E)/ h + o(1). 2π h j =1

k,j

It follows that the spectrum in Jk mod o(h5/2 ) determines the actions S0 (E), k,j S2 (E) and  S0k (E) and  S1k (E) on Jk . This provides the separation of the data for the Nk wells and the half well. Then, under Assumption 7, we proceed with k,j k,j reconstructing μˆ from the functions S0 (E), S2 (E) for any k and j ≤ Nk and  S0k (E).

4.3 Reconstruction of the Profile from the Actions: Love Case Using the semiclassical diagonalization of the Rayleigh operator away from the boundary and condition (20), the reconstruction procedure for the matrix Rayleigh operators can be partially reduced to the recovery of the scalar Love type operator. Therefore, we start by describing the reconstruction procedure for the scalar Love operator. In the next section, we discuss how to extend the method to the Rayleigh operator. We restrict ourselves to the simplest case of a single well, with barrier and decreasing profile. We summarize the procedure:  1 , that is connected to the Step 1. We start by constructing the half well, W boundary between E0 and E1 . For E ∈]E0 , E1 [, there is only one (half) well,  1 (E), of order 1 with W  1 (E1 ) = [Z  , 0]. W 1 ˆ S˜01 (E) (or counting function N(h, E)), E ∈ We are given E0 = μ(0), [E0 , E1 ]. Here S˜01 (E) = π hN (h, E) + o(1), N(h, E) = #{λj (h) ≤ E}. Since  1 (E1 ), we may reconstruct μˆ on this interval ]Z  , 0] μˆ is strictly decreasing in W 1 as inverse to   E 4 √ d2 d ˜1 f (E) = u 2A (22) S (u) du, π du du 0 E0  where Ag(E) =

E

√

E − u g(u) du.

E0

Step 2. We note that Z2 in this case is the Z ∗ defined in Assumption 5. We consider E ∈ ]E1 , E2 [ which corresponds to wells of order k = 2 with Nk = 1 (one connected component for Z < 0 separated from the boundary). The two  2 (E) with W 2 (E2 ) = [Z− , Z2 ] and W  2 (E2 ) = [Z2 , 0]. wells are W12 (E) and W 1 Here, Z− is the unique point in [ZI , Z1 ] such that E2 = μ(Z ˆ − ). S02 (and  S22 ). We can get the local minimum E1 We are given S02,1 , S22,1 and  from observing the bottom of the associated (second) cluster or, more precisely, by

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finding the lowest singularity E1 of the counting function or  S02 (E) above E0 so that  S02 (E) is smooth in ]E1 − , E1 ], with  > 0. We get the local maximum E2 as the next singularity of the counting function. As in Sect. 4.2, from these singularities of  S02 (E) we can read off the values μ(Z ˆ 1 ), μˆ  (Z1 ) (though the point Z1 is yet to be determined). We continue to reconstruct μˆ from [Z1 , 0] to [Z2 , 0] from  S02 by inverting (22). For the reconstruction of μˆ on the interval I = [Z− , Z2 ], more effort is needed. We note that, up to this point, I itself cannot be determined yet. We introduce Φ(E) = f+ (E) − f− (E)

Ψ (E) =

and

1 f+ (E)

−

1

, f− (E)

(23)

where functions f± are illustrated on Fig. 1. We have 

(S02,1 ) (E) = T g(E),

T g(E) =

E

E1

g(u) Φ(u) du with g(u) = √ . √ u E−u

The inversion formula for the Abel transform d 2 T g(E) = πg(E) dE √ E d T ◦ (S02,1 ) (E). yields Φ(E) for E ∈ [E1 , E2 [: Φ(E) = π dE 1 d BΨ (E) suppleWe recover BΨ (E) from S22,1 (E) by solving S22,1 (E) = − 12 dE   mented with condition BΨ (E1 ) = 2π E1 μˆ (Z1 ). We obtain Ψ (E) for E ∈ [E1 , E2 [ as solution to second-order inhomogeneous ordinary differential equation π d2 (T ◦ BΨ )(E) = E 2 Ψ  (E) + 4EΨ  (E) − Ψ (E). E 3/2 dE 2 supplemented with the “initial” conditions Ψ (E1 ) = 0,

lim

E↓E1

  E − E1 Ψ  (E) = 2μˆ  (Z1 ),

where μˆ  (Z1 ) is obtained from the limiting behavior of the counting function (or S02,1 (E)) as E ↓ E1 . We use that the period of small oscillations of pendulum associated to the local minimum of μˆ at Z1 is given by (S02,1 ) (E) and get

 =

f+ (E) f− (E)

6 

dZ μ(E ˆ − μ) ˆ

=π

2 + o(1) E1 μˆ  (Z1 )

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μˆ  (Z1 ) =

E1 + o(1) 2(hN(h, E))2

as E ↓ E1 , h ↓ 0.

(Z1 , Z2 are still unknown.) With ±f± (E) > 0 for E ∈ ]E1 , E2 [, we then find 5 2f±

= ±Φ + s Φ 2 − 4

Φ Ψ

(24)

with s = sign(f+ + f− ) is not (yet) determined, and only if the well is mirrorsymmetric with respect to its vertex then f+ + f− = 0 and the square root in (24) vanishes. By Assumption 7, the function s = s(E) is constant for E ∈ ]E1 , E2 [. Hence, in what follows we may exchange s with ±. So we proceed by computing two versions and then choose the right version which glues smoothly to the rest of the profile. We have 1 f+ (E) = Z1 + 2 f− (E) = Z1 +

1 2



! Φ Φ ± Φ2 − 4 dE, Ψ ! 5 Φ −Φ ± Φ 2 − 4 dE. Ψ 5

E

E1



E

E1

Since f+ (E2 ) = Z2 and f− (E2 ) = Z− , we find that 1 Z2 = Z1 + 2 Z− = Z1 +

1 2



E2

E1



E2

E1

! Φ Φ ± Φ2 − 4 dE, Ψ ! 5 Φ 2 −Φ ± Φ − 4 dE. Ψ 5

Hence, the distance, Z2 − Z1 , between the two critical points is recovered (modulo mirror symmetry of Z1 with respect to 2c ). Since f± are both monotonic on ]E1 , E2 [, μˆ can be recovered (up to mirror symmetry) on I . With this result, the reconstructions on [Z1 , 0] and I can be smoothly glued together, and the uncertainty in the translation of I and the “orientation” of μˆ on I are eliminated. Thus, μˆ is uniquely determined on the interval [Z− , 0]. Step 3. On the interval [ZI , Z− ] we may use the Weyl asymptotics again to recover μ. ˆ The counting function in the interval [E2 , E3 ] is obtained from  S03 2 which corresponds with Area({(Z, ζ ) : μ(Z)(1 ˆ + ζ ) ≤ E}) = A1 (E) + A2 (E), where A1 (E) = Area({(Z, ζ ) : μ(Z)(1 ˆ + ζ 2 ) ≤ E, Z− ≤ Z ≤ 0}) is already known, and  A2 (E) = 2

Z− f (E)

6

E − μˆ dZ, μˆ

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ZI ≤ f (E) < Z− since E2 ≤ E ≤ E3 = μˆ I . Thus, we may recover μˆ on the interval [ZI , Z− ] where μˆ is decreasing by (22). The two profiles for μˆ on [ZI , Z− ] and on [Z− , Z2 ] are then glued together at Z = Z− which is already known. This completes the reconstruction procedure.

4.4 Reconstruction of the Profile from the Actions: Rayleigh Case By Assumption 3 and using Theorem 3.2 we can follow the procedure for Love waves with a few modifications coming only from the change of symbol of the scalar Love-type operator of order h2 . Then only the second action S22,1 is changed, which affects the reconstruction of a semiclassical well separated from the boundary. By abuse of notation, we keep using the same symbols as for the Love case. The well Wjk might be a new well. Then we define the functions f± : [Ek−1 , Ek [→ I so that Wjk (E) = [f− (E), f+ (E)] for any E ∈ [Ek−1 , Ek [. Case II. The well Wjk might also be joining two wells of order k −1, or extending a single well of order k − 1. Note that the profile under Ek−1 has already been recovered. The smooth joining of two wells can be carried out under Assumption 7. We consider now functions f− (E) and f+ (E) for E ∈ [Ek−1 , Ek ] such that Wjk is the union of three connected intervals, Case I.

Wjk (Ek ) = [f− (Ek ), f− (Ek−1 ) × [∪ [f− (Ek−1 ), f+ (Ek−1 )] ∪ ]f+ (Ek−1 ), f+ (Ek )]. For an illustration, see Fig. 1. For either case, we define Φ(E), Ψ (E) by (23). As for Love waves, the function k,j Φ can be recovered from S0 (E), on ]Ek−1 , Ek [. From (different from the Love k,j case) S2 (E), we recover BΨ (E) =

E









E  √ du Ek−1 (7E − 6u)Ψ (u) − 2 u − 1 Ψ (u) u(E−u)    E 1 E−u  − Ek−1 36Ψ (u) − 24σ u Ψ (u) arctan u du,

Ek−1 < E < Ek , where σ = 8ν(1 − 2ν). We introduce operator T according to  T g(E) =

E

Ek−1

g(u) du. √ E−u

By setting E = z2 , we end up with a third-order inhomogeneous ordinary differential equation,

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2 2 d3 z (T ◦BΨ )(z2 ) = 16z6 Ψ  (z2 )−192z4 Ψ  (z2 )+96(2−σ )z2 Ψ  (z2 )−96σ Ψ (z2 ) π dz3 √ √ for Ψ (z2 ) nonsingular on the interval [ Ek−1 , Ek [ . This equation is supplemented with “initial” conditions: For Case I, Ψ (Ek−1 ) and the asymptotic behaviors of Ψ  (E) and Ψ  (E) for E in a neighborhood of Ek−1 can be extracted from T ◦ BΨ (E) and its derivatives at Ek−1 . Clearly, Ψ (Ek−1 ) = 0. Using the derivatives and Ψ (Ek−1 ) = 0,

lim

E↓Ek−1

  E − Ek−1 Ψ  (E) = 2μˆ  (Zk−1 ),

we obtain, for E > Ek−1 close to Ek−1 ,   2 d 4EΨ  (E) − (T ◦ BΨ )(z2 ) = 0 E↓Ek−1 π dz lim

yielding the asymptotic behavior of Ψ  (E), and  lim

E↓Ek−1

−108E

1/2



Ψ (E) + 8E

3/2

 d2 2 Ψ (E) − 2 (T ◦ BΨ )(z ) = 0 dz 

yielding the asymptotic behavior of Ψ  (E). With these, the solution to the thirdorder inhomogeneous ordinary differential equation is unique. For Case II, Ψ (Ek−1 ), Ψ  (Ek−1 ) and Ψ  (Ek−1 ) are all nonsingular. That is, if Ek−1 is a local maximum, Ψ and all its derivatives are smooth from above and below, and therefore they can be recovered from the reconstruction on Jk−1 through one-sided limits. We note that in case Ek−1 is a local maximum in the middle of two wells in Jk−1 the two different Ψ s for each well are not smooth below Ek−1 , but it does not matter as in Jk (above Ek−1 ) we use f± from the monotonically increasing slopes continued from Jk−1 . Thus, the solution to the third-order inhomogeneous ordinary differential equation is also unique. With the recovery of Φ and Ψ we can recover f± and then μˆ as in the case of Love waves, again, subject to a gluing procedure.

References 1. Y. Colin de Verdière, Semiclassical analysis and passive imaging. Nonlinearity 22, R45–R75 (2009) 2. Y. Colin de Verdière, A semiclassical inverse problem II: reconstruction of the potential. Geometric Asp. Anal. Mech. 292, 97–119 (2011) 3. S. Crampin, Higher modes of seismic surface waves: Phase velocities across Scandinavia. J. Geophys. Res. 69, 4801–4811 (1964)

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4. M.V. De Hoop, A. Iantchenko, G. Nakamura, J. Zhai, Semiclassical Analysis of Elastic Surface Waves (2017). arXiv:1709.06521 5. M.V. de Hoop, A. Iantchenko, R.D. Van der Hilst, J. Zhai, Semiclassical Inverse Spectral Problem for Elastic Love Surface Waves in Isotropic Media. arXiv:1908.10529 6. M.V. de Hoop, A. Iantchenko, R.D. Van der Hilst, J. Zhai, Semiclassical Inverse Spectral Problem for Elastic Rayleigh Waves in Isotropic Media. arXiv:1908.11698 7. A. Derode, E. Larose, M. Tanter, J. de Rosny, A. Tourin, M. Campillo, M. Fink, Recovering the Green’s function form field-field correlations in an open scattering medium. J. Acoust. Soc. Am. 113, 2973–2976 (2004) 8. M. Dimassi, J. Sjöstrand, in Spectral Asymptotics in the Semiclassical Limit. London Mathematical Society Lecture Note Series, vol. 268 (Cambridge Univ. Press, Cambridge, 1999) 9. N.A. Haskell, The dispersion of surface waves on multilayered media. Bull. Seismol. Soc. Am. 43, 17–34 (1953) 10. B. Helffer, J. Sjöstrand, Multiple wells in the semi-classical limit I. Commun. PDE 1, 1934– 1944 (1984) 11. B. Helffer, in Semi-Classical Analysis for the Schrödinger Operator and Applications. Lecture Notes in Mathematics, vol. 1336 (Springer, 1989) 12. L. Knopoff, Green’s function for eigenvalue problems and the inversion of dispersion data. Geophys. J. Int. 4, 161–173 (1961) 13. S. Lebedev, R.D. van der Hilst, Global upper-mantle tomography with the automated multimode inversion of surface and S-wave forms. Geophys. J. Int. 173, 505–518 (2008) 14. A. Love, Some Problems of Geodynamics (Cambridge Univ. Press, Cambridge, 1967) 15. R.G. Newton, Scattering Theory of Waves and Particles, 2 edn. (Dover, New York, 2002) 16. G. Nolet, Higher Rayleigh modes in Western Europe. Geophys. Res. Lett. 2, 60–62 (1975) 17. J. Rayleigh, On waves propagated along the plane of an elastic solid. Proc. Lond. Math. Soc. 17, 4–11 (1885) 18. N. Shapiro, M. Campillo, L. Stehly, M. Ritzwoller, High resolution surface wave tomography from ambient seismic noise. Science 307, 1615–1618 (2005) 19. K. Wilmanski, in Surface Waves in Geomechanics: Direct and Inverse Modelling for Soils and Rocks., vol. 481. chapter Elastic Modelling of Surface Waves in Single and Multicomponent Systems, pp. 203–276 (Springer, Vienna, 2005) 20. Y. Yang, M.H. Ritzwoller, A.L. Levshin, N.M. Shapiro, Ambient noise Rayleigh wave tomography across Europe. Geophys. J. Int. 168, 259–274 (2007) 21. H. Yao, R.D. van Der Hilst, M.V. De Hoop, Surface-wave array tomography in SE Tibet from ambient seismic noise and two-station analysis. Geophys. J. Int. 166, 732–744 (2006)

Persistence of Point Spectrum for Perturbations of One-Dimensional Operators with Discrete Spectra César R. de Oliveira

and Mariane Pigossi

MSC (2010) 47B39, 81Q15, 47N50

1 Introduction We study spectral and dynamical properties of Schrödinger operators obtained by suitable perturbations of the unbounded self-adjoint operator H1 , acting in the Hilbert space 2 (Z), with purely discrete spectrum, whose eigenvalues {λm }m∈Z satisfy |λm − λn |  β,

∀ m = n,

(1)

for some β > 0. The corresponding normalized eigenfunctions are denoted by {ψm }m∈Z . This is a rather common situation in (quantum) physics. We consider two kinds of perturbations, autonomous one M and time-periodic Q(t). We present sufficient conditions for the persistence of the pure point spectrum and the presence of dynamical localization after perturbations; due to (1), a general argument (see Theorem 4.1 in [8]) implies that bounded perturbations of H1 have discrete spectra, however, since for the proof of dynamical localization one usually needs certain control of eigenfunctions and matrix elements of the diagonalization operator, the linear operator version of the KAM technique will be employed, without mention the usual occurrence of dense point spectra in the time-periodic case. This work is a generalization of recent results by the same authors [7, 8], where H1 was the particular case of the discrete 1D Schrödinger operator for a particle under a uniform electric field. Since there are relatively few cases for which this

C. R. de Oliveira () Departamento de Matemática, UFSCar, São Carlos, SP, Brazil e-mail: [email protected] M. Pigossi Departamento de Matemática, UFES, Vitória, ES, Brazil © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 P. Miranda et al. (eds.), Spectral Theory and Mathematical Physics, Latin American Mathematics Series, https://doi.org/10.1007/978-3-030-55556-6_7

125

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C. R. de Oliveira and M. Pigossi

version of the KAM technique works, in this manuscript we have tried to push the arguments to a more general version of previous results [7, 8]; in particular, (1) is the only spectral hypothesis on the unperturbed system (specific hypotheses on its eigenfunctions appear in the theorems below). The introduction of suitable spaces and conditions on the autonomous perturbation, including the statement of our main result in this case, is the subject of Sect. 1.1, whereas the corresponding statements for the time-periodic case appear in Sect. 1.2. The proofs are discussed in Sects. 2 (autonomous case) and 3 (time-periodic case).

1.1 Autonomous Case Consider the one-dimensional Schrödinger operator H , acting in the Hilbert space 2 (Z), with action (H ψ)(n) := (H1 ψ)(n) + (Mψ)(n),

ψ ∈ l 2 (Z),

(2)

with both H1 and M self-adjoint operators and H1 as discussed above. We are particularly interested in studying the dynamical property called dynamical localization (see [19] and references therein), which is probed through the moments 7 8 p rψ (t) := e−itH ψ, |X|p e−itH ψ of order p > 0, where |X| denotes the absolute value of the position operator so that (|X|p ψ)(n) = |n|p ψ(n); they measure how fast the dynamics spread over the lattice Z, in particular whether such spreading ceases or not. Let (en ) be the usual basis of 2 (Z), that is, en (m) = δnm . One says that H has dynamical localization (DL) if, for all initial conditions en , each moment is uniformly bounded in time, i.e., for each p > 0 and n ∈ Z, p

sup ren (t) < ∞ .

(3)

t

This is the usual notion of absence of transport for Schrödinger operators in Z and, physically, the system would be considered an insulator. It is known that dynamical localization requires pure point spectrum of the Hamiltonian H ; however, there are examples of systems with unusual behavior, that is, pure point spectrum without dynamical localization [9, 10, 18] (see also [1] for new results in case of operators with dense point spectrum and generic initial conditions). In [8] we have shown that H has discrete spectrum since M ∈ ∞ (Z). This general argument for point spectrum of H does not allow us to conclude the presence of dynamical localization. We need more detailed information on its eigenfunctions. Our main goal now is to show that for Mr small enough, DL

Persistence of Point Spectrum for Perturbations of One-Dimensional Operators. . .

