Mathematical Challenges of Zero-Range Physics: Models, Methods, Rigorous Results, Open Problems (Springer INdAM Series, 42) 3030604527, 9783030604523

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Springer INdAM Series 42

Alessandro Michelangeli  Editor

Mathematical Challenges of Zero-Range Physics Models, Methods, Rigorous Results, Open Problems

Springer INdAM Series Volume 42

Editor-in-Chief Giorgio Patrizio, Università di Firenze, Florence, Italy Series Editors Giovanni Alberti, Università di Pisa, Pisa, Italy Filippo Bracci, Università di Roma Tor Vergata, Rome, Italy Claudio Canuto, Politecnico di Torino, Turin, Italy Vincenzo Ferone, Università di Napoli Federico II, Naples, Italy Claudio Fontanari, Università di Trento, Trento, Italy Gioconda Moscariello, Università di Napoli “Federico II”, Naples, Italy Angela Pistoia, Sapienza Università di Roma, Rome, Italy Marco Sammartino, Universita di Palermo, Palermo, Italy

This series will publish textbooks, multi-authors books, thesis and monographs in English language resulting from workshops, conferences, courses, schools, seminars, doctoral thesis, and research activities carried out at INDAM - Istituto Nazionale di Alta Matematica, http://www.altamatematica.it/en. The books in the series will discuss recent results and analyze new trends in mathematics and its applications. THE SERIES IS INDEXED IN SCOPUS

More information about this series at http://www.springer.com/series/10283

Alessandro Michelangeli Editor

Mathematical Challenges of Zero-Range Physics Models, Methods, Rigorous Results, Open Problems

Editor Alessandro Michelangeli Institute for Applied Mathematics, and Hausdorff Center for Mathematics University of Bonn Bonn, Germany

ISSN 2281-518X ISSN 2281-5198 (electronic) Springer INdAM Series ISBN 978-3-030-60452-3 ISBN 978-3-030-60453-0 (eBook) https://doi.org/10.1007/978-3-030-60453-0 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 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 is a follow-up initiative to the third international meeting ‘Mathematical Challenges of Zero-Range Physics: Rigorous Results and Open Problems’ held in Rome in July 2018 under the auspices of the INdAM and presents an up-to-date collection of some of the most relevant results and challenging open problems in the field. There has been since long over the decades a large transversal community of mathematicians grappling with the sophisticated challenges of the rigorous modelling and the spectral and scattering analysis of quantum systems of particles subject to an interaction so much localised to be considered with zero range. As witnessed in the recent series of the ‘zero-range meetings’ like ours in 2018, such a community is experiencing very fruitful and inspiring exchanges with experimental and theoretical physicists involved in the subject. The present volume reflects this spirit, with a diverse range of contributions by some of the speakers in Rome and additional experts. It has been conceived with the deliberate twofold purpose of serving, on the one hand, as an updated reference for the most recent results, the mathematical tools, and the vast related literature and, on the other hand, as a bridge towards several key open problems that will surely form the forthcoming research agenda in this field. I am grateful to all contributors, who so effectively managed to shape their own chapters in such a spirit. Besides, this book owes much to the INdAM scientific board for providing that unique and stimulating opportunity represented by the INdAM meetings, and it could not have been completed without the precious support of the INdAM administrative staff and the Springer publishing team, as well as the anonymous reviewers for their careful work and the quality of their reports. I extend my gratitude to all of them. Bonn, Germany May 2020

Alessandro Michelangeli

v

Contents

Thermodynamic Properties of Ultracold Fermi Gases Across the BCS-BEC Crossover and the Bertsch Problem . . . . . .. . . . . . . . . . . . . . . . . . . . Martina Iori, Tommaso Macrì, and Andrea Trombettoni

1

Scattering Theory for Delta-Potentials Supported by Locally Deformed Planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Claudio Cacciapuoti, Davide Fermi, and Andrea Posilicano

35

The Massless Nelson Hamiltonian and Its Domain. . . . . . .. . . . . . . . . . . . . . . . . . . . Julian Schmidt A Note on the Dirac Operator with Kirchoff-Type Vertex Conditions on Metric Graphs .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . William Borrelli, Raffaele Carlone, and Lorenzo Tentarelli

57

81

Dirac Operators and Shell Interactions: A Survey .. . . . . .. . . . . . . . . . . . . . . . . . . . 105 Thomas Ourmières-Bonafos and Fabio Pizzichillo Ultraviolet Properties of a Polaron Model with Point Interactions and a Number Cutoff . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 133 Jonas Lampart Zero Modes and Low-Energy Resolvent Expansion for Three Dimensional Schrödinger Operators with Point Interactions .. . . . . . . . . . . . . . 149 Raffaele Scandone Spectral Properties of Point Interactions with Fermionic Symmetries . . . . 163 Andrea Ottolini Born-Oppenheimer Type Approximation for a Simple Renormalizable System. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 177 Haci Akbas and O. Teoman Turgut Spectral Isoperimetric Inequality for the δ  -Interaction on a Contour . . . . 215 Vladimir Lotoreichik vii

viii

Contents

Quantum Confinement in α-Grushin Planes. . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 229 Eugenio Pozzoli Kre˘ın-Višik-Birman Self-Adjoint Extension Theory Revisited . . . . . . . . . . . . . 239 Matteo Gallone, Alessandro Michelangeli, and Andrea Ottolini Translation and Adaptation from Russian of M. Sh. Birman, “On the Theory of Self-Adjoint Extensions of Positive Definite Operators”, Math. Sb. 28 (1956), 431–450 . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 305 Mikhail Khotyakov and Alessandro Michelangeli