127

is present. For this, as we mentioned above, we will employ a diagonalization method based on the well-known iterative procedure due originally to Kolmogorov, Arnold and Moser (KAM), and initially implemented in the quantum setting of timeperiodic perturbations in [2, 4]. The above spectral and dynamical questions are technically challenging since they involve perturbations of unbounded operators; this is another justification for using the KAM technique. If the perturbation M, in this autonomous case, is a potential, then we may have two kinds of “smallness” conditions, either through a specific norm  · r (4) or the l ∞ (Z) norm (see Remark 1.2). We underline that a condition in terms of the l ∞ norm is quite natural and was a novelty in [8] that can be replicated here. Denote by Ar (r ≥ 0) the Banach space of linear operators a in 2 (Z) with matrix representation a = {anm } in the orthonormal basis {ψm } of eigenvectors of H1 , that is, anm := ψn , aψm , n, m ∈ Z, and finite norm ar := sup n



|anm | er|n−m| < ∞.

(4)

m

A linear operator a is diagonal (with respect to the basis {ψm }) if anm = 0 for n = m; for a general linear operator a we denote its diagonal by diag a, that is, by replacing anm with zero if n = m. [a, c] := ac − ca is the commutator of the 9 −(j +2)/2j +1 . Our main result in this setting operators a and c. Let γ := 5−1 ∞ j =0 2 is the following: Theorem 1.1 Fix r > 0. Let the unperturbed operator H1 , {ψm }, β and γ be as above. If the perturbation M satisfies Mr < min

β 2

,γ ,

then there exists a unitary operator P ∈ Ar such that P −1 (H1 + M)P is diagonal with discrete and simple spectrum and whose eigenfunctions are {P ψm }m∈Z . Furthermore, if there are c1 > 0, ρ > 0, so that |ψm (n)|  c1 e−ρ|n−m| ,

∀n, m,

then H = H1 + M has dynamical localization. Remark 1.2 If M is a potential, that is, a real-valued multiplication operator (Men = M(n)en , with M(n) a real number, for all n), then the hypothesis about the “size” of the allowed perturbation may also be given in terms of the l ∞ (Z) norm (instead of the  · r norm), that is, in this case the conclusions of Theorem 1.1 hold if

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C. R. de Oliveira and M. Pigossi

M∞ = sup |M(n)|  n

β , 2 d0

(5)

where d is the linear operator with matrix elements dnm =



e−ρ(|n−l|+|m−l|) .

l∈Z

The proof in this case is an adaptation of the proof of Theorem 1.1 and will be omitted here; for details how to perform such adaptation, see the proof of the main result of [8]. Remark 1.3 Dynamical localization in Theorem 1.1 will follow by checking that the eigenfunctions of the perturbed operator satisfy the SULE condition [10, 19], that is, there exist α > 0 and, for each eigenfunction P ψm , an nm ∈ Z so that, for any δ > 0, |P ψm (n)|  C(δ) eδ|nm |−α|n−nm | ,

∀n,

with the same parameter C(δ) for all m.

1.2 Time-Periodic Case Now we pass to time-periodic perturbations Q(t) of H1 in the Howland’s Formalism [5, 16, 17], which considers the space K := L2 ([0, 2π ], dt) ⊗

2

(Z)

(6)

and the quasi-energy operator K = −iτ ∂t + H1 + Q(t)

(7)

acting in K; the time variable t is incorporated as a “spatial variable” and the operator depends on the frequency τ > 0 of the perturbation in original variables (the original period is 2π/τ ), which in (6)–(7) appears as 2π -periodic (a change of variable). The differential operator −iτ ∂t acts in L2 ([0, 2π ], dt) with periodic boundary conditions, whose eigenvalues√are kτ , k ∈ Z, and the corresponding normalized eigenvectors are χk (t) = eikt / 2π . Denote by K0 the (free) quasi-energy operator obtained with the particular choice Q = 0, that is, K0 = −iτ ∂t + H1 ,

(8)

Persistence of Point Spectrum for Perturbations of One-Dimensional Operators. . .

129

which has a complete set of orthonormal eigenvectors eikt ϕm,k (t, n) := √ ⊗ ψm (n), 2π

m, k ∈ Z,

(9)

whose eigenvalues are λm,k = kτ + λm , m, k ∈ Z (in may instances a dense set in R). ρ To formulate our second main result, we need to introduce the space Bσ 0 (0, ∞), for ρ0  0, σ  0, of the set of maps h from (0, ∞) to the space of bounded linear operators on K such that, for each τ1 ∈ (0, ∞), h is represented by the matrix hn,m,k1 ,k2 (τ1 ), n, m, k1 , k2 ∈ Z, with [4] hρ0 ,σ :=

sup

sup

   hn,m,k

τ1 ,τ2 ∈(0,∞) n∈Z m∈Z k ∈Z 1 k2 ∈Z τ1 =τ2

1 ,k2

    ˜ n,m,k1 ,k2 (τ1 , τ2 ) + ⇔ (τ1 ) + ∂h

× (1 + |n + m|)σ eρ0 (|k1 −k2 |+|n−m|) < ∞ . The above matrix representation of the operator h in the basis {ϕm,k } of eigenvectors   of K0 is hn,m,k1 ,k2 (τ ) := ϕn,k1 , h ϕm,k2 , where ·, · denotes the inner product in K and ∂˜ designates the discrete derivative ˜ n,m,k1 ,k2 (τ1 , τ2 ) := hn,m,k1 ,k2 (τ1 ) − hn,m,k1 ,k2 (τ2 ) . ∂h τ1 − τ2 As in [4, 7], we have used the symbol ⇔ to indicate |hm,n,k2 ,k1 (τ1 )| + ˜ m,n,k2 ,k1 (τ1 , τ2 )|; however, due to symmetry, we will not explicitly consider |∂h these terms in estimates. Since the concepts of dynamical localization and SULE are not standard in the time-periodic case, we dwell on the discussion in what follows. The adaptation [7] of the concept of (spatial) dynamical localization for the operator K is that for every ikt p > 0 and each initial condition of the form ζ m,k = √e ⊗ em , 2π

p

sup r(m,k) (u) < ∞,

(10)

u

  p where r(m,k) (u) := e−iuK ζ m,k , |X|p e−iuK ζ m,k . Such dynamical localization will follow from a SULE condition for K [10, 19] (see also the specific discussion in [7] for the time-periodic case), that is, there exist α > 0 and, for each eigenfunction ϕm,k of K, an nm ∈ Z so that, for any δ > 0,     ϕk,m (t, n) = eikt ⊗ ψm (n) ≤ C(δ) eδ|nm |−α|n−nm | ,

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with the same C(δ) for all k, n, t, m. This will be our strategy to prove DL in the time-periodic setting. In what follows, |I | denotes the Lebesgue measure of the (measurable) set I ⊂ R. Theorem 1.4 Let H1 and β > 0 be as above. Given ρ0 > 0, σ > 1, pick 0 < r < ρ0 with er  σ . If for some  > 0 there exist  > 0 and 1 > 0 so that 

Qρ0 ,4σ

 1 1 , , (β − 1 ) , < min 4 2

(12)

1

then there is I ⊂ (0, ∞) with |I | < 12  2 so that, for τ ∈ (0, ∞) \ I , there is a ρ −r unitary operator P ∈ B0 0 (0, ∞) implying that P −1 K P

(13)

has pure point spectrum. The eigenvectors of K are of the form P ϕm,k , m, k ∈ Z. Furthermore, if the eigenvectors ϕm,k of K0 satisfy   ϕm,k (j, n)  c e−s|m−n| ,

∀m, k, j, n,

(14)

for some c > 0 and s > 0 with s > ρ0 − r, then K has dynamical localization. Remark 1.5 Note that Theorem 1.4 does not hold for β = 0, and for β = 0 the condition β > 1  is always satisfied for  small enough. Explicitly, one may take =

80 L(σ )2

r 10σ +1 

 −j −1 (8σ +1)  , ∞ 2−j −1 (10σ +3) 22−j (j +1)(8σ +1) er2−2j −1 32−j (σ +1) σ 2 j =0 2 e

9

1 = with L(σ ) =

& k

 e r 4σ +1 e−2r , 16 L(σ )2 22σ 3σ +2 σ

(1 + |k|)−σ

&

n,m (1 + |n − m|)

−σ (1 + |n + m|)−2σ

.

2 Proof of Theorem 1.1 We refer the readers interested in recalling the general idea of the KAM method in operator setting to the references, in particular [2, 8, 14]. Lemma 2.1 gives a suitable solution to a “commutator equation” that is successively applied in the KAM iteration scheme; its proof is very similar to the proof of Lemma 5.3 in [8] and will not be repeated here. Lemma 2.1 Let G be a diagonal operator and B ∈ Ar with (G + diag B)nn = gn + Bnn ∈ R, and

Persistence of Point Spectrum for Perturbations of One-Dimensional Operators. . .

|(gn + Bnn ) − (gm + Bmm )| > K,

∀m = n,

131

(15)

for some K > 0. Given B ∈ Ar , there exists W ∈ Ar solving   G + diag B, W + B − diag B = 0 ,

(16)

with Wnn = 1 for all n. Furthermore, W − 1r  C Br ,

(17)

with C = 1/K. The following well-known remark is simple and important in the proofs below. If a is a bounded linear operator on a Banach space with a − 1 < 1, then a is invertible in the set of bounded operators and  −1  a − 1 ≤

a − 1 . 1 − a − 1

(18)

2.1 Implementation 2.1.1

First Step k = 0

Put D = D 0 = H1 , M 0 = M and P 0 = 1; thus

 



(P 0 )−1 D + M P 0 = D 0 + M 0 = D 0 + diag M 0 + M 0 − diag M 0 .

(19)

D 0 +diag M 0 is a diagonal operator with (D 0 +diag M 0 )nn = λn +(diag M 0 )nn =: λ0n and clearly   0  λ − λ 0  > 1 , n m for some 0 < 1  1. By Lemma 2.1, for each fixed r > 0, there exists a solution W 0 ∈ Ar to   D 0 + diag M 0 , W 0 + M 0 − diag M 0 = 0 , (20) with

 diag W 0 − 1 = 0, and for C0 = 1/1 ≥ 1,

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  0   W − 1  C0 M 0  . r r

(21)

−1

 Then, W 0 (P 0 )−1 D + M P 0 W 0 equals

−1 0

−1 0   = W0 D + diag M 0 W 0 + W 0 M − diag M 0 W 0  −1

0  (20) 0 −1 0 0 W D + diag M 0 − W 0 M − diag M 0 = W

−1 0  + W0 M − diag M 0 W 0 =: D 1 + M 1 .

(19)

Hence, (P 1 )−1 (D + M)P 1 = D 1 + M 1 , with ⎧ 1 0 0 ⎪ ⎪P = P W ⎪ ⎪ ⎪ ⎨ D 1 = D 0 + diag M 0 ⎪ ⎪ ⎪ ⎪ ⎪ 

 ⎩ 1 0 −1 0 M − diag M 0 W 0 − 1 . M = W By (18), W 0 is invertible and  0  0   W − 1 W − 1 r   r  1 − W 0 − 1r 1 − C0 M 0 r   ≤ 2W 0 − 1r < 1 ,

 0 −1   W − 1r ≤

in case   1 C0 M 0 r < < 1; 2 this will be verified ahead. One has  1  



 M  =  W 0 −1 M 0 − diag M 0 W 0 − 1  r r  0 −1   0   0   M  W − 1 . ≤ W r r r Since          0 −1   W  =  W 0 −1 − 1 + 1   W 0 −1 − 1 + 1 < 2 , r r r r

(22)

Persistence of Point Spectrum for Perturbations of One-Dimensional Operators. . .

133

one then defines θ1 through        1 M   2M 0  W 0 − 1  2 C0 M 0 2 =: θ 21 . 1 r r r r 2.1.2

(k + 1)th Step

Suppose we can find, for some k ∈ N, (i) An operator M k ∈ Ar satisfying  k M   θ 2k , k r with θk =

k−1 "

2−j −1 (j +1)  0  M  , 2Cj r

(23)

j =0

and “initial condition”     M  = M 0   θ0 . r r

(24)

k (ii) A diagonal operator D k with (D k +diag M k )nn = λkn = λk−1 n +Mnn satisfying, for some Kk > 0,

 k  λ − λk  > k+1 , n m

∀ n = m ,

(25)

so that 

k −1

D + M P k = Dk + M k , P

(26)

with P k = W 0 W 1 · · · W k−1 . In the following we show that (i)–(ii) hold true for k + 1. By Lemma 2.1, there exists a solution W k ∈ Ar to 

 D k + diag M k , W k + M k − diag M k = 0 ,

with

 diag W k − 1 = 0 . Furthermore, with Ck = 1/k ≥ 1,

(27)

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C. R. de Oliveira and M. Pigossi

  k   W − 1  Ck M k  . r r

(28)



−1 k −1

P D + M P k W k equals Then, W k  

−1 k

−1 k D + diag M k W k + W k M − diag M k W k = Wk  −1

k  (27) k −1 k k W D + diag M k − W k M − diag M k = W

−1 k + Wk (M − diag M k )W k =: D k+1 + M k+1 .

(19)

Hence, 

k+1 −1

D + M P k+1 = D k+1 + M k+1 , P with ⎧ ⎪ P k+1 = P k W k ⎪ ⎪ ⎪ ⎪ ⎨ D k+1 = D k + diag M k ⎪ ⎪ ⎪ ⎪ ⎪ 

 ⎩ k+1 k −1 k M − diag M k W k − 1 . M = W By (18), W k is invertible and  k  k   W − 1 W − 1 r     r 1 − W k − 1r 1 − Ck M k r   ≤ 2 W k − 1 r < 1 ,

 k −1   W − 1r ≤

in case   1 Ck M k r < < 1 ; 2 this will be verified ahead. One has  k+1   



 M  =  W k −1 M k − diag M k W k − 1 | r r  k −1   k   k   M  W − 1 . ≤ W r r r Since          k −1   W  =  W k −1 − 1 + 1   W k −1 − 1 + 1 < 2 , r r r r

(29)

Persistence of Point Spectrum for Perturbations of One-Dimensional Operators. . .

135

one then defines θk+1 through      k+1  M  ≤ 2M k  W k − 1  2 Ck θ 2k+1 =: θ 2k+1 . k k+1 r r r 2.1.3

Convergence

In order to show that the above iterative process converges as k → ∞, we verify (I) θk → θ∞ . (II) P k is a Cauchy sequence in Ar , so that there exists P ∈ Ar with P k → P , P = 0. (III) M k → 0 in Ar . Verification of (I ) We verify that there exists 0 < θ∞ < ∞ so that θk → θ∞ as k → ∞. Note that we may suppose that 1 ≤ Ck increases with k, thus θk is increasing as well. By (23) and (24), it follows that θk =

k−1 k−1 "

"

2−j −1  0  2−j −1 M  ≤ θ0 . 2Cj 2C j r j =0

j =0

So, if Cj does not grow too fast as j → ∞, θk converges to some θ∞ < ∞; for instance, this happens if 1 ≤ Cj  2j +1 (and so Kj = 2j1+1 ). So, from now on we choose Cj = 2j +1 . Pick 0 < h < 1/5 so that ∞ "

2−j −1 h 2Cj < , θ0

j =0

and naturally γ from the statement of Theorem 1.1 shows up, since we need to select θ0 so that (recall that Mr ≤ θ0 ) θ0
0, for all n = m and all 0 ≤ k ≤ ∞). k     Since Mnn ≤ M r and θ0 = Mr , by (32), for all n we have

  k     k 2k −1 M k  ≤ M   θ h = M h2 −1 . 0 r nn r k

k

k

Now it is enough to impose that β Mr < & . 2k −1 2 k≥0 h

k

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C. R. de Oliveira and M. Pigossi

&  2k −1 = 1, if M < β/2 it is sufficient to take 0 < h Since, the limh↓0 h r k≥0 small enough for (34) be satisfied, and the convergence of the whole process is justified. Remark 2.2 The fact that the operator P can be chosen unitary follows by the proof of the Corollary of Theorem 2.1 in [4]. If for some c, ρ > 0 the eigenfunctions {ψm } of H1 satisfy |ψm (n)|  c e−ρ|n−m| ,

∀m, n ∈ Z ,

(35)

we can show that the H operator has dynamical localization. Indeed, since P ∈ Ar there is and C > 0 so that |Pnm |  Ce−r|n−m| ,

∀m, n,

and by (35) one has 

|ψm (n)|  C1

e−ρ1 |m −n| , (1 + |m − n|2 )

for any ρ1 < ρ, for all m , n ∈ Z and for some C1 > 0. By picking 0 < ρ1 < r, for {P ψm } one has   en , ψk ψk , P ψm  |(P ψm ) (n)| = |en , P ψm | =  k





|Pkm | |ψk (n)| 

k



C e−r|k−m|

k



Ce−ρ1 |k−m|

k

 C2



 k



e−ρ1 |m −n| (1 + |m − n|2 )

C1 e−ρ1 |n−k| (1 + (n − k)2 )

 e−ρ1 |n−m| C2 = e−ρ1 |n−m| . 2 (1 + (n − k) ) (1 + (n − k)2 ) k

Since the last sum is convergent and its value is independent of m, n, one has  e−ρ1 |n−m| , |(P ψm ) (n)|  C a particular case of SULE. The proof of Theorem 1.1 is complete.

(36)

Persistence of Point Spectrum for Perturbations of One-Dimensional Operators. . .

139

3 Proof of Theorem 1.4 3.1 Initial Lemmas Let (λnk1 (τ ))n,k1 ∈Z be the sequences of functions (0, ∞)  τ −→ λnk (τ ) := λn + k1 τ + Qn,n,k,k (τ ), where Qn,n,k1 ,k1 (τ ) are the diagonal matrix elements of the operator Q, also denoted by diag Q (here, an operator a is diagonal if an,m,k1 ,k2 = 0 if n = m or k1 = k2 ). Note that for all n, m, k1 , the matrix elements     Qn,m,k1 ,k1 = ϕn,k1 , Qϕm,k1 = ψn , Qψm = Qn,m are independent of k1 , so that we may write Qn,m for the matrix elements of the time average Q. Also note that all Qn,m,k1 ,k2 depend only on (k1 − k2 ). We will need to estimate λnk1 (τ ) − λmk2 (τ ) = λn − λm + τ (k1 − k2 ) + Qn,n,k1 ,k1 − Qm,m,k2 ,k2 , which uses rather different approaches depending on k1 = k2 or k1 = k2 . Some values of the parameter τ are then discarded during iterations, and the measure of the discarded set will be controlled; that is the goal of next lemma. Lemma 3.1 If, for some σ > 1, Qρ0 ,4σ
0, there exists 0 < K < 1/2, so that the set   I = τ ∈ (0, ∞) | ∃ k1 = k2 ∈ Z, λnk1 (τ ) − λmk2 (τ ) <

K , =: η nmk (1 + |n − m|)σ (1 + |k1 − k2 |)σ (1 + |n + m|)2σ 1

where k = k1 − k2 = 0, has Lebesgue measure smaller than  2 . Proof It is enough to consider sets Inmk of the form

  Inmk (ηnmk ) = τ ∈ (0, +∞) | λnk1 (τ ) − λmk2 (τ ) < ηnmk ,  since I ⊂ k =0,n,m Inmk . By taking τ1 , τ2 ∈ Inmk (ηnmk ), with τ1 = τ2 , we have

k = 0 ,

140

C. R. de Oliveira and M. Pigossi

2ηnmk ηnmk + ηnmk = |τ1 − τ2 | |τ1 − τ2 |     λnk (τ1 ) − λmk (τ1 ) λnk (τ2 ) − λmk (τ2 ) 1 2 1 2 > + |τ1 − τ2 | |τ1 − τ2 |     Qn,n,k1 ,k1 (τ2 ) − Qn,n,k1 ,k1 (τ1 )  k1 − k2  − |τ1 − τ2 |   Qn,m,k ,k (τ1 ) − (Q)n,m,k ,k (τ2 ) 2 2 2 2 − |τ1 − τ2 |     ˜  ˜  ∂Q − = |k1 − k2 | − ∂Q (τ , τ ) (τ , τ )   n,n,k1 ,k1 1 2 n,m,k2 ,k2 1 2  =: |k| − 2Q(τ1 , τ2 ) ; hence, 2ηnmk  |τ1 − τ2 | . |k| − 2Q(τ1 , τ2 ) Since 2Q(τ1 , τ2 ) < end of Sect. 3.2),

1 2

(38)

(it is sufficient that the condition (37) holds true; see the

|τ1 − τ2 | 

2ηnmk |k| −

 4 ηnmk ,

1 2

and so |Inmk (ηnmk )|  4ηnmk . By denoting i = n − m and j = n + m, |I | 



|Inmk (ηnmk )| 

k =0,n,m

 k,i,j

4K (1 + |k|)σ (1 + |i|)σ (1 + |j |)2σ

= 4K L(σ ). 1

Since σ > 1, L(σ ) converges. Taking K =  2 /(4L(σ )), the result follows.

 

As in the proof of Theorem 1.1, the proof of the following “commutator equation” (i.e., Lemma 3.2) will not be repeated here, since it is analogous to the proof of Lemma 4.3 in [7]. ρ

Lemma 3.2 Fix ρ0 > 0. Let G be a diagonal operator and Q ∈ B4σ0 (0, ∞) with (G + diag Q)n,n,k1 ,k1 (τ1 ) = gnk1 (τ1 ) := λn + k1 τ1 + Qn,n,k1 ,k1 satisfying, for each τ1 ∈ (0, ∞) and σ > 1,   gnk (τ1 ) − gmk (τ1 ) > 1 2

(1 + |n − m|)σ (1 + |k

K σ 2σ 1 − k2 |) (1 + |n + m|)

(39)

Persistence of Point Spectrum for Perturbations of One-Dimensional Operators. . .

141

for k1 = k2 and some K > 0 (by (40), if k1 = k2 and m = n, one has (43)). Given ˜ if 0 < r < ρ0 with r ≤ eσ and β > ,  Qρ0 ,4σ < min with ˜ 2 = K 2 3−(2σ +1) 22σ e2r solution to

er 4σ +1 σ

 1 1 ˜ , , (β − ) 4 2

(40) ρ −r

, then there exists W ∈ B0 0

  G + diag Q, W + Q − diag Q = 0 ,

(0, ∞),

(41)

with Wn,n,k1 ,k1 (τ1 ) = 1 for all n, k1 and τ1 > 0. Furthermore, W − 1ρ0 −r,0 

32σ +2 22σ 2r  σ 4σ +1 Qρ0 ,4σ . e er K2

(42)

Remark 3.3 In case k = k1 − k2 = 0 and n = m, one has, by (40), |λnk1 (τ ) − λmk1 (τ )| = |λn − λm + Qn,n − Qm,m |  β − 2Qρ0 ,4σ > ˜ > 0 . (43) So, restrictions to the values of the frequency τ only occur for k = 0, as described in Lemma 3.1.

3.2 Implementation Now we implement the iteration process for time-periodic perturbations.

3.2.1

First Step p = 0

Put D = D 0 = K0 , Q = Q0 , B0 = (0, ∞) and P 0 = 1; thus

 



(P 0 )−1 D + Q P 0 = D 0 + Q0 = D 0 + diag Q0 + Q0 − diag Q0 .

(44)

D 0 + diag Q0 is the diagonal operator with (D 0 + diag Q0 )n,n,k,k = λ0nk1 (τ ) = En + k1 τ + Q0nnkk (τ ) and it is assumed that      0  λnk1 (τ ) − λ0mk2 (τ ) > K0 (1 + |k1 − k2 | + |n − m|)−σ (1 + |k1 + k2 | + n + m)−2σ , with τ ∈ B0 . By Lemma 3.2, for some K0 > 0, 0 < r0 < ρ0 and τ ∈ B1 = B0 − I1 , where

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C. R. de Oliveira and M. Pigossi

I1 = τ > 0 | ∃k1 , k2 ∈ Z, k1 = k2 ;     0 λnk1 (τ ) − λ0mk2 (τ )
0, ¯ = ∇ × A(h) ¯ = (0, 0, (h)B), ¯ (h) ¯ = h¯ q for a suitable chosen positive constant q (see below). 1 (iii) V (x) = − |x| is the Coulomb potential and

214

C. Pérez-Estrada and C. Villegas-Blas

  ∂ ∂ (iv) the operator h¯ L3 = −ı h¯ x1 ∂x − x 2 ∂x1 is the component of the angular 2 momentum operator hL along the direction of the magnetic ¯ = x×(−ı h)∇ ¯ field B(h). ¯ We now follow the mechanism implemented in reference [14] and explained above by taking h¯ = N 1+1 in order to get well defined clusters of eigenvalues of HV (h, ¯ B) around E = −1/2 for N sufficiently large. Here the main problem is that we are dealing with a non-bounded perturbation of the hydrogen atom Hamiltonian HV (h). ¯ The key point is that by taking the strength of the magnetic field (h)B ¯ sufficiently small, we can guarantee the existence of those clusters through a stability theorem. Let us describe briefly the analysis (see [2] for details): Let us introduce the unitary dilation operators defined on L2 (R3 ) for each α > 0: (Dα f )(x) = α 3/2 f (αx),

f ∈ L2 (R3 ).

(37)

Then we have Dh¯ 2 HV (h, ¯ B)Dh¯ −2 ; < 2

3 h¯ (h)B 1 1 h¯ 3 (h)B 1 ¯ ¯ 2 2 + (x1 + x2 ) − L3 = 2 − Δ− 2 |x| 8 2 h¯ =:

1 h¯ 2

SV (λ(h, ¯ B)),

(38)

where λ(h, ¯ B) = h¯ 3 (h)B ¯ and the operator SV (λ) is given by: 1 λ2 1 λ + (x12 + x22 ) − L3 . SV (λ) = − Δ − 2 |x| 8 2

(39)

Note that the operator SV (λ) is written as a magnetic perturbation (of strength λ) 1 of the hydrogen atom Hamiltonian operator SV := − 12 Δ − |x| with h¯ = 1 whose eigenvalues are EN = − 2(N 1+1)2 , N ∈ N. The authors of reference [4] show that the operator SV (λ) does not converge in norm sense to the operator SV . However, they show that to associate the magnetic perturbation with the Laplacian term (i.e. 1 without the Coulomb potential V = − |x| ) turns out to be a key idea. Thus the following lemma is the basis in order to establish the stability theorem: Lemma 1 Let S0 = − 12 Δ and S0 (λ) = − 12 Δ + z ∈ [0, ∞).

λ2 2 8 (x1

+ x22 ) − λ2 L3 .Consider

1. We have the following convergence in norm: V (S0 (λ) − z)−1 → V (S0 − z)−1 , as λ → 0.

(40)

Semiclassical Clusters of the Zeeman (Stark) Hydrogen Atom

215

2. Consider λ = λ(h) ¯ with h¯ = 1/(N +1) and (h) ¯ = h¯ q , q > 3/2. For |z−EN | = −3 O(N ) we have   2q−3   − 5 −1 −1 , V (S0 (λ(h)) − z) − V (S − z) = O N ¯ 0

(41)

as N → ∞. Here and thereafter, we are using the notation O(N a ), a ∈ R, to denote a bounded operator whose norm is bounded by CN a with N ≥ N0 for some positive constants C and N0 . Let ρ(N) be the distance between EN and next eigenvalue neighbors in the spectrum of SV . We actually have ρ(N) = O(N −3 ). Let r(N ) = cρ(N ) with c < 1/2. Let ΓN be the circle with center EN and radius r(N ). Theorem 3.1 (Stability Theorem) Take (h) ¯ = h¯ q with q > 9. The following spectral projectors are well-defined for N sufficiently large: 1 PN = − 2π ı ΠN = −

1 2π ı

 

(SV (λ(h = 1/(N + 1), B)) − z)−1 dz,

(42)

(SV − z)−1 dz.

(43)

ΓN

ΓN

Moreover, these projectors are orthogonal and satisfy PN − ΠN  = O(N −

2q−33 5

).

(44)

For q > 33/2, the difference of the orthogonal projectors PN − ΠN converges in norm to zero. Consequently, the spectrum of SV (λ(h¯ = 1/(N + 1), B)) inside the circle ΓN consist of a cluster CN of eigenvalues with total multiplicity dN provided N is sufficiently large. Note that the eigenvalue clusters of the operator HV (h, ¯ B) around E = −1/2 are the same as those of the operator SV (λ(h¯ = 1/(N +1), B)) around EN but scaled by a factor of 1/h¯ 2 = (N + 1)2 (see Eq. (38)). Let us denote by {EN,j (1/(N + 1), B)}, j = 1, 2, . . . , dN , the elements of the eigenvalue cluster of HV (h, ¯ B) around E = −1/2. The eigenvalue shifts for the operator SV (λ(h¯ = 1/(N + 1), B)) in the cluster CN are the eigenvalues of the operator PN (SV (λ) − EN )PN . Thus writing PN = ΠN + (PN − ΠN ) and using the stability theorem 3.1 we have the following Theorem 3.2 (M. Avendaño-Camacho, P. Hislop, C. Villegas-Blas) Let B > 0 be fixed, and let ρ be a continuous function on R. Let (h) = hq with q > 33/2, and take h = 1/(N + 1), with N ∈ N. Then we have

216

C. Pérez-Estrada and C. Villegas-Blas

  dN EN,j (1/(N + 1), B) − (−1/2) 1  lim ρ N →∞ dN (1/(N + 1)) j =1

 B = ρ − 3 (x, p) dλ(x, p), 2 Σ(−1/2)    B ρ − 3 (α) dμΓ (α), = 2 α∈Γ (−1/2) 



(45) (46)

where 3 (x, p) = x1 p2 − x2 p1 is the component of the classical angular momentum vector  = x×p along the direction of the magnetic field B(h) on the energy surface Σ(−1/2) with collision orbits treated as in [14]. Given α ∈ Γ (−1/2), 3 (α) dentes the value of 3 at the Kepler orbit α in Σ(−1/2). We remark that in order to prove Theorem 3.2 we need to show the estimates ΠN L3 ΠN = O(N) and ΠN (x12 + x22 )k ΠN = O(N 4k ) for N → ∞ and k ∈ N. The last estimate is by no means immediate. Suitable coherent states for the eigenspace of the hydrogen atom Hamiltonian SV associated to the eigenvalue EN are used in reference [2] in order to show it. The physical intuition comes from the Kepler problem where the maximum apogee distance rmax (E) for an orbit in configuration space of negative energy E is 1/|E| (including collision and non-collision orbits). Thus rmax (E = −1/(2(N +1)2 )) = 2(N +1)2 . Therefore we expect, semiclassically speaking, that for a Kepler orbit in configuration space, x12 + x22 = O(N 4 ). Hence we can show the following: Proposition 1 Let h¯ = 1/(N + 1) and q > 33/2. Let σ = (2q − 33)/5. Then   PN (SV (λ) − EN )PN B = ΠN − hL3 ΠN + O(N −σ ) (47) 2 h2 (h) and for any polynomial Q, we have   dN EN,j (1/(N + 1), B) − (−1/2) 1  = Q dN (1/(N + 1)) j =1

.

    B 1 Tr Q ΠN − hL3 ΠN + O(N −σ ). (48) dN 2

 Since ΠN − B2 hL3 ΠN  = O(1) then Eq. (47) implies that the norm and then N )PN the spectral radius of PN (SVh(λ)−E is O(1). This is the reason why we scale the 2 (h) eigenvalue shifts of HV (h, ¯ B) around E = −1/2 by the factor (h) ¯ in Theorem 3.2. Next, we approximate the test function ρ by a polynomial Q uniformly on a compact interval (using the Weierstrass and then we use   approximation theorem) & E (1/(N +1),B)−(−1/2) in the left hand side Eq. (48) to approximate d1N dj N=1 ρ N,j (1/(N +1))

Semiclassical Clusters of the Zeeman (Stark) Hydrogen Atom

217

of Eq. (45). Finally,

a Szegö-type   theorem is obtained in order to evaluate the limit of d1N Tr Q ΠN − B2 hL3 ΠN when N → ∞ by using localization and resolution of the identity properties of suitable coherent states. This completes the proof of Theorem 3.2. Now we want to describe briefly how we can find the explicit expression for the weak limit measure dμB . We have two different methods to do it. The first one consists of using the explicit expression for the measure dμΓ (see Eq. (34)) in the right hand side of Eq. (46) in Theorem 3.2. Since 3 (x, p) = cos(ψ) cos(θ ) and then independent of (φ, γ ), then doing the change of variables u = cos(ψ) cos(θ ), v = cos(ψ) sin(θ ), with (u, v) in the half-disk u2 + v 2 ≤ 1, v ≥ 0 we get:   B ρ − 3 (α) dμΓ (α) 2 α∈Γ (−1/2)   π/2  θ =π  B ρ − cos(ψ) cos(θ ) cos(ψ) sin(ψ) sin(θ )dψdθ = 2 ψ=0 θ =0   v=√1−u2  u=1  v B = ρ − u dvdu, √ 2 u2 + v 2 u=−1 v=0 $  %  1   B 2 B 2 2 = ρ − u [1 − |u| ] du = ρ(x) 1 − |x| dx. 2 B − B2 B −1



(49)

The second method consists of using approximations for the eigenvalue shifts of the operator SV (λ(h¯ = 1/(N + 1), B)) in the cluster CN around EN = − 2(N 1+1)2 . From Eq. (47), we see that if σ > 1 then the error term in Eq. (47) is of size smaller than the separation O(1/N) between consecutive eigenvalues {− B2 Nm+1 | m =

 −N, . . . , N } of the operator ΠN − B2 hL ¯ 3 ΠN . Thus we need to require a magnetic field even weaker than the one in Theorem 3.2 (where q > 33/2) as a result from the condition σ > 1. Proposition 2 Assume q > 19 (i.e. σ > 1). The eigenvalues of SV (λ) inside the cluster CN around EN can be written in the following way: For N sufficiently large and m = −N, . . . , N , we have EN,m,k = EN −

B m h¯ 2 (h) ¯ + G(N, m, k), 2 N +1

(50)

where, given m, the index k = 1, . . . , N + 1 − |m| and the error term G(N, m, k) = O(N −σ )h¯ 2 (h). ¯ We remark that Proposition 2 implies the existence of sub-clusters of eigenvalues of SV (λ(h¯ = 1/(N +1), B)) in the cluster CN around the numbers EN − B2 Nm+1 h¯ 2 (h), ¯ m = −N, . . . , N . The study of those sub-clusters and a corresponding LEDT is provided in [3].

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Using the eigenvalue approximation in Proposition 2, we can evaluate the left hand side of Eq. (45) in Theorem 3.2 obtaining a Riemann sum whose limit can be identified with the last integral in Eq. (49):   dN EN,j (1/(N + 1), B) − (−1/2)) 1  F dN (1/(N + 1)) j =1

m=N 1  = dN

N +1−|m| 

m=−N

1 = dN =

m=N  m=−N

m=N  m=−(N +1)

k=1

  G(N, m, k) B m + F − 2 N +1 h2 (h)

  B m + O(N −σ ) (N + 1 − |m|) F − 2 N +1 

   |m| B m 1 1− F − + O(N −σ ), (51) N +1 2 N +1 N +1

where we have used the mean value theorem, that F  is bounded on a fixed compact &m=N interval, and the fact that m=−N (N + 1 − |m|) = dN . The first term in the last row in Eq. (51) can be thought of as a Riemann sum  associated to [−1,1] F (− B2 u) (1 − |u|) du with the partition of such interval given

N by −1, N−N , . . . , , 1 . +1 N +1

4 The Stark Hydrogen Atom Problem The Stark hydrogen atom Hamiltonian in a constant electric field (0, 0, F (h)) ¯ is the following operator densely defined on L2 (R3 ): h¯ 2 1 β + (h)F H˜ V (h, ¯ x3 , (h) ¯ F ) = − ΔR 3 − ¯ = h¯ , β > 0. 2 |x| For F > 0, the spectrum of H˜ V (h, ¯ F ) is absolutely continuous and equal to the whole real line R. So the electric case is quite different than the magnetic one. Namely, even for a weak electric field, the eigenvalues of the hydrogen atom Hamiltonian do not split into clusters of eigenvalues. However, there is a splitting into clusters of resonances due to a stability theorem introduced by I. Herbst [5] using a complex dilation technique. We describe below, very briefly, the work of Herbst adapted to our case (the Coulomb potential). See references [5] and [7] for details. As in the magnetic case, we consider the following re-scaling of the operator H˜ V (h, ¯ F ):

Semiclassical Clusters of the Zeeman (Stark) Hydrogen Atom

1 Dh¯ 2 H˜ V (h, ¯ F )Dh¯ −2 = 2 h¯



219

1 1 − ΔR 3 − + h¯ 4 (h)F ¯ x3 2 |x|

1 1 where S˜h¯ (F ) = A + W˜ h¯ (F ), A = ΔR3 − , 2 |x|

 =

1 ˜ Sh¯ (F ), (52) h¯ 2

W˜ h¯ (F ) = h¯ 4 (h)F ¯ x3 . (53)

For θ ∈ R, consider the following dilation of the operator S˜h¯ (F ): S˜h¯ (F, θ ) = Deθ S˜h¯ (F )De−θ = A(θ ) + W˜ h¯ (F, θ ), where A(θ ) = −

e−θ e−2θ θ ΔR 3 − , W˜ h¯ (F, θ ) = h¯ 4 (h)e ¯ F x3 . 2 |x|

(54) (55)

Then consider θ ∈ C in Eqs. (54) and (55). Write S˜h¯ (F, θ ) = H˜ 0 (h, ¯ F, θ )+V (θ ) −θ with the dilated Coulomb potential V (θ ) = − e|x| and the Stark operator e H˜ 0 (h, ¯ F, θ ) = −

−2θ

2

e−2θ θ G(2h¯ 4 (h)e ¯ F) 2 G(α) = −ΔR3 + αx3 , α ∈ C.

θ ΔR3 + h¯ 4 (h)e ¯ F x3 =

with

(56)

Consider the operator G(α) defined on the Schwartz space S(R3 ). Then I. Herbst [5] proves the following relevant facts concerning the existence and theory of resonances for the Stark hydrogen atom. First, Herbst studies the operator G(α) with α = 0 and proves that its spectrum is empty: Theorem 4.1 (Herbst) For α = 0 the following properties hold: • The numerical range, T (G(α)), is the open half plane {z ∈ C | 1z > (1α/α) z} • G(α) is closable. Denote its closure by G(α). • The spectrum of G(α) is empty • The resolvent (z − G(α))−1 is jointly analytic in the variables (z, α) for α = 0 and z ∈ C. Next, Herbst studies the operator S˜h¯ (F, θ ) with 0 < θ < π/3 that allows the precise definition of resonances that we consider in the article: Theorem 4.2 (Herbst) For 0 < θ < π/3 • S˜h¯ (F, θ ) is closed • The family of operators {S˜h¯ (F, θ ) | 0 < θ < π/3} is an analytic family of type A, • The spectrum of S˜h¯ (F, θ ) is discrete (isolated eigenvalues of finite multiplicity) and independent of θ . Moreover, the multiplicity of each eigenvalue is independent of θ .

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We will call the eigenvalues of 12 S˜h¯ (F, θ ) (independent of θ ) the resonances of the h¯ Stark hydrogen atom problem. One of the key ideas in the work of Herbst is to consider the operator Kh¯ (F, θ, z) defined as follows. Write

 Sh¯ (F, θ ) − z = I + Kh¯ (F, θ, z) (H0 (h, ¯ F, θ ) − z) ,

(57)

where we define −1 Kh¯ (F, θ, z) = V (θ ) (H0 (h, ¯ F, θ ) − z) .

(58)

Herbst [5] proves that, for F > 0, the operator Kh¯ (F, θ, z) is compact and jointly analytic in (θ, z) in the region z ∈ C and 0 < |θ | < π3 . Then using analytic Fredholm theory Herbst shows Theorem 4.2 (see reference [5] for all the very interesting details). Moreover, Herbst proves the norm convergence of Kh¯ (F, θ, z)  −2θ −1 to the operator K0 (θ, z) = V (θ ) − e 2 ΔR3 − z in the weak field limit for compacts in a suitable region for (θ, z). For purposes of this article, it will be relevant to have an estimate of that convergence for 0 < θ < π/3 and z in a circle γN with center EN = − 2(N 1+1)2 , N ∈ N, and radius 1/(8N 3 ) for N sufficiently large (remember that weak field limit means h¯ → 0 or, equivalently, N → ∞). Proposition 3 For F > 0, 0 < θ < π/3 and z ∈ γN , the operator Kh¯ (F, θ, z) converges to the operator K0 (θ, z) when N → ∞ with the following estimate Kh¯ (F, θ, z) − K0 (θ, z) = O((h)). ¯

(59)

We remark that the estimate in Eq. (59) can be shown by using the following estimates: Lemma 2 For z ∈ γN and 0 < θ < π/3 we have  −1  H˜ 0 (h,  = O(N 2 ), ¯ F, θ ) − z

 −2θ −1 e ΔR 3 − z  −  = O(N 2 ), 2

−1 e−2θ ΔR 3 − z  = O(N), V (θ ) − 2  −2θ −1 e ΔR 3 − z  = O(N 2 ), e−2θ p3 − 2 

where p3 = −ı

∂ . ∂x3

(60)

Using estimates in Proposition 3 and Lemma 2 we can establish a stability theorem which will ensure us the existence of dN = (N + 1)2 eigenvalues of S˜h¯ (F, θ ) inside the circle γN with center EN for N sufficiently large. We remark that the eigenvalues

Semiclassical Clusters of the Zeeman (Stark) Hydrogen Atom

EN of the operator A = 12 ΔR3 − A(θ ) = −

e−2θ 2

ΔR 3 −

e−θ |x|

1 |x|

221

remain as eigenvalues of the dilated operator

with the same multiplicity dN . Let

PN (θ ) =

1 2π ı

( γN

1 ΠN (θ ) = 2π ı

 −1 z − S˜h¯ (F, θ ) dz,

(61)

(z − A(θ ))−1 dz.

(62)

( γN

Theorem 4.3 (Stability Theorem) For F > 0, 0 < θ < π/3 and β > 3 with (h) ¯ = h¯ β we have PN (θ ) − ΠN (θ ) = O(N 6−β ), h¯ =

1 . N +1

(63)

Consequently, if ΠN (θ )⊥ = I − ΠN (θ ) and PN (θ )⊥ = I − PN (θ ) ΠN (θ )⊥ PN (θ ) = O(N 6−β ), PN (θ )⊥ ΠN (θ ) = O(N 6−β ).

(64)

In particular, if we require β > 6 then for N sufficiently large dim Ran PN (θ ) = dim Ran ΠN (θ ) = dN .

(65)

The stability theorem 4.3 allows us to study clusters of resonances of the Stark hydrogen atom close to −1/2 in the sense of Theorems 1.1, 1.2 and 3.2. Let us denote those resonances by zN,j (h, ¯ F ) with j = 1, . . . , dN (remember the scaling in Eq. (52)). Theorem 4.4 (P. Hislop and C. Villegas-Blas) Let F > 0 be fixed, and let ρ be a function analytic in a disk of radius 3F about z = 0. Let (h) ¯ = h¯ β , with β > 6, and take h¯ = 1/(N + 1). For the resonance cluster {zN,j (h, ¯ F ) | j = 1, . . . , dN } near −1/2, we have   dN zN,j (h, 1  ¯ F ) − (−1/2) ρ lim N →∞ dN (h) ¯ j =1





=

ρ Σ(−1/2)

 =

α∈Γ (−1/2)

  2π 1 F (Φ t (x, p))3 dt dλ(x, p), 2π 0   ρ Fˆ (α) dμΓ (α),

(66) (67)

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C. Pérez-Estrada and C. Villegas-Blas

where Φ t is the Hamiltonian flow for the Kepler problem on the energy surface Σ(−1/2), and (Φ t (x, p))3 is the projection of this flow onto the third coordinate axis x3 , the direction of the electric field. The measure dλ is the normalized Liouville measure on the energy surface Σ(−1/2). The function Fˆ : Γ (−1/2) → R is defined by 1 Fˆ (α) = 2π



2π

F (Φ t (x, p))3 dt,

(68)

0

where (x, p) is any element of the Kepler orbit α. The scaling of the resonance shifts zN,j (h, ¯ F ) − (−1/2) is a consequence of the stability theorem 4.3 together with the following estimates whose prove can be given by using dilated coherent states for the dilated hydrogen atom Hamiltonian A(θ ) (see Eq. (55) for the definition of the operator W˜ h¯ (F, θ )): Lemma 3 There exist a constant r0 > 1, independent of θ with |θ | < any k ∈ N, we have

π 4,

so that for

−1 k ΠN (θ )DN 2 x DN 2 ΠN (θ ) −1 k = ΠN (θ )DN 2 x χx≤r0 DN 2 ΠN (θ ) + ΠN (θ )RN ,

(69)

where χx≤r0 is the characteristic function on the set {x ∈ R3 | x ≤ r0 } and RN  = O(N −∞ ). As a consequence, we have the following estimates: 2 ΠN (θ )W˜ h¯ (F, θ )ΠN (θ ) = O((h) ¯ h¯ ), 2 ΠN (θ )W˜ h¯ (F, θ ) = O((h) ¯ h¯ ).

(70)

Thus we have from Theorem 4.3 and Eqs. (70) in Lemma 3: 2 6−β ), (S˜h¯ (F, θ ) − EN )PN (θ ) = ΠN (θ )W˜ h¯ (F, θ )ΠN (θ ) + O((h) ¯ h¯ N

(71)

where we have used ΠN (θ )(S˜h¯ (F, θ ) − EN )ΠN (θ ) = ΠN (θ )W˜ h¯ (F, θ )ΠN (θ ), ΠN (θ )A(θ )ΠN (θ )⊥ = 0 and (S˜h¯ (F, θ ) − EN )PN (θ ) = PN (θ )W˜ h¯ (F, θ )ΠN (θ ) [I + (ΠN (θ ) − PN (θ ))]−1 . (72)

ΠN (θ)W˜ h¯ (F,θ)ΠN (θ) −1 = F eθ ΠN (θ )DN 2 x3 DN 2 ΠN (θ ) then from Eq. (69) in (h¯ )h¯ 2    ΠN (θ)W˜ h¯ (F,θ)ΠN (θ)  Lemma 3 we see that   is O(1). Thus from Eq. (71) with β > 6 (h¯ )h¯ 2 and writing the resonances shifts zN,j (h, ¯ F ) − (−1/2) as the eigenvalues of the (S˜h¯ (F,θ)−EN ) PN (θ ) we find the scale indicated in Eq. (66). operator PN (θ ) h¯ 2

Since

Since ρ is analytic in a disk and the Kepler orbits with fixed energy in configuration space are bounded then it is enough to prove Theorem 4.4 for a

Semiclassical Clusters of the Zeeman (Stark) Hydrogen Atom

223

monomial ρ(z) = zq , q ≥ 0. The first step of the proof consists of writing the sum in the left hand side of Eq. (66) as the trace of the corresponding operator: q dN  zN,j (h, 1  ¯ F ) − (−1/2) dN (h) ¯ j =1

1 = Tr dN

PN (θ )(S˜h¯ (F, θ ) − EN )q PN (θ )

!

(h¯ 2 (h)) ¯ q

.

(73)

We remind the reader that the operator S˜h¯ (F, θ ) is not self-adjoint so Eq. (73) is not an immediate fact. Next we approximate the right hand side of Eq. (73) by the scaled trace of the perturbation term conjugated by the projector ΠN (θ ): Theorem 4.5 For q ∈ N and β > 6 dN z˜ N,j (h, 1  ¯ F ) − E˜ N dN h2 (h) ¯ i=1

!q

1 = Tr dN

ΠN W˜ h¯ (F ) ΠN h¯ 2 (h) ¯

!q + O(N −δ ). (74)

The proof of Theorem 4.5 involves the stability theorem 4.3 and the estimates indicated  above in order to first approximate the right hand side of Eq. (73) by Π (θ) W˜ (F,θ) Π (θ)

q

h¯ N N Tr with 0 < θ < π/4 . Then considering the analytical h¯ 2 (h¯ ) function ξ(θ ) ≡ T r(ΠN (θ )Wh¯ (F, θ )ΠN (θ ))q , which is constant on the real line, we obtain     1 ΠN (θ )Wh¯ (F, θ )ΠN (θ ) q 1 ΠN Wh¯ (F )ΠN q Tr = Tr . (75) dN dN h¯ 2 (h) h¯ 2 (h) ¯ ¯

1 dN

Finally, we use the following Szegö-type theorem (L. Thomas, C. Villegas-Blas [13]) in order to evaluate the limit N → ∞ of the right hand side of Eq. (74): Theorem 4.6 Let V a multiplication operator with V = V (x) a polynomially bounded continuous function on configuration space R3 . Then for ρ : R → R a continuous function    1 −1 = T r ΠN ρ ΠN D(N V D 2 ΠN 2 (N +1) +1) N →∞ dN     2π 1 t ρ V ((Φ (x, p))x )dt dλ(x, p), 2π 0 Σ(−1/2) lim

where (Φ t (x, p))x is the projection of Φ t (x, p) to the position vector x.

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5 An Explicit Expression for the Weak Limit Measure for the Stark Hydrogen Atom Problem In this section we consider the right hand side of Eq. (66) in Theorem 3.2 and obtain an explicit expression for the weak limit measure for the Stark hydrogen atom problem. Namely, we show Theorem 5.1 Let F > 0. For ρ a continuous function on R: % $  3F   2 2 2 |x| dx, ρ Fˆ (α) dμΓ (α) = ρ(x) 1 − 3F − 3F2 3F α∈Γ (−1/2)



(76)

where dx denotes the Lebesgue measure on R. Theorem 5.1 shows that the real part of the normalized resonance shifts of the Stark hydrogen atom  problem around (−1/2) distribute following the probability measure 2 2 3F dμF = 3F 1 − 3F |x| dx whose support is the interval [− 3F 2 , 2 ]. Proof We first compute the time average of the third component of the position vector along the Kepler orbits on the energy surface Σ(−1/2). L×A From Eqs. (12) to (14) we obtain |L||A| = cos(γ )v − sin(γ )u. Hence writing $ % A L×A x = |x| cos(ϑ) + sin(ϑ) |A| |L||A| = |x| [cos(ϑ + γ )u + sin(ϑ + γ )v]

(77)

we obtain from Eq. (10) that x3 = −

cos2 (ψ) sin(ϑ + γ ) sin(θ ) . 1 + sin(ψ) cos(ϑ)

(78)

Now using that |L| = |x|2 dϑ dt and Eqs. (78), (10) and (11) we have: 1 2π

 0

2π

1 cos5 (ψ) sin(θ ) x3 (t) dt = − 2π



2π

sin(ϑ + γ )

dϑ (1 + sin(ψ) cos(ϑ))3  2π 1 1 ∂ cos4 (ψ) sin(θ ) sin(γ ) = dϑ 4π ∂ψ 0 (1 + sin(ψ) cos(ϑ))2 =

0

3 sin(ψ) sin(θ ) sin(γ ). 2

(79)

Thus we are actually obtaining that the time average of x3 (t) is equal to − 32 A3 where A3 is the projection of the Runge-Lenz vector A along the direction of the electric field.

Semiclassical Clusters of the Zeeman (Stark) Hydrogen Atom

225

Next, since the time average of x3 (t) is independent of the angle φ then by using the expression for the measure dμΓ in Eq. (34) we have: 

  ρ Fˆ (α) dμΓ (α)

α∈Γ (−1/2)

1 = 2π =

1 2π

=

1 2π

 3F sin(ψ) sin(θ ) sin(γ ) cos(ψ) sin(ψ) sin(θ )dψdθ dγ ρ 2    1 3F ζ  ρ dζ dadb 2 2 B1 ζ + a 2 + b2 !    1  1 3F ζ  ρ dadb dζ, (80) 2 Dζ −1 ζ 2 + a 2 + b2 



where we have considered the change of variables given by ζ = sin(ψ) sin(θ ) sin(γ ), a = sin(ψ) sin(θ ) cos(γ ), b = sin(ψ) cos(θ ), B1 is the unit ball in R3 and Dζ is the disk of radius ζ in R2 . From last equation we can conclude the proof.  Theorem 4.4 can now be rewritten in the following way: Theorem 5.2 Under the same hypothesis of Theorem 4.4, we have:  %  $  3F dN 2 zN,j (h, 2 1  2 ¯ F ) − (−1/2) = |x| dx. ρ ρ(x) 1 − N →∞ dN (h) 3F − 3F2 3F ¯ lim

j =1

(81) As in the magnetic case [2], we can also prove Theorem 5.2 using the approximation for h2 e θ F Π

(S˜h¯ (F,θ)−EN )PN (θ) (h¯ )h¯ 2

in terms of the operator

ΠN (θ)W˜ h¯ (F,θ)ΠN (θ) (h¯ )h¯ 2

=

large. Namely, since ¯ N (θ )x3 ΠN (θ ) given by Eq. (71) taking β sufficiently  for z ∈ C fixed the function det eθ ΠN (θ )x3 ΠN (θ ) − zI is analytic as a function of θ and equal to the constant det (ΠN x3 ΠN − zI ) for θ ∈ R , then the eigenvalues of eθ ΠN (θ )x3 ΠN (θ ) are the same as the eigenvalues of ΠN x3 ΠN . The eigenvalues of the operator ΠN x3 ΠN are well known (see [8], section 77, chapter X) and equal to ΔN (n1 , n2 ) =

3 (N + 1)(n1 − n2 ), 2

(82)

where, for N given, the quantum number m associated to the projection of the angular momentum along the direction of the electric field is allowed to take the values m = −N, . . . N and the quantum numbers n1 , n2 are such that N = n1 + n2 + |m| and n1 , n2 = 0, 1, . . . , N − |m|. In particular, we have that the separation between consecutive eigenvalues of the operator

ΠN (θ)W˜ h¯ (F,θ)ΠN (θ) (h¯ )h¯ 2

is O(1/N). Thus taking β > 7 we can

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C. Pérez-Estrada and C. Villegas-Blas

show that the spectrum of intervals of size

CN 6−β

ΠN (θ)W˜ h¯ (F,θ)ΠN (θ) . (h¯ )h¯ 2

(S˜h¯ (F,θ)−EN )PN (θ) (h¯ )h¯ 2

is contained in the union of disjoint

(for some constant C) centered around the eigenvalues of

One can show that given k = −N, . . . , N, the eigenvalue 3(N 2+1)k of the operator ΠN x3 ΠN has multiplicity N + 1 − |k|. Moreover, given N and k there (S˜ (F,θ)−E )P (θ)

N N in the interval are as many as N + 1 − |k| eigenvalues of h¯ (h¯ )h¯ 2   3k 3k 6−β , 6−β . Thus we have for the left hand side of 2(N +1) − CN 2(N +1) − CN Eq. (81) in Theorem 5.2 that

  dN zN,j (h, 1  ¯ F ) − (−1/2) ρ dN (h) ¯ j =1

=

N 1  dN

N −|m|

m=−N n1 =0

=

 ρ

  3F (n1 − n2 ) + O N 6−β 2(N + 1)

 N    1 3F k 6−β + O N (N + 1 − |k|)ρ 2(N + 1) (N + 1)2 k=−N

=

N  k=−(N +1)

 1−

     |k| 3F k 1 ρ + O N 6−β , N +1 2(N + 1) N + 1

(83)

where, given N, m, n1 , we are taking n2 = N − n1 − |m|. Note that we are using in the third equality the mean value theorem and that ρ is smooth with bounded derivative on a compact interval. Finally, we consider the limit N → ∞ of the Riemann sum on the right hand side 1 of Eq. (83) in order to get −1 ρ( 3F2 η ) [1 − |η|] dη and then Eq. (81) in Theorem 5.2. We remark that the proof of Theorem 5.2 by using a suitable approximation for the real part of the scaled resonance shifts completely misses the relevant geometry of the problem (the role of the time averages of the coordinate x3 along the Kepler orbits on the energy surface Σ(−1/2)). Acknowledgments The authors want to thank Salvador Pérrez-Esteva and Oscar Chavez-Molina for discussions on the content of the paper. The authors want to thank the projects PAPIIT-UNAM IN105718 and CONACYT Ciencia Básica 283531 for partial financial support. C. Villegas-Blas thanks the organizers of the “Conference on Spectral Theory and Mathematical Physics, Santiago 2018” for their invitation and financial support to attend the conference through the Chilean project CONICYT REDI17056.

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References 1. R. Abraham, J. Marsden, Foundations of Mechanics, 2nd edn. (AMS Chelsea Publishing, New York, 1987) 2. M. Avendaño Camacho, P.D. Hislop, C. Villegas-Blas, Semiclassical Szegö Limit of eigenvalue clusters for the hydrogen atom Zeeman Hamiltonian. Ann. Henri Poincaré 18(12), 3933–3973 (2017) 3. M. Avendaño Camacho, P.D. Hislop, C. Villegas-Blas, On a limiting eigenvalue distribution theorem for sub-clusters of the hydrogen atom in a constant magnetic field (2019, in preparation) 4. J. Avron, I. Herbst, B. Simon, Schrödinger operators with magnetic fields. I. General interactions. Duke Math. J. 45(4), 847–883 (1978) 5. I.W. Herbst, Dilation analyticity in constant electric field. I. The two body problem. Commun. Math. Phys. 64(3), 279–298 (1979) 6. G. Hernández-Dueñas, S. Pérez Esteva, A. Uribe, C. Villegas-Blas, Perturbations of the Landau Hamiltonian: asymptotics of eigenvalue clusters. Preprint, arXiv:1911.08989 (2019) 7. P.D. Hislop, C. Villegas-Blas, Semiclassical Szego limit of resonance clusters for the hydrogen atom Stark Hamiltonian. Asymptot. Anal. 79(1–2), 17–44 (2011) 8. L.D. Landau, E.M. Lifshitz, Quantum Mechanics (Non Relativistic Theory), 3rd edn. revised and enlarged (Butterworth Heinemann, Oxford, 2000) 9. A. Martinez, An Introduction to Semiclassical and Microlocal Analysis (Springer, Berlin, 2002) 10. D. Ojeda-Valencia, C. Villegas-Blas, On limiting eigenvalue distribution theorems in semiclassical analysis, in Spectral Analysis of Quantum Hamiltonians. Operator Theory: Advances and Applications, vol. 224 (Birkhauser, Springer Basel AG, Basel, 2012), pp. 221–252 11. A. Pushnitski, G. Raikov, C. Villegas-Blas, Asymptotic density of eigenvalue clusters for the perturbed Landau Hamiltonian. Commun. Math. Phys. 320, 425–453 (2013) 12. L.E. Thomas, C. Villegas-Blas, Singular continuous limiting eigenvalue distributions for Schrödinger operators on a 2-sphere. J. Funct. Anal. 141(1), 249–273 (1996) 13. L.E. Thomas, C. Villegas-Blas, Asymptotics of Rydberg states for the hydrogen atom. Commun. Math. Phys. 187(3), 623–645 (1997) 14. A. Uribe, C. Villegas-Blas, Asymptotics of spectral clusters for a perturbation of a hydrogen atom. Commun. Math. Phys. 280(1), 123–144 (2008) 15. C. Villegas-Blas, The Laplacian on the n-sphere, the hydrogen atom and the Bargmann space representation. Ph.D. Thesis, Mathematics Department, University of Virginia, 1996 16. A. Weinstein, Asymptotics of eigenvalue clusters for the Laplacian plus a potential. Duke Math. J. 44(4), 883–892 (1977)

The Negative Spectrum of the Robin Laplacian Nicolas Popoff

1 The Robin Laplacian in Corner Domains In this article, we consider the Laplacian in a bounded corner domain, with a Robin boundary condition ∂ν u − αu = 0, and we review various recent results around the asymptotics of its first eigenvalues as α → +∞. We present the analysis establishing the asymptotic behavior of the first eigenvalue in a corner domain, including an a priori two-side estimate, and the analysis of the essential spectrum of the tangent operators, which are Laplacians in cones with a normalized Robin boundary condition. We also review our results in the case where the weight function in the Robin boundary condition can vanish at a given point, leading to a symmetric but non self-adjoint operator. Next, we give a more precise asymptotics of the lowlying eigenvalues in a regular domain, using an effective operator, defined on the boundary, and involving the mean curvature. These results are then applied to obtain a reverse Faber-Krahn inequality for the Robin Laplacian.

1.1 An Eigenvalue Problem in Corner Domains We consider the Robin eigenvalue problem on a domain ! ⊂ Rn : 

− u = λu on !, ∂ν u − αu = 0 on ∂!,

(1)

N. Popoff () Institut de Mathématiques de Bordeaux, Talence, Gironde, France e-mail: [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 P. Miranda et al. (eds.), Spectral Theory and Mathematical Physics, Latin American Mathematics Series, https://doi.org/10.1007/978-3-030-55556-6_12

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where α is a real parameter, and ∂ν is the outward derivative normal. In order to define an operator associated with this eigenvalue problem, we introduce the quadratic form Q! α: 2 2 Q! α (u) := ∇uL2 (!) − αuL2 (∂!) ,

u ∈ H 1 (!).

(2)

We now introduce the corner domains (together with their tangent cones) that we consider in this article. Let M be a Riemannian manifold without boundary. The class of corner domains of M, denoted by D(M), and admissible cones of Rn , denoted by Pn , are defined recursively, following [17]: Initialization: P0 has one element, {0}. D(S0 ) is formed by all non-empty subsets of S0 = {±1}. Recurrence: For n ≥ 1, 1. A cone ' ⊂ Rn (i.e. a dilation invariant set) belongs to Pn if and only if the section ' ∩ Sn−1 belongs to D(Sn−1 ), 2. ! ⊂ M satisfies ! ∈ D(M) if and only if ! is bounded, connected, and for any x ∈ !, there exists a tangent cone 'x ∈ Pn to ! at x. The definition of a tangent cone is the classical one : we say that 'x is the tangent cone to ! at x when there exist a neighborhood U ⊂ M of x, a neighborhood V ⊂ Rn of 0, and a C ∞ diffeomorphism ψ : U → V, such that ψ(x) = 0,

ψ(U ∩ !) = V ∩ 'x and ψ(U ∩ ∂!) = V ∩ ∂'x .

In dimension 2, the cones in P2 are half-planes, sectors and the full plane. The corner domains in D(R2 ) are (curvilinear) polygons of the plane, with a finite number of vertices, each one of opening in (0, π ) ∪ (π, 2π ). This includes of course regular domains. Corners of opening angle π (for example: curvature jumps) are excluded because we have asked the local map ψ to be C ∞ . From this local definition of corner domain, nice global properties of stratification are satisfied. We describe them now for further use. Let ' ∈ Pn , and let  ∈ Pd be its reduced cone. This means that in some suitable coordinates, we may write ' = Rn−d × 

(3)

with  ∈ Pd and d minimum for such a writing. The integer d is called the dimension of the reduced cone of '. When d = n, the cone ' is said to be irreducible. Now, let ! be a corner domain, and for each x ∈ !, we denote by d(x) the dimension of the reduced cone of 'x . We classify the points x ∈ ! as follows: for each fixed integer 0 ≤ k ≤ n, we denote by Ak = {x ∈ !, d(x) = k}.

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Then, as claimed in [17] and proved in details in [7], the connected component of Ak are submanifolds of codimension k. In particular, as expected, the vertices of ! (i.e. the points x ∈ ! such that d(x) = n) form a finite set. The strata of ! are the connected components of each set Ak for k ∈ {1, . . . , n}. A corner domain is not necessarily Lipschitz, but it is a union of Lipschitz domains, therefore, when ! ∈ D(Rn ), the trace map from H 1 (!) into L2 (∂!) is compact (see [17, Lemma A.A.9]) and the quadratic form Q! α is lower semibounded. We define the self-adjoint operator L! α as its Friedrichs extension, whose spectrum is a sequence of eigenvalues (λj (α, !))j ≥1 (shortened to λj (α) when there is no ambiguity on the domain !), in particular λ1 (α) is called the principal eigenvalue of the system (1).

1.2 The Principal Eigenvalue of the Robin Laplacian and Its Asymptotics As proved in [37, Section 2.3], the function λ1 (·) is decreasing and concave. Moreover, the case α = 0 corresponds to the Neumann problem, therefore λ1 (0) = 0. The limit as α → −∞ is linked to the Dirichlet problem, in the sense that limα→−∞ λ1 (α) is the first Dirichlet eigenvalue on !. We refer to [21] for a more precise asymptotics. But the limit as α → +∞ appears to be more singular, in the sense that it is not clear what limit problem would drive the asymptotics, and therefore, does not enter the framework of regular perturbation theory. Using the constant test function u = 1! in (2), we deduce from the min-max principle the basic upper bound λ1 (α) ≤ −α |∂!| |!| , and therefore λ1 (α) → −∞ as α → +∞. We refer to [36, 49] for more precise geometrical estimates valid for all α ≥ 0. This asymptotic regime for the Robin Laplacian has several applications in reaction diffusion systems [37], surface superconductivity [2, 24, 41], and the study of its spectrum has received a lot of interest since then [11, 14, 15, 19, 22, 27, 29, 32, 33, 38, 43]. Our objective in this section is to gather various results on the asymptotic regime α → +∞. Aside from the operator L! α on bounded domains, it is of interest to define the Robin Laplacian on unbounded domains. We are particularly interested in the tangent cones to a bounded corner domain, i.e. open sets in the class Pn . For such a cone ' ∈ Pn , the operator L' α can be defined in the same way as above, although it is not anymore with compact resolvent. Using a scaling, we have that 2 ' L' α is unitarily equivalent to α L1 , that is why we introduce the normalized model ' ' operator L := L1 , corresponding to α = 1 and the bottom of its spectrum, E('). At this stage, it is not clear how these quantities are defined, because it is not clear why the associated quadratic form Q' := Q' 1 is bounded from below. We will clarify this point in the next section. We will then prove that E(') > −∞ when ' is a cone in Pn , but it does not necessarily correspond to a discrete eigenvalue, and the question of its nature is interesting in itself.

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If we accept the fact that the eigenfunctions associated with λj (α) tend to be localized near some part of the boundary as α → +∞, then, according to the above scaling, the general expected result would be E (!)α 2 ,

(4)

E (!) = inf E('x ),

(5)

λ1 (α)

∼

α→+∞

where x∈∂!

'x being the tangent cone at x ∈ !. For smooth domains, there is only one possible kind of tangent geometries: halfspaces. We denote by Rn+ the model half-space, then the variables can be decoupled and therefore E(Rn+ ) = E(R+ ). This last quantity is found by solving explicitly the spectral problem −y  = Ey on R+ with the boundary condition y  (0) = −y(0), and we have E(R+ ) = −1. We deduce that E (!) = −1 when ! is regular. Then, in this case, (4) is proved (see [37, 39] and [16] for higher eigenvalues), with various improvements depending on the geometry of the boundary, see Sect. 2.1. The sectors of opening θ ∈ (0, 2π ), denoted by Sθ , enjoy an explicit expression:  E(Sθ ) =

− sin−2

θ 2

if θ ∈ (0, π ) ,

− 1 if θ ∈ [π, 2π ) .

(6)

For planar polygonal domains with corners of opening (θk )k=1,...,N , it is conjectured in [37] that (4) holds, with in that case E (!) = − max (1, sin−2 0 −∞. Then 1. The Robin Laplacian, L' , is well defined as the Friedrichs extension of Q' , with form domain H 1 ('), and the bottom of its spectrum satisfies E(') > −∞. 2. Assume moreover that ' is irreducible. Then the bottom of the essential spectrum of L' is E (ω). This theorem relies on the fact that    1 Q' (u) = |∂r u|2 + 2 Qωr (u(r, ·)) r n−1 dr, r r>0

(9)

where (r, θ ) ∈ (0, +∞) × ω denotes spherical coordinates in the cone '. We now see that in ', the Robin Laplacian in the region r ≈ R can be bounded from below by the Robin Laplacian in the cylinder ω × R, with the parameter in the boundary condition equal to R (see Fig. 1). As R gets large, using a localization formula, we can combine (9) and the hypothesis that E (ω) is finite in order to have a lower bound for the quadratic form, related to the operator defined on the section ω. This is here that the recursive construction of corner domains enters the game. The Persson’s Lemma provides the bottom of the essential spectrum. Let us notice that the quantity giving the bottom of the essential spectrum is in fact an infimum over singular chains of a corner domain, and that its structure reminds of the HVZ’s Theorem for the Nbody problem, see [23] for a recent approach. The full proof of this theorem relies on a fine analysis of the ground state energy of the Robin Laplacian, defined on the singular chains of a corner domain, for which we prove continuity and monotonicity, once the set of singular chains has been endowed with a suitable topology and partial order, see [7]. Once Theorem 1.2 is proved, the first assertion in Theorem 1.1 can be proved by recursion. To prove the estimates of the second assertion, we perform a multiscale analysis to get a remainder in the asymptotics of λ1 (α). The lower bound comes from a standard localization formula, based on a multiscale partition of the unity of the domain. The upper bound comes from a construction of test functions localized near a minimizer of x → E('x ). Both these strategies need original treatment, due to the recursive structure of the corner domain, and provide a priori estimates which

The Negative Spectrum of the Robin Laplacian

235

Fig. 1 The cone ' with its section ω. In the zone |x| = R, the variables decouple at first order as R → +∞

Π

ω R∞

are not sharp in the general case. We do not enter these details here since they are quite technical. Given an irreducible cone ', one may ask for a condition guaranteeing that the inequality E(') ≤ E (ω) is strict, i.e. whether there exist discrete eigenvalues below the essential spectrum of L' . This question, interesting in itself, receives an answer in the following cases: • If ' is a sector of opening θ , the answer is yes if and only if θ ∈ (0, π ), i.e. if and only if it is a convex sector, see (6). Moreover, it is proven in [33] that when θ ∈ (0, π ), there exist only a finite number of eigenvalues below the essential spectrum. • If the complement of the cone is convex, then the answer is no [45, Corollary 3]. • If the mean curvature is positive at one point of the boundary of ω, then the answer is yes. Moreover, there exists an infinite number of eigenvalues below the essential spectrum [45, Theorem 6]. • In [10], we assume that n = 3 and that ω is smooth. Denote by κ the geodesic curvature of ∂ω, and by κ+ its positive part. Then E (') = −1, and the number of eigenvalues of L' in (−∞, −1 − λ) behaves, as λ → 0+ , as 1 8π λ

 ∂ω

κ+ (k)ds.

• When ω is convex, upper and lower bound on E(') are given in [38, Section 5], using geometrical quantities. These bounds become sharp when ' is polygonal and ω admits an inscribed circle. More generally, to describe E (ω), the first threshold in the spectrum in L' , and to define and prove the existence of resonances in a neighborhood of E (ω) is an interesting challenge.

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1.4 Extensions with a Weight Function For the Robin Laplacian, we can add a smooth positive weight function G in the boundary conditions. These conditions become ∂ν u = αG(x)u. In our analysis, for x ∈ ∂! fixed, we change α into αG(x) and clearly, the results are still true by replacing E (!) by: EG (!) := inf G(x)2 E('x ). x∈∂!

For the Robin Laplacian, such cases were already considered in [38] and [13]. Note that the Laplacian with a Robin-type boundary condition a∂ν u = u, involving a variable function a which vanishes at a point x0 , can be very different. In dimension 2, denote by s an arclength parameter of ∂! near x0 , and assume that a vanishes at order 1 (i.e. a(s0 ) = 0 and a  (s0 ) = 0). The associated quadratic form u → ∇u2L2 (!) − |a|−1/2 u2L2 (∂!) is not bounded from below in H 1 , and it is not clear how to define a self-adjoint operator. This problem has been noticed in the model case of a half-disc [4, 40]. Using the Kondratiev theory [35], we are able to show in [42] that the Robin Laplacian has deficiency indices equal to (1, 1) [48]. Therefore, the spectrum of its adjoint covers the whole complex plane C, and we are able to describe its self-adjoint extensions as a one-parameter family. The situation where a vanishes to other orders may be very different, and may not be covered by the theory of Kondratiev. We plan to investigate these cases.

2 Effective Hamiltonian in the Regular Case: The Role of the Curvature 2.1 Next Terms in the Asymptotics We now assume that ! is a C 2 domain, and we describe how to get a more precise asymptotics of the low-lying eigenvalues. As the analysis of the last section has shown, the minimizer of the local energy will govern the first term of the asymptotics. To understand the next term of the asymptotics, the strategy is to localize the operator near the place where this minimum is reached. But for a regular domain, E('x ) = −1 for all x ∈ ∂!. If one thinks of the local energy as an effective potential, then it would say in that case that its extremum is reached on the whole boundary and it is expected that an effective operator, defined on ∂!, will lead the asymptotics of the low-lying eigenvalues of L! α , as α → +∞. In the literature, initially the results were in dimension n = 2: if we denote by γ : ∂! → R the curvature of the boundary, and γmax its maximum, then it is proved in [19, 43] that, as α → +∞, for fixed j ∈ N,

The Negative Spectrum of the Robin Laplacian

λj (α) = −α 2 − γmax α + O(α 2/3 ).

237

(10)

Note that in case of balls and spherical shells, these asymptotics are obtained in [22] through analytical ODEs and exploit known asymptotics of Bessel functions. The asymptotics (10) rises several questions: what is the analogue of this result in higher dimension, and is it possible to see the influence of j ∈ N in the next term of the asymptotics? In [26], the authors assume that ! ⊂ R2 is C ∞ , and that γ admits a unique non-degenerate maximum. They prove that this maximum acts as a well (this point will be enlightened below) and they provide a full asymptotic expansion of the eigenvalues (whose first terms are given in (12) below). For a regular domain ! ⊂ Rn , we denote by H : ∂! → R the mean curvature of its boundary and by −S the (positive) Laplace-Beltrami operator on the hypersurface ∂!. Then, we prove in [47]: Theorem 2.1 Assume that ! is C 3 . Denote by Leff α the operator −S −(n−1)αH , acting on H 2 (∂!), and μj (α) its j -th eigenvalue. Let j ∈ N be fixed, then, as α → +∞: λj (α) = −α 2 + μj (α) + O(1).

(11)

As α → +∞, the operator Leff α has the form of a semiclassical Schrödinger operator on ∂!, the potential being proportional to −H . In particular its eigenvalues satisfy, as α gets large, μj (α) ∼ −(n−1)αHmax , where Hmax denotes the maximum of H . More precise asymptotics enters the framework of harmonic approximation, see [18, 28, 50]: hypotheses on H near its maximum will imply more structure in the asymptotics. The most common framework is the case where H reaches its maximum in a unique point, called a curvature’s well. In particular: Corollary 2.2 There holds, as α → +∞: λj (α) = −α 2 − (n − 1)αHmax + o(α) Assume moreover that ! is C ∞ , and that H admits a unique global maximum at s0 and that the Hessian of −(n − 1)H at this maximum is positive-definite. Denote by tk its eigenvalues and set V=

 n−1 5 k=1

  tk

2nk − 1 , nk ∈ N . 2

Let us sort the elements of V in the increasing order, repeating the terms if they appear multiple times, and denote by ej the j -th element. Then for each j ∈ N there holds, as α → +∞, λj (α) = −α 2 − (n − 1)αHmax + ej α 1/2 + O(α 1/4 ).

(12)

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Moreover, if ej is of multiplicity one, the remainder estimate can be improved to O(1). Notice that various improvements of the above result exist, depending on some changes in the assumptions: • An analogue of the result holds when ! is not bounded, provided it has a good structure at infinity, see [47, Definition 1.1], in the spirit of [12]. • In dimension 2, if H has a unique maximum at s0 which is degenerate, in the sense that there exists p > 1 such that near s0 : H (s) = Hmax − Cp (s − s0 )2p + o((s − s0 )2p+1 ),

Cp > 0,

then a three-terms asymptotics is still available: 1

1

λj (α) = −α 2 − αHmax + e˜j α p+1 + O(α 2(p+1) ), where e˜j is the j -th eigenvalue of −∂s2 + Cp s 2p , acting on L2 (R). • If the domain ! is only C 2 , (11) still holds with a remainder in O(log α) instead of O(1). • In dimension n = 2, assuming that there exists a unique non-degenerate curvature well, (12) has been obtained in [26], with a full asymptotic expansion in power of α 1/2 . In the case of a symmetric domain with two curvature wells, the tunneling effect is analyzed in [29], in particular it is proved that the first spectral R eff gap λR 2 (α) − λ1 (α) is well approximated by the one of Lα , which can be seen 1 as a Schrödinger operator acting on S with a double-wells potential. Following this analysis, the question of a refinement of (7) (and of an asymptotics for the higher eigenvalues) when the domain has a non-smooth boundary is a challenging question. For polygons, this is the object of [31]. Let us describe the asymptotics in the case of a polygon with straight edges, the case of a curvilinear polygon being done in the same references with additional technical difficulties. Following the ideas of [5, 6], this problem is treated in [32]: denote by (vj )j =1,...,N the N vertices of !. With each of these vertices is associated a model problem on the infinite sector Sθi , where θi ∈ (0, 2π ) \ {π } is the opening angle of ! at the vertex vj . The model problem at vj is given by the Robin Laplacian LSθi . The bottom of its spectrum is given by (6), and its essential spectrum is [−1, +∞). Moreover, we know that there is only a finite number the & Ki of eigenvalues below ! essential spectrum, see [33]. Then the first K := N i=1 Ki eigenvalues of Lα , the Robin Laplacian in !, will be “attracted” by the corners, with an exponentially small interaction, in the following sense: label the K eigenvalues below the essential spectrum of the N model problems in an increasing sequence ( j )j =1,...,K . Then there exists c > 0 such that for 1 ≤ j ≤ K: λj (α) =

jα

2

+ o(e−cα ).

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239

The asymptotics expansion of the eigenvalues λj , with j > K, requires a more global approach, since the sides of the polygons will now contribute. This is done in [34] (see also [44]), under an additional hypothesis on the essential spectrum of the model problems: in a polygon with straight edges, once the corners have “attracted” the K first eigenvalues, the j -th eigenvalue of the Robin Laplacian satisfies for j > K: λj (α) = −α 2 + νj −K + O

log α  , α

where νj is the j -th eigenvalue of the Laplacian on the graph formed by the sides of the boundary, with a Dirichlet boundary condition at each junction. Note that this approach is inspired by Grieser [25] for a similar problem.

2.2 Application to a Faber-Krahn Inequality The optimization of eigenvalues under geometrical constraints has received a lot of interest since more than a century, originating from Lord Rayleigh’s Theory of sound. He conjectures the following property, now classical: among bounded domains ! of fixed volume, the first eigenvalue of the Dirichlet Laplacian should be minimized by the ball. This has been proven by Faber and Krahn in the 1920s. This kind of questions have been extended to numerous optimization problems, such as the second Neumann eigenvalue and higher Dirichlet eigenvalues, forming the family of isoperimetric spectral inequalities. We refer to [30, Sections 3–7] for an overview containing the historical references. In this part, we denote by λ1 (α, !) the first eigenvalue of the Robin Laplacian, in order to emphasize the dependency on the domain. It has been proved in [8] in dimension 2, and in [14] in any dimension, that λ1 (α, !) is also minimized by a ball when α is fixed in (−∞, 0). When α = 0, we have ∂α λ1 (0, !) = − |∂!| |!| , therefore when ! is not a ball, and B is a ball of same area than !, we have from the isoperimetric inequality ∂α λ1 (0, !) < ∂α λ1 (0, B). Supported by this consideration, it has been conjectured in [3] that the ball should become the maximizer for λ1 (α, !) when α > 0 is fixed. This reverse Faber-Krahn inequality has been proved in dimension n = 2 for α small enough [22, Section 4], but, surprisingly, this conjecture has been disproved for large α: in [22, Section 3], it is proved that if B and A are a ball and a spherical shell with the same volume, then, for α large enough, λ1 (α, B) < λ1 (α, A). Such a counter-example relies on explicit calculations, using the spherical invariance of the domains in order to express the eigenvalues as roots of special functions.

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N. Popoff

In view of Corollary 2.2, the maximization of λ1 (α, !) leads to the following question: Set the volume of !, how to minimize Hmax ?

(13)

The counter example to the Faber-Krahn inequality shows also that this problem has no solution without additional constraint, indeed a thin annulus of large radius can have a fixed volume, but its mean curvature can be arbitrary small. In [46], we prove: Theorem 2.3 Let ! ⊂ Rn be a bounded star-shaped regular domain, and B a ball of same volume. Then Hmax (!) ≥ Hmax (B), moreover there is equality if and only if ! is a ball. The above theorem relies on the standard isoperimetric inequality, combined with a Minkowski type equality:  |∂!| =

p(s)H (s)dS, ∂!

where p(s) = s · ν(s) is the support function of a star-shaped domain, ν(s) being the exterior normal. Using the normalized curvature flow, this Theorem has been improved in dimension n = 2 to the more restrictive class of simply connected domain (note that spherical shells are simply connected only when n ≥ 3). One may ask whether the result holds true in any dimension among domains with connected boundary, but a counter example has been shown in [20]: the authors construct a family of nodoids, diffeomorphic to a ball, in dimension 3, whose mean curvature is arbitrarily small, with a fixed volume. As a consequence of Theorem 2.3, we get: Corollary 2.4 Let ! be a domain which is simply connected if n = 2, or star shaped if n ≥ 3. Assume that ! is not a ball and let B be a ball of same volume. Then, for all j ∈ N, there exists α0 > 0 such that ∀α ∈ (α0 , +∞),

λj (α, !) < λj (α, B).

It has been shown recently that in dimension 3, the fixed perimeter constraint is a relevant geometric constraint for this optimization constraint: in that case, for a fixed α > 0, the ball appear to be the optimizer among convex domains [51], as conjectured in [1]. Acknowledgments The author would like to thank the referee who provided useful and detailed comments in order to improve the contents and the presentation of the manuscript.

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26. B. Helffer, A. Kachmar, Eigenvalues for the Robin Laplacian in domains with variable curvature. Trans. Am. Math. Soc. 369(5), 3253–3287 (2017) 27. B. Helffer, K. Pankrashkin, Tunneling between corners for Robin Laplacians. J. Lond. Math. Soc. (2) 91(1), 225–248 (2015) 28. B. Helffer, J. Sjöstrand, Multiple wells in the semiclassical limit. I. Commun. Partial Differ. Equ. 9(4), 337–408 (1984) 29. B. Helffer, A. Kachmar, N. Raymond, Tunneling for the Robin Laplacian in smooth planar domains. Commun. Contemp. Math. 19(1), 1650030, 38 (2017) 30. A. Henrot, Extremum Problems for Eigenvalues of Elliptic Operators. Frontiers in Mathematics (Birkhäuser Verlag, Basel, 2006) 31. M. Khalile, l’Université Paris-Saclay. PhD thesis, Université de Grenoble, 2018 32. M. Khalile, Spectral asymptotics for Robin Laplacians on polygonal domains. J. Math. Anal. Appl. 461(2), 1498–1543 (2018) 33. M. Khalile, K. Pankrashkin, Eigenvalues of robin Laplacians in infinite sectors. Math. Nachr. 291(5–6), 928–965 (2018) 34. M. Khalile, T. Ourmières-Bonafos, K. Pankrashkin, Effective operator for robin eigenvalues in domains with corners. Preprint, arXiv:1809.04998 (2018) 35. V.A. Kondrat’ev, Boundary value problems for elliptic equations in domains with conical or angular points. Trudy Moskov. Mat. Obšˇc. 16, 209–292 (1967) 36. H. Kovaˇrík, On the lowest eigenvalue of Laplace operators with mixed boundary conditions. J. Geom. Anal. 24(3), 1509–1525 (2014) 37. A.A. Lacey, J.R. Ockendon, J. Sabina, Multidimensional reaction diffusion equations with nonlinear boundary conditions. SIAM J. Appl. Math. 58(5), 1622–1647 (1998) 38. M. Levitin, L. Parnovski, On the principal eigenvalue of a Robin problem with a large parameter. Math. Nachr. 281(2), 272–281 (2008) 39. Y. Lou, M. Zhu, A singularly perturbed linear eigenvalue problem in C 1 domains. Pac. J. Math. 214(2), 323–334 (2004) 40. M. Marlettta, G. Rozenblum, A Laplace operator with boundary conditions singular at one point. J. Phys. A Math. Theor. 42(12), 125204 (2009) 41. E. Montevecchi, J.O. Indekeu, Effects of confinement and surface enhancement on superconductivity. Phys. Rev. B 62(21), 14359 (2000) 42. S.A. Nazarov, N. Popoff, Self-adjoint and skew-symmetric extensions of the Laplacian with singular Robin boundary condition. C. R. Math. Acad. Sci. Paris 356(9), 927–932 (2018) 43. K. Pankrashkin, On the asymptotics of the principal eigenvalue for a Robin problem with a large parameter in planar domains. Nanosyst. Phys. Chem. Math. 4(4):474–483 (2013) 44. K. Pankrashkin, On the Robin eigenvalues of the Laplacian in the exterior of a convex polygon. Nanosyst. Phys. Chem. Math. 6, 46–56 (2015) 45. K. Pankrashkin, On the discrete spectrum of Robin Laplacians in conical domains. Math. Model. Nat. Phenom. 11(2), 100–110 (2016) 46. K. Pankrashkin, N. Popoff, Mean curvature bounds and eigenvalues of Robin Laplacians. Calc. Var. Partial Differ. Equ. 54(2), 1947–1961 (2015) 47. K. Pankrashkin, N. Popoff, An effective Hamiltonian for the eigenvalue asymptotics of the Robin Laplacian with a large parameter. J. Math. Pures Appl. (9) 106(4), 615–650 (2016) 48. M. Reed, B. Simon, Methods of Modern Mathematical Physics. IV. Analysis of Operators (Academic [Harcourt Brace Jovanovich Publishers], New York, 1978) 49. A. Savo, Optimal eigenvalue estimates for the Robin Laplacian on Riemannian manifolds. Preprint, arXiv:1904.07525 (2019) 50. B. Simon, Semiclassical analysis of low lying eigenvalues. I. Nondegenerate minima: asymptotic expansions. Ann. Inst. H. Poincaré Sect. A (N.S.) 38(3), 295–308 (1983) 51. A.V. Vikulova, Parallel coordinates in three dimensions and sharp spectral isoperimetric inequalities. Preprint, arXiv:1906.11141 (2019)

On Some Integral Operators Appearing in Scattering Theory, and their Resolutions Serge Richard and Tomio Umeda

2010 Mathematics Subject Classification 47G10

1 Introduction Investigations on the wave operators in the context of scattering theory have a long history, and several powerful technics have been developed for the proof of their existence and of their completeness. More recently, properties of these operators in various spaces have been studied, and the importance of these operators for nonlinear problems has also been acknowledged. A quick search on MathSciNet shows that the terms wave operator(s) appear in the title of numerous papers, confirming their importance in various fields of mathematics. For the last decade, the wave operators have also played a key role for the search of index theorems in scattering theory, as a tool linking the scattering part of a physical system to its bound states. For such investigations, a very detailed understanding of the wave operators has been necessary, and it is during such investigations that several integral operators or singular integral operators have appeared, and that their resolutions in terms of smooth functions of natural selfadjoint operators have been provided. The present review paper is an attempt to gather some of the formulas obtained during these investigations. Singular integral operators are quite familiar to analysts, and refined estimates have often been obtained directly from their kernels. However, for index theorems or for topological properties these kernels are usually not so friendly: They can hardly fit into any algebraic framework. For that reason, we have been looking

S. Richard () Graduate School of Mathematics, Nagoya University, Chikusa-ku, Nagoya, Japan e-mail: [email protected] T. Umeda Department of Mathematical Sciences, University of Hyogo, Shosha, Himeji, Japan e-mail: [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 P. Miranda et al. (eds.), Spectral Theory and Mathematical Physics, Latin American Mathematics Series, https://doi.org/10.1007/978-3-030-55556-6_13

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for representations in which these singular integral operators have a smoother appearance. It turns out that for several integral operators this program has been successful, and suitable representations have been exhibited. On the other hand, let us stress that even though these formulas are necessary for a C ∗ -algebraic approach, it seems that for pure analysis they do not lead to any new refined estimates. Let us now be more precise for a few of these operators, and refer to the subsequent sections for more information. The Hilbert transform is certainly one of the most famous and ubiquitous singular integral operators. Its explicit form in L2 (R) is recalled in (2.1). A nice representation of this singular operator does not take place directly in L2 (R), but by decomposing this Hilbert space into odd and even functions, then it becomes possible to obtain an expression for the Hilbert transform in terms of the operators tanh(π A) and cosh(π A) where A denotes the generator of dilations in L2 (R+ ). Such an expression is provided in Proposition 2.1. Similarly, the Hankel transform whose definition in L2 (R+ ) is recalled in (2.3) is an integral operator whose kernel involves a Bessel function of the first kind. This operator is not invariant under the dilation group, and so does the inversion operator J defined on f ∈ L2 (R+ ) by [Jf ](x) = x1 f ( x1 ). However, the product of these two operators is invariant under the dilation group and can be represented by a smooth function of the generator of this group, see Proposition 2.2 for the details. For several integral operators on R+ or on Rn , the dilation group plays an important role, as emphasized in Sect. 2. On the other hand, on a finite interval (a, b) this group does not play any role (and is even not defined on such a space). For singular operators on such an interval, the notion of rescaled energy representation can be introduced, and then tools from natural operators on R can be exploited. This approach is developed in Sect. 3. As a conclusion, this short expository paper does not pretend to be exhaustive or self-contained. Its (expected) interest lies on the collection of several integral operators which can be represented by smooth functions of some natural self-adjoint operators. It is not clear to the authors if a general theory will ever be built from these examples, but gathering the known examples in a single place seemed to be a useful preliminary step.

2 Resolutions Involving the Dilation Group The importance of the dilation group in scattering theory is well-known, at least since the seminal work of Enss [6]. In this section we recall a few integral operators which appeared during our investigations on the wave operators. They share the common property of being diagonal in the spectral representation of the generator of the dilation group. Before introducing these operators, we recall the action of this group in L2 (Rn ) and in L2 (R+ ). In L2 (Rn ) let us set A for the self-adjoint generator of the unitary group of dilations, namely for f ∈ L2 (Rn ) and x ∈ Rn :

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[eitA f ](x) := ent/2 f (et x). There also exists a representation in L2 (R+ ) which is going to play an important role: its action is given by [eitA f ](x) := et/2 f (et x) for f ∈ L2 (R+ ) and x ∈ R+ . Note that we use the same notation for these generators independently of the dimension, but this slight abuse of notation will not lead to any confusion.

2.1 The Hilbert Transform Let us start by recalling that the Hilbert transform is defined for f in the Schwartz space S(R) by the formula 1 [Hf ](x) := P.v. π

 R

i 1 f (y) dy = − √ x−y 2π

Here P.v. denotes the principal value, kˆ := the Fourier transform of f defined by 1 [F f ](k) := √ 2π

k |k|

 R

 R

eikx kˆ [F f ](k) dk.

(2.1)

for any k ∈ R∗ and F f stands for

e−ikx f (x) dx.

(2.2)

It is well known that this formula extends to a bounded operator in L2 (R), still denoted by H. In this Hilbert space, if we use the notation X for the canonical self-adjoint operator of multiplication by the variable, and D for the self-adjoint d realization of the operator −i dx , then the Hilbert transform also satisfies the equality H = −i sgn(D), with sgn the usual sign function. Based on the first expression provided in (2.1) one easily observes that this operator is invariant under the dilation group in L2 (R). Since this group leaves the odd and even functions invariant, one can further decompose the Hilbert space in order to get its irreducible representations, and a more explicit formula for H. Thus, let us introduce the even / odd representation of L2 (R). Given any function ρ on R, we write ρe and ρo for the even part and the odd part of ρ. We then introduce the unitary map U : L2 (R)  f → Its adjoint is given on

h1  h2

  √ fe ∈ L2 (R+ ; C2 ). 2 fo

∈ L2 (R+ ; C2 ) and for x ∈ R by

 ∗ h1   1  U h2 (x) := √ h1 (|x|) + sgn(x)h2 (|x|) . 2

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A new representation of the Hilbert transform can now be stated. We refer to [13, Lem. 3] for the initial proof, and to [19, Lem. 2.1] for a presentation corresponding to the one introduced here. Proposition 2.1 The Hilbert transform H in L2 (R) satisfies the following equality:  0 tanh(π A) − i cosh(π A)−1 U H U = −i . tanh(π A) + i cosh(π A)−1 0 ∗



Note that this representation emphasizes several properties of the Hilbert transform. For example, it makes it clear that the Hilbert transform exchanges even and odd functions. By taking the equality tanh2 + cosh−2 = 1, one also easily deduces that the norm of H is equal to 1. This latter property can certainly not be directly deduced from the initial definition of the Hilbert transform based on the principal value. The Hilbert transform appears quite naturally in scattering theory, as emphasized for example in the seminal papers [4, 26, 27]. The explicit formula presented in Proposition 2.1 is used for the wave operators in [19, Thm. 1.2]. A slightly different version also appears in [13, Eq. (1)].

2.2 The Hankel Transform Let us recall that the Hankel transform is a transformation involving a Bessel function of the first kind. More precisely, if Jm denotes the Bessel function of the first kind with m ∈ C and 1(m) > −1, the Hankel transform Hm is defined on f ∈ Cc∞ (R+ ) by  [Hm f ](x) =

R+

√ xy Jm (xy) f (y) dy.

(2.3)

Note that slightly different expressions also exist for the Hankel transform, but this version is suitable for its representation in L2 (R+ ). Let us also introduce the unitary and self-adjoint operator J : L2 (R+ ) → L2 (R+ ) defined for f ∈ L2 (R+ ) and x ∈ R+ by [Jf ](x) =

1 1 f . x x

It is now easily observed that neither Hm nor J are invariant under the dilation group. However, the products J Hm and Hm J are invariant, and thus have a representation in terms of the generator of dilations. The following formulas have been obtained in [5, Prop. 4.5] based on an earlier version available in [2, Thm. 6.2]. Note that in the statement the notation  is used for the usual Gamma function.

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Proposition 2.2 For any m ∈ C with 1(m) > −1 the map Hm continuously −1 . In extends to a bounded invertible operator on L2 (R+ ) satisfying Hm = Hm addition, the equalities J Hm = &m (A)

and

Hm J = &m (−A),

hold with &m (t) := ei ln(2)t

( m+1+it ) 2 ( m+1−it ) 2

.

Let us mention that the function t → &m (t) has not a very interesting asymptotic behavior for large |t|. However, by taking the asymptotic behavior of the Gamma function into account, one can observe that the product of two such functions has a much better behavior, namely for any m, m ∈ C with 1(m) > −1 and 1(m ) > −1 the map t → &m (−t)&m (t)

 π  belongs to C [−∞, ∞] and one has &m (∓∞)&m (±∞) = e∓i 2 (m−m ) . Note that such a product of two function &m appears at least in two distinct contexts: For the wave operators of Schrödinger operators with an inverse square potential, see [2, Thm. 6.2], [5, Eq. (4.24)] and also [8, 9], and for the wave operators of an Aharonov-Bohm system [16, Prop. 11]. Let us also mention that additional formulas in terms of functions of A have been found in [16, Thm. 12] as a result of a transformation involving a Bessel function of the first kind and a Hankel function of the first kind. Remark 2.3 Let us still provide a general scheme for operators in L2 (R+ ) which can be written in terms of the dilation group. If K denotes an integral operator with a kernel K(·, ·) satisfying for any x, y, λ ∈ R+ the relation K(λx, λy) =

1 K(x, y), λ

(2.4)

then this operator commutes with the dilation group. As a consequence, K can be rewritten as a function of the generator A of the dilation group in L2 (R+ ), and one has K = ϕ(A) with ϕ given by 

∞

ϕ(t) = 0

1

K(1, y) y − 2 +it dy.

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Note that this expression can be obtained by using the general formula 

1 ϕ(A)f = √ 2π

R

ϕ(t) ˇ e−itA f

in conjunction with the homogeneity relation (2.4).

2.3 A Three Dimensional Example When dealing with scattering theory for Schrödinger operators in L2 (R3 ) one more operator commuting with the generator of dilations appears quite naturally, see [14, Sec. 3]. Let us set F for the Fourier transform in R3 defined for f ∈ S(R3 ) and k ∈ R3 by [F f ](k) =

1 (2π )3/2

 R3

e−ik·x f (x) dx.

Then we can define for f ∈ S(R3 ) and x = 0 the integral operator 1 [Tf ](x) = −i √ 2π

 R+

eiκ |x| [F f ](κ x) ˆ κ 2 dκ κ |x|

(2.5)

x where we have again used the notation xˆ := |x| ∈ S2 . An easy computation shows that this operator is invariant under the action of the dilation group. It is thus natural to express the operator T in terms of the operator A. In fact, the operator T can be further reduced by decomposing the Hilbert space with respect to the spherical harmonics. Let us set h := L2 (R+ , r 2 dr) and consider the spherical coordinates (r, ω) ∈ R+ × S2 . For any ∈ N = {0, 1, 2, . . .} and m ∈ Z satisfying − ≤ m ≤ , let Y m denote the usual spherical harmonics. Then, by taking into account the completeness of the family {Y m } ∈N,|m|≤ in L2 (S2 , dω), one has the canonical decomposition

L2 (R3 ) =

C

H

m

,

(2.6)

∈N,|m|≤

some g ∈ h}. For fixed where H m = {f ∈ L2 (R3 ) | f (rω) = g(r)Y m (ω) a.e. forD ∈ N we denote by H the subspace of L2 (R3 ) given by − ≤m≤ H m . Let us finally observe that since the dilation group acts only on the radial coordinate, its action is also reduced by the above decomposition. In other terms, this group leaves each subspace H m invariant. As a final ingredient, let us recall that the Fourier transform F also leaves the subspace H m of L2 (R3 ) invariant. More precisely, for any g ∈ Cc∞ (R+ ) and for

On Some Integral Operators Appearing in Scattering Theory, and their Resolutions

249

(κ, ω) ∈ R+ × S2 one has  [F (gY m )](κω) = (−i) Y m (ω)

R+

r2

J

+1/2 (κr)

√ κr

g(r) dr ,

(2.7)

where Jν denotes the Bessel function of the first kind. So, we naturally set F : Cc∞ (R+ ) → h by the relation F (gY m ) = F (g)Y m (it is clear from (2.7) that this operator does not depend on m). Similarly to the Fourier transform in L2 (R3 ), this operator extends to a unitary operator from h to h. Remark 2.4 If V : L2 (R+ , r 2 dr) → L2 (R+ , dr) is the unitary map defined by [Vf ](r) := rf (r) for f ∈ L2 (R+ , r 2 dr), then the equality VF V ∗ = (−i) H +1/2 holds, where the r.h.s. corresponds to the Hankel transform defined in (2.3). By taking the previous two constructions into account, one readily observes that the operator T is reduced by the decomposition (2.6). As a consequence one can look for a representation of the operator T in terms of the dilation group in each subspace H m . For that purpose, let us define for each ∈ N the operator T acting on any g ∈ Cc∞ (R+ ) as 1 [T g](r) = −i √ 2π

 R+

eiκ r [F g](κ) κ 2 dκ . κr

The following statement has been proved in [14, Prop. 3.1]. Proposition 2.5 The operator T extends continuously to the bounded operator

 ϕ (A) in H m with ϕ ∈ C [−∞, ∞] given explicitly for every x ∈ R by 1 ϕ (x)= e−iπ 2

/2



   12 ( +3/2+ix)  12 (3/2−ix)

1  1  1+ tanh(π x)−i cosh(π x)−1  2 ( +3/2−ix)  2 (3/2+ix)

and satisfying ϕ (−∞) = 0 and ϕ (∞) = 1. Furthermore, T defined

the operator  in (2.5) extends continuously to the operator ϕ(A) ∈ B L2 (R3 ) acting as ϕ (A) on H . Let us finally observe that in the special case = 0, one simply gets ϕ0 (A) =

 1

1 + tanh(π A) − i cosh(π A)−1 . 2

This operator appears in particular in the expression for the wave operators in R3 , see [12, Sec. 2.1] and [20, Thm. 1.1]. The same formula but for one-dimensional system can be found for example in [12, Sec. 2.4] and in [18, Thm. 2.1], see also [7, Thm. 1.1]. The adjoint of this operator also appears for one-dimensional system in [12, Sec. 2.3] and in the expression for the wave operator in a periodic

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 setting [22, Thm. 5.7]. The related formula 12 1 + tanh(π A/2) is used for systems in dimension 2 as for example in [12, Sec. 2.2] or in [21, Thm. 1.1]. Remark 2.6 Let us mention that operators similar to the one presented in (2.5) appear quite often in the context of scattering theory, and then such expressions can be reformulated in a way similar to the one presented in Proposition 2.5. For example, such kernels can be exhibited from the asymptotic expansion of the generalized eigenfunctions of the relativistic Schrödinger operators in dimension 2 [25, Thm. 6.2] or in dimension 3 [24, Thm. 10.2].

3 Resolutions in the Rescaled Energy Representation In the previous section, the dilation group played a special role since all operators were invariant under its action. For many other singular kernels appearing in scattering theory, this is no more true, and in many settings there is no analog of the dilation group. However, a replacement for the operator A can often be found by looking at the rescaled energy representation, see for example [1, Sec. 2.4] and [23, Sec. 3.1]. The main idea in this approach is to rescale the underlying space such that it covers R, and then to use the canonical operators X and D on R. Let us stress that here the energy space corresponds to the underlying space since the following operators are directly defined in the energy representation.

3.1 The Finite Interval Hilbert Transform In this section we consider an analog of the Hilbert transform but localized on a finite interval. More precisely, let us consider the interval % := (a, b) ⊂ R. For any f ∈ Cc∞ (%) and λ ∈ % we consider the operator defined by [Tf ](λ) :=

 1 1 P.v. f (μ) dμ. π % λ−μ

(3.1)

This operator corresponds to the Hilbert transform but restricted to a finite interval. In order to get a better understanding of this operator, let us consider the Hilbert space L2 (R) and the unitary map U : L2 (%) → L2 (R) defined on any f ∈ L2 (%) and for x ∈ R by 5 [U f ](x) :=

 a + be2x  1 b−a f . 2 cosh(x) 1 + e2x

On Some Integral Operators Appearing in Scattering Theory, and their Resolutions

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The adjoint of this map is given for h ∈ L2 (R) and λ ∈ % by 5 ∗

[U h](λ) =

1 λ − a  1 b−a . h ln √ 2 (λ − a)(b − λ) 2 b − λ

Let us now denote by L the operator of multiplication by the variable in L2 (%) and set ρ(L) for the operator of multiplication in L2 (%) by a function ρ ∈ L∞ (%). Then, a straightforward computation leads to the following expression for its representation in L2 (R): ρ(X) ˜ := U ρ(L) U ∗ is the operator of multiplication

2x  by the function x → ρ(x) ˜ = ρ a+be . In particular, by choosing ρ(λ) = λ 1+e2x 2X

one obtains that U L U ∗ is the operator ρ(X) ˜ = a+be . Note that the underlying 1+e2X function is strictly increasing on R and takes the asymptotic values ρ(−∞) ˜ = a and ρ(∞) ˜ = b. Let us now perform a similar conjugation to the operator T . A straightforward computation leads then to the following equality for any h ∈ Cc∞ (R) and x ∈ R: [U T U ∗ h](x) =

 1 1 P.v. h(y) dy . π sinh(x − y) R

d Thus if we keep denoting by D the self-adjoint operator corresponding to −i dx in 2 L (R), and if one takes into account the formula



 e−ixy i P.v. dy = tanh π x/2 π R sinh(y) one readily gets: Proposition 3.1 The following equality holds

 U T U ∗ = −i tanh π D/2 .

(3.2)

Such an operator plays a central role for the wave operator in the context of the Friedrichs-Faddeev model [11, Thm. 2]. Let us also emphasize one of the main interest of such a formula. Recall that X and D satisfy the usual canonical commutation relations in L2 (R). Obviously, the same property holds for the selfadjoint operators X% := U ∗ X U and D% := U ∗ D U in L2 (%). More interestingly for us is that for any functions ϕ ∈ L∞ (R) and ρ ∈ L∞ (%) the operator ϕ(D) ρ(X) ˜ in L2 (R) is unitarily equivalent to the operator ϕ(D% ) ρ(L) in L2 (%). In particular,

 this allows us to define quite naturally isomorphic C ∗ -subalgebras of B L2 (R) and 

2 of B L (%) , either generated by functions of D and X, or by functions of D% and L. By formula (3.2), one easily infers that the singular operator T defined in (3.1) belongs to such an algebra.

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3.2 The Finite Interval Hilbert Transform with Weights The operator considered in this section is associated with a discrete adjacency operator on Z. Once considered in its energy representation, this operator leads to a Hilbert transform on a finite interval multiplied by

 some weights. We consider the Hilbert space L2 (−2, 2) and the weight function β : (−2, 2) → R given by 1/4

. β(λ) := 4 − λ2

 For any f ∈ Cc∞ (−2, 2) and for λ ∈ (−2, 2) we define the operator 1 P.v. [Tf ](λ) := 2π i



2

−2

β(λ)

1 β(μ)−1 f (μ) dμ. λ−μ

(3.3)

Clearly, this operator has several singularities: on the diagonal but also at ±2. In order to get a better understanding of it, let us introduce

 the unitary transformation U : L2 (−2, 2) → L2 (R) defined on f ∈ L2 (−2, 2) by [U f ](x) :=

√

2

 1 f 2 tanh(x) . cosh(x)

The action of its adjoint is given on h ∈ L2 (R) by ∗

[U h](λ) = √

√ 2 4 − λ2

 h arctanh(λ/2) .

 We also introduce the multiplication operators b± (X) ∈ B L2 (R) defined by the real functions b± (x) :=

ex/2 ± e−x/2 . (ex + e−x )1/2

The function b+ is continuous, bounded, non-vanishing, and satisfies limx→±∞ b+ (x) = 1. The functions b− is also continuous, bounded, and satisfies limx→±∞ b− (x) = ±1. With these notations, the following statement has been proved in [15]. Proposition 3.2 One has U T U∗ = −

 1 b+ (X) tanh(π D)b+ (X)−1 − ib− (X) cosh(π D)−1 b+ (X)−1 . 2 (3.4)

On Some Integral Operators Appearing in Scattering Theory, and their Resolutions

253

Note that a slightly simpler expression is also possible, once a compact error is accepted. More precisely, since the functions appearing in the statement of the previous proposition have limits at ±∞ the operator in the r.h.s. of (3.4) can be rewritten as  1 tanh(π D) − i tanh(X) cosh(π D)−1 + K (3.5) 2

 with K ∈ K L2 (R) , see for example [3] for a justification of the compactness of the commutators. Note that this expression can also be brought back to the initial representation by a conjugation with the unitary operator U . Let us also mention that the operators obtained above play an important role for the wave operator of discrete Schrödinger operators on Zn . Such operators have been studied in [10] and in [15]. −

3.3 The Upside Down Representation In this section we deal with a singular kernel which is related to a one-dimensional Dirac operator. Compared to the operators introduced so far, its specificity comes from its matrix-values. Dirac operators depend also on a parameter m which we choose strictly positive. The following construction takes already place in the energy representation of the Dirac operator, namely on its spectrum. Let us define the set  := (−∞, −m) ∪ (m, +∞) and for each λ ∈ Σ the 2 × 2 matrix 1 B(λ) = √ diag 2

5 4

λ−m , λ+m

5 4

! λ+m . λ−m

Clearly, for any λ ∈ Σ the matrix B(λ) is well defined and invertible, but it does not have a limit as λ 7 m or as λ 8 −m. For f ∈ Cc∞ (Σ; C2 ) we consider the singular operator T defined by  1 1 −1 B(μ) f (μ) dμ. [Tf ](λ) := B(λ) P.v. π λ − μ Σ The trick for this singular operator is to consider the following unitary transformation which sends the values ±m at ±∞, while any neighbourhood of the points ±∞ is then located near the point 0. More precisely, let us define the unitary operator U : L2 (Σ; C2 ) → L2 (R; C2 ) given for f ∈ L2 (Σ; C2 ) and x ∈ R by

254

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 ex + 1  √   ex/2 f m x . U f (x) := 2m x e −1 e −1 The adjoint of the operator U is provided for h ∈ L2 (R; C2 ) and λ ∈ Σ by the expression 

√  U h (λ) = 2m ∗

5

  λ + m  λ+m 1 h ln . λ−m λ+m λ−m

We shall now compute the kernel of the operator U T U ∗ , and observe that this new kernel has a very simple form. For that purpose, we keep the notations X and D for the canonical self-adjoint operators on L2 (R), and denote by F the Fourier transform in L2 (R; C2 ), namely two copies of the Fourier transform (2.2). One then checks, by a direct computation, that for any measurable function ρ : Σ → M2 (C) one has 

 eX + 1 U ρ(L) U = ρ m X . e −1 ∗

Furthermore, for any f = (f1 , f2 ) ∈ Cc∞ (R; C2 ) and x ∈ R, it can be obtained straightforwardly that [U T U ∗ h](x)  1 P.v. = 4π R

1 sinh((y−x)/4)

− 0

1 cosh((y−x)/4)

1 sinh((y−x)/4)

0 +

! 1 cosh((y−x)/4)

h(y) dy.

By summing up the information obtained so far one obtains: Proposition 3.3 For any m > 0 one has   0 tanh(2π D) + i cosh(2π D)−1 UT U = i . 0 tanh(2π D) − i cosh(2π D)−1 ∗

We refer to [17, Sec. III.D] for the details of the computation, and for the use of this expression in the context of one-dimensional Dirac operators. Note that in Section IV of this reference the C ∗ -algebraic properties mentioned at the end of Sect. 3.1 are exploited and the construction leads naturally to some index theorem in scattering theory. Acknowledgments S. Richard thanks the Department of Mathematics of the National University of Singapore for its hospitality in February 2019. The authors also thank the referee for suggesting the addition of Remark 2.3. Its content is due to him/her.

On Some Integral Operators Appearing in Scattering Theory, and their Resolutions

255

The author S. Richard was supported by the grant Topological invariants through scattering theory and noncommutative geometry from Nagoya University, and by JSPS Grant-in-Aid for scientific research (C) no 18K03328, and on leave of absence from Univ. Lyon, Université Claude Bernard Lyon 1, CNRS UMR 5208, Institut Camille Jordan, 43 blvd. du 11 novembre 1918, F69622 Villeurbanne cedex, France. The author T. Umeda was supported by JSPS Grant-in-Aid for scientific research (C) no 18K03340.

References 1. J. Bellissard, H. Schulz-Baldes, Scattering theory for lattice operators in dimension d ≥ 3. Rev. Math. Phys. 24(8), 1250020, 51 pp. (2012) 2. L. Bruneau, J. Derezi´nski, V. Georgescu, Homogeneous Schrödinger operators on half-line. Ann. Henri Poincaré 12(3), 547–590 (2011) 3. H.O. Cordes, On compactness of commutators of multiplications and convolutions, and boundedness of pseudodifferential operators. J. Funct. Anal. 18, 115–131 (1975) 4. P. D’Ancona, L. Fanelli, Lp -boundedness of the wave operator for the one dimensional Schrödinger operator. Commun. Math. Phys. 268(2), 415–438 (2006) 5. J. Derezi´nski, S. Richard, On Schrödinger operators with inverse square potentials on the halfline. Ann. Henri Poincaré 18, 869–928 (2017) 6. V. Enss, Geometric methods in scattering theory, in New Developments in Mathematical Physics (Schladming, 1981). Acta Phys. Austriaca Suppl. XXIII (Springer, Vienna, 1981), pp. 29–63 7. H. Inoue, Explicit formula for Schroedinger wave operators on the half-line for potentials up to optimal decay. J. Funct. Anal. 279(7), 108630, 23 pp. (2020) 8. H. Inoue, S. Richard, Index theorems for Fredholm, semi-Fredholm, and almost periodic operators: all in one example. J. Noncommut. Geom. 13(4), 1359–1380 (2019) 9. H. Inoue, S. Richard, Topological Levinson’s theorem for inverse square potentials: complex, infinite, but not exceptional. Rev. Roum. Math. Pures App. LXIV(2–3), 225–250 (2019) 10. H. Inoue, N. Tsuzu, Schroedinger wave operators on the discrete half-line. Integr. Equ. Oper. Theory 91(5), Paper No. 42, 12 pp. (2019) 11. H. Isozaki, S. Richard, On the wave operators for the Friedrichs-Faddeev model. Ann. Henri Poincaré 13, 1469–1482 (2012) 12. J. Kellendonk, S. Richard, Levinson’s theorem for Schrödinger operators with point interaction: a topological approach. J. Phys. A Math. Gen. 39, 14397–14403 (2006) 13. J. Kellendonk, S. Richard, On the structure of the wave operators in one dimensional potential scattering. Math. Phys. Electron. J. 14, 1–21 (2008) 14. J. Kellendonk, S. Richard, On the wave operators and Levinson’s theorem for potential scattering in R3 . Asian-Eur. J. Math. 5, 1250004-1–1250004-22 (2012) 15. H.S. Nguyen, S. Richard, R. Tiedra de Aldecoa, Discrete Laplacian in a half-space with a periodic surface potential I: resolvent expansions, scattering matrix, and wave operators. Preprint, arXiv 1910.00624 16. K. Pankrashkin, S. Richard, Spectral and scattering theory for the Aharonov-Bohm operators. Rev. Math. Phys. 23, 53–81 (2011) 17. K. Pankrashkin, S. Richard, One-dimensional Dirac operators with zero-range interactions: spectral, scattering, and topological results. J. Math. Phys. 55, 062305-1–062305-17 (2014) 18. S. Richard, Levinson’s theorem: an index theorem in scattering theory, in Proceedings of the Conference Spectral Theory and Mathematical Physics, Santiago 2014. Operator Theory Advances and Applications, vol. 254 (Birkhäuser, Basel, 2016), pp. 149–203 19. S. Richard, R. Tiedra de Aldecoa, New formulae for the wave operators for a rank one interaction. Integr. Equ. Oper. Theory 66, 283–292 (2010)

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20. S. Richard, R. Tiedra de Aldecoa, New expressions for the wave operators of Schrödinger operators in R3 . Lett. Math. Phys. 103, 1207–1221 (2013) 21. S. Richard, R. Tiedra de Aldecoa, Explicit formulas for the Schrödinger wave operators in R2 . C. R. Acad. Sci. Paris Ser. I. 351, 209–214 (2013) 22. S. Richard, R. Tiedra de Aldecoa, Spectral and scattering properties at thresholds for the Laplacian in a half-space with a periodic boundary condition. J. Math. Anal. Appl. 446, 1695– 1722 (2017) 23. H. Schulz-Baldes, The density of surface states as the total time delay. Lett. Math. Phys. 106(4), 485–507 (2016) 24. T. Umeda, Generalized eigenfunctions of relativistic Schrödinger operators I. Electron. J. Differ. Equ. 127, 46 pp. (2006) 25. T. Umeda, D. Wei, Generalized eigenfunctions of relativistic Schrödinger operators in two dimensions. Electron. J. Differ. Equ. 143, 18 pp. (2008) 26. R. Weder, The Wk,p -continuity of the Schrödinger wave operators on the line. Commun. Math. Phys. 208(2), 507–520 (1999) 27. K. Yajima, The Lp boundedness of wave operators for Schrödinger operators with threshold singularities I, The odd dimensional case. J. Math. Sci. Univ. Tokyo 13(1), 43–93 (2006)

The Strong Scott Conjecture: the Density of Heavy Atoms Close to the Nucleus Heinz Siedentop

1 Introduction: The Ground State Problem in Physics Given an neutral atom of atomic number Z, we are interested in its groundstate energy E(Z) and the corresponding electron density ρ. This question can be posed within the context of various models for the atom ranging from effective models like simple density functionals like the Thomas-Fermi model, more advanced effective models like Hartree-Fock or Müller theory, or full quantum models. In the following we shall review the status for some of those models. The presentation is based on joint results with Weikard, Frank, Warzel, Handrek, Merz, Simon, and Chen.

2 Nonrelativistic Models 2.1 The Thomas-Fermi Functional One of the radically elementary approximations in non-relativistic quantum mechanics is the theory developed by Thomas [76] and Fermi [20, 21]. It is given by the so called Thomas-Fermi functional (Lenz [41])

H. Siedentop () Mathematisches Institut, Ludwig-Maximilians-Universität München, Theresienstraße 39, 80333, München, Germany e-mail: [email protected] http://www.math.lmu.de/~hkh © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 P. Miranda et al. (eds.), Spectral Theory and Mathematical Physics, Latin American Mathematics Series, https://doi.org/10.1007/978-3-030-55556-6_14

257

258

H. Siedentop



 EZTF (ρ)

:=

R3

5 3 3 10 γρ(x)

  1 ρ(x)ρ(y) Zρ(x) dx + dxdy − |x| 2 R3 ×R3 |x − y| ? @A B

(1)

=:D[ρ]

where γ := (3π 2 )2/3 is the Thomas-Fermi constant. The three terms represent the kinetic energy, the potential energy in the field of the nucleus, and the electronelectron repulsion. The functional is naturally defined for all non-negative ρ ∈ 5 L 3 (R3 ) with D[ρ] < ∞ (see Simon [67]), i.e., electron densities with finite kinetic energy and finite electron-electron repulsion. The electron-nucleus attraction is finite automatically. (The long range part is dominated by the electron-electron repulsion and the short range part by the kinetic energy.) Among others the following facts are known about the Thomas-Fermi functional (see, e.g., Lieb and Simon [45]): The ground state energy of the Thomas-Fermi functional is 7

E TF (Z) := inf EZTF (ρ) = E TF (1)Z 3 . ρ

Moreover it has a unique minimizer ρZTF in the above set. This minimizer has the particle number  N :=

ρZTF = Z

(2)

and has the scaling behavior ρZTF (x) = Z 2 ρ1TF (Z 1/3 x).

(3)

The corresponding mean-field potential, the Thomas-Fermi potential, has the following scaling behavior ϕZTF (x) :=

Z − ρZTF ∗ | · |−1 (x) = Z 4/3 ϕ TF (Z 1/3 x). |x|

(4)

It is obvious from the definition and from the Thomas-Fermi equation γ TF 2/3 2 ρZ (x)

= ϕZTF (x)

(5)

that ϕZTF (x) ≤

Z |x|

(6)

Strong Scott Conjecture

259

or, written in the density, ρZTF (x) ≤

23/2 Z 3/2 . 3π 2 |x|3/2

(7)

In fact equality holds up to an error of order o(|x|−3/2 ) as x → 0. Sommerfeld [72] showed that ϕ S (x) :=

34 π 2 1 23 |x|4

(8)

fulfills the Thomas-Fermi differential equation 1 Δϕ = (2/γ )3/2 ϕ 3/2 4π

(9)

for x = 0. The function ϕ S is called the Sommerfeld solution of the Thomas-Fermi equation. It is also an upper bound ϕZT F ≤ ϕ S with the corresponding bound on the density ρZTF (x) ≤

35 π 1 . 23 |x|6

(10)

Again equality holds up to an error term o(|x|−6 ) for x → ∞.

2.2 Thomas-Fermi-Weizsäcker Functional The intuitive derivation of the Thomas-Fermi functional assumes that the potential is locally a constant. Leading corrections to the energy should stem from those regions where this is not the case, i.e., close to the nucleus. Weizsäcker [80] added a correction that penalizes rapid changes of the density. The Weizsäcker functional reads  √ E TFW (ρ) := A2 dx|∇ ρ(x)|2 + EZTF (ρ) (11) R3

√ which is naturally defined for densities ρ such that ρ ∈ H 1 (R3 ). Weizsäcker chose the value of the constant A to be one. Later derivations proposed different constants, e.g., the gradient expansion of Kirzhnits [40] suggests A = 1/9. An important feature is that the ground state energy of an atom equals to leading order the Thomas-Fermi energy but has a correction of order Z 2 , i.e., E TFW (Z) = E TF (Z) + γA Z 2 + o(Z 2 ) as Z → ∞

(12)

260

H. Siedentop

(Lieb [43] and Lieb and Liberman [44]); in particular the constant can be adjusted to the values of in non-relativistic quantum mechanics. The occurrence of a Z 2 correction in the context of quantum mechanics was argued for by Scott [57] and is called the Scott correction.

2.3 Quantum Mechanics Thomas and Fermi developed their theory in order to approximate the groundstate energy and the ground state density of atoms. That it actually approximates the Nparticle quantum mechanics is a seminal success of theory. To formulate the result we need someE notation. Write Q for the form domain of the one-particle energy and write QN := N n=1 Q for the form domain of the N -particle energy. The quantum energy of an atom with atomic number Z with N electrons in the state ψ ∈ QN is given as E[ψ] :=

N  

ψ, (T (pn ) −

n=1

 Z )ψ + |xn |



 ψ,

1≤m3/2 ∀r∈R+ |U (r)| ≤ Cχ(0,1) (r)/r + Cχ(1,∞) (r)/r α . Theorem 3.3 (Relativistic Strong Scott Correction: Chandrasekhar) Fix Z/c ∈ (0, 2/π ). Then, for all U ∈ U  lim

Z→∞ R3

dxU (|x|)c−3 ρZC (x/c) =

 R3

dxU (|x|)ρ CH (x).

(35)

We expect that these test functions are in a sense optimal as far as their singularity at the origin and their decay at infinity is concerned. Actually, a more refined version for the densities in each angular momentum channel and an even large class of test functions are possible. These results were proven by Frank et al. [25] and are based on a new Sobolev type inequality [24].

Strong Scott Conjecture

267

3.3 No-Pair Operators: The Furry Picture The idea that an electron cannot occupy all one-particle states but only those orthogonal to an already filled sea dates back to Dirac [8]. In today’s language it is called the Dirac sea. Brown and Ravenhall [4] implemented this by assuming that the Dirac sea is the negative spectral subspace of the free Dirac operator D0 := cα · p + c2 β,

(36)

where we have the four 4 × 4 matrices  α :=

0σ σ 0



– σ the three Pauli matrices—are the three Dirac, and β :=

  1 0 , 0 −1

i.e., the functional (28) is defined on QBR := [χR3 (D0 )]H 1/2 (R3 : C4 ) and T (p) := D0 − c2 . The boundedness from below of this functional follows from Evans et al. [11] (see also Tix [77, 78]), if and only if γ := Z/c ≤ 2/(2/π + π/2) which covers all known elements. The groundstate energy shows also a Scott correction [23]. However, it is numerically known that the energies, although larger than the corresponding Chandrasekhar functional (a consequence of [28]), are still too low compared to physical values. There are, however, other choices of the one-particle space possible. This has been discussed in great detail in the literature (Sucher [73–75]). A choice which is known to reproduce the one-particle energies of the Dirac operator and to yield— numerically—chemical accuracy is the Furry picture (see Furry and Oppenheimer [26], Reiher and Wolf [53], and Saue [54]): in this model the kinetic energy is—as in all no-pair models—chosen to be D0 , however, QF := [χR+ (D0 − Z/|x|)]H 1/2 (R3 : C4 ).

(37)

Similar to the previous cases we add a superscript F for the energies and densities. This model is bounded from below, if Z/c ≤ 1. Recently the following results on its ground-state properties were obtained: The Energy In the following we will also use the rescaled hydrogen Dirac operator Dγ := α · p + β − 1 − γ /|x|.

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Theorem 3.4 (Relativistic Scott Correction: Furry Picture) Fix γ := Z/c ∈ (0, 1). Then, as Z → ∞, 7

E C (Z) = E TF (1)Z 3 + ( 12 − s F (γ ))Z 2 + o(Z 2 ) .

(38)

%     $ p2 γ 2 en − ⊗ C − en (Dγ ) s (γ ) := 2 |x| n

(39)

Here F

where the eigenvalues are labeled in increasing order. This result was proven by Handrek and Siedentop [29] and compared with measured and numerical data. The agreement is unreasonably good. The Density Guided by the energetic results we expect no change on the ThomasFermi scale Z −1/3 . In fact this can be proven completely analogously to the Chandrasekhar case (see Merz and Siedentop [50]). On the Scott scale Z −1 we expect a change, namely the convergence to the sum of the absolute squares of the hydrogenic Dirac orbitals again following the spirit of Lieb’s original conjecture. To formulate it we write ϕn for an orthonormal basis of eigenfunctions of Dγ , i.e., Dγ ϕn = en (Dγ )ϕn . We define ρ DH (x) :=

4  

|ϕn (x, σ )|2 .

(40)

σ =1 n

Although the Dirac hydrogen orbitals are known explicitly (see, e.g., Bethe [3], a summation formula analogous to the one obtained by Lieb and Heilmann [30] is not known. Nevertheless it turns out that the series in (40) converges [48]. Again, we focus on the Scott scale and define for fixed γ ρ¯ZF (x) := c−3 ρZF (x/c).

(41)

As expected, one proves the following convergence result (Merz and Siedentop [48]). Theorem 3.5 (Relativistic Strong Scott Correction: Furry Picture) Fix Z/c ∈ (0, 1). Then for U ∈ U  lim

Z→∞ R3

dx U (|x|)c−3 ρZF (x/c)

 =

R3

dx U (|x|)ρ DH (x).

(42)

Again this result can be refined to resolve angular momenta and to an even large class of test functions [48].

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269

References 1. V. Bach, A proof of Scott’s conjecture for ions. Rep. Math. Phys. 28(2), 213–248 (1989) 2. R.D. Benguria, M. Loss, H. Siedentop, Stability of atoms and molecules in an ultrarelativistic Thomas-Fermi-Weizsäcker model. J. Math. Phys. 49(1), 012302 (2008) 3. H. Bethe, Quantenmechanik der Ein- und Zwei-Elektronenatome, in Handbuch der Physik, XXIV.1, Buch 2, chap. 3, 2nd edn. ed. by H. Geiger, K. Scheel (Springer, Berlin, 1933), pp. 273–560 4. G.E. Brown, D.G. Ravenhall, On the interaction of two electrons. Proc. R. Soc. Lond. A. 208, 552–559 (1951) 5. S. Chandrasekhar, The maximum mass of ideal white dwarfs. Astrophys. J. 74, 81–82 (1931) 6. H. Chen, On the excess charge problem in relativistic quantum mechanics. PhD thesis, LudwigMaximilians-Universität München, July 2019 7. H. Chen, H. Siedentop, On the excess charge of a relativistic statistical model of molecules with an inhomogeneity correction. arXiv:1912.00205 [math-ph] (2019) 8. P.A.M. Dirac, A theory of electrons and protons. Proc. R. Soc. Lond. A 126, 360–365 (1930) 9. E. Engel, R.M. Dreizler, Field-theoretical approach to a relativistic Thomas-Fermi-DiracWeizsäcker model. Phys. Rev. A 35, 3607–3618 (1987) 10. E. Engel, R.M. Dreizler, Solution of the relativistic Thomas-Fermi-Dirac-Weizsäcker model for the case of neutral atoms and positive ions. Phys. Rev. A 38, 3909–3917 (1988) 11. W.D. Evans, P. Perry, H. Siedentop, The spectrum of relativistic one-electron atoms according to Bethe and Salpeter. Commun. Math. Phys. 178(3), 733–746 (1996) 12. C.L. Fefferman, L.A. Seco, An upper bound for the number of electrons in a large ion. Proc. Natl. Acad. Sci. USA 86, 3464–3465 (1989) 13. C.L. Fefferman, L.A. Seco, Asymptotic neutrality of large ions. Commun. Math. Phys. 128, 109–130 (1990) 14. C. Fefferman, L. Seco, Eigenfunctions and eigenvalues of ordinary differential operators. Adv. Math. 95(2), 145–305 (1992) 15. C.L. Fefferman, L.A. Seco, Aperiodicity of the Hamiltonian flow in the Thomas-Fermi potential. Rev. Math. Iberoam. 9(3), 409–551 (1993) 16. C. Fefferman, L. Seco, The density of a one-dimensional potential. Adv. Math. 107(2), 187– 364 (1994) 17. C. Fefferman, L. Seco, The eigenvalue sum of a one-dimensional potential. Adv. Math. 108(2), 263–335 (1994) 18. C. Fefferman, L. Seco, On the Dirac and Schwinger corrections to the ground-state energy of an atom. Adv. Math. 107(1), 1–188 (1994) 19. C. Fefferman, L. Seco, The density in a three-dimensional radial potential. Adv. Math. 111(1), 88–161 (1995) 20. E. Fermi, Un metodo statistico per la determinazione di alcune proprietá dell’atomo. Rend. Accad. Naz. Lincei 6(12), 602–607 (1927) 21. E. Fermi, Eine statistische Methode zur Bestimmung einiger Eigenschaften des Atoms und ihre Anwendung auf die Theorie des periodischen Systems der Elemente. Z. Phys. 48, 73–79 (1928) 22. R.L. Frank, H. Siedentop, S. Warzel, The ground state energy of heavy atoms: relativistic lowering of the leading energy correction. Commun. Math. Phys. 278(2), 549–566 (2008) 23. R.L. Frank, H. Siedentop, S. Warzel, The energy of heavy atoms according to Brown and Ravenhall: the Scott correction. Doc. Math. 14, 463–516 (2009) 24. R.L. Frank, K. Merz, H. Siedentop, Equivalence of Sobolev norms involving generalized Hardy operators. Int. Math. Res. Notices rnz135 (2019). https://doi.org/10.1093/imrn/rnz135 25. R.L. Frank, K. Merz, H. Siedentop, B. Simon, Proof of the strong Scott conjecture for Chandrasekhar atoms. Pure Appl. Funct. Anal. Preprint, arXiv:1907.04894 (2019) 26. W.H. Furry, J.R. Oppenheimer, On the theory of the electron and positive. Phys. Rev. II. Ser. 45, 245–262 (1934)

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