Thermodynamic Properties of Ultracold Fermi Gases Across the BCS-BEC Crossover and the Bertsch Problem Martina Iori, Tommaso Macrì, and Andrea Trombettoni

Abstract In this chapter we review the thermodynamic properties of ultracold Fermi gases in which the strength of the interaction is continuously varied. The system features a crossover between a state described by the BCS theory of superconductivity and a Bose-Einstein condensate. A discussion of the Bertsch problem is presented. Keywords Fermi gases · Unitary limit · BCS-BEC crossover · Bertsch parameter

1 Introduction The properties of fermions in the so-called unitary limit were at the center of several presentations in the Rome’s Workshop “Mathematical Challenges of Zero Range Physics”. Although most of the rigorous results have been obtained so far for few particles, as reviewed in other seminars, a perduring attention has been devoted to the opposite, many-body, limit, where the number of fermions is much larger than one and it suitably scales with the size of the system. A particular attention focused during the last two decades on the so-called Bertsch

M. Iori Institute of Economics, Sant’Anna School of Advanced Studies, Pisa, Italy e-mail: [email protected] T. Macrì Departamento de Física Teórica e Experimental, Universidade Federal do Rio Grande do Norte, Natal, RN, Brazil e-mail: [email protected] A. Trombettoni () Department of Physics, University of Trieste, Trieste, Italy CNR-IOM DEMOCRITOS Simulation Center and SISSA, Trieste, Italy e-mail: [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. Michelangeli (ed.), Mathematical Challenges of Zero-Range Physics, Springer INdAM Series 42, https://doi.org/10.1007/978-3-030-60453-0_1

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M. Iori et al.

problem, posed by G.F. Bertsch during the 10th Conference on Advances in ManyBody Theory (Seattle, 1999). The challenge was (and still is) to determine the ground-state properties of the many-body system composed of spin-1/2 fermions interacting via a zero-range, infinite scattering-length, contact interaction. A related, quantitative question is to determine at zero temperature the chemical potential of the unitary Fermi gas (UFG), expressing it in units of the Fermi energy of the ideal Fermi gas. The resulting dimensionless number is the so-called Bertsch parameter, whose knowledge determines the thermodynamic properties of the UFG in the zero temperature limit. The goal of the seminar given by one of the authors was to stimulate a discussion on the connection between few-body physics and the many-body system. Since other chapters in the present book already discuss rigorous mathematical results for few fermions at the unitary limit, and given the fact that many properties of the UFG have already well reviewed in the literature [1, 2] (to which we refer for an extensive list of references), we focus in this chapter on a, hopefully pedagogical, presentation of results on the BEC-BCS crossover in the many-body limit to provide a guide to basic properties of such systems of interacting fermions. We hope that such presentation may stimulate further connections between mathematical results on the UFG and topics of many-body physics subject of current investigation. Finally, we thank the participants of the Workshop for many inspiring discussions on the mathematical literature of fermions at the unitary limit. The link between a Bose-Einstein condensate (BEC) state and superconductivity was first discussed by Ogg [3] and by Schafroth in [4]. They suggested that a pair of fermions with attractive interaction could be seen as a bosonic compound. This idea pointed to a possible explanation of superconductivity as a kind of BEC formed by pairs of fermions, the Cooper pairs [5]. With the appearance of the the BCS theory [6], however, the differences between boson condensates and Cooper pairs were emphasized. With the introduction of the concept of off-diagonal long range order (ODLRO) [7], a link between the two phenomena was put forward, with a generalization of BEC to fermionic systems. In 1969 Eagles discussed how in certain cases an electron gas can be so diluted that the distance between the pairs of fermions is much smaller than the interparticle one [8]. It is therefore possible to see Cooper pairs as a species of molecules. In this approximation it was possible to write equations for the energy gap and for the density. Subsequently, Leggett presented a first discussion of the BEC-BCS crossover [9]. In this paper the attention is focused on the properties of the groundstate at T = 0 of a diluted gas with two fermionic species. In the limit where as , the scattering length in s-wave, is much larger than r0 , the radius of the potential, the only relevant dimensionless quantity is as kF , where kF is the Fermi wavevector. It is therefore expected that at zero temperature the gap parameter Δ and the chemical potential μ are functions only of this variable. Disregarding the Hartree-Fock terms, Leggett arrived at the usual equation for the energy gap of the BCS theory, with a potential Vkk  , plus an equation for the density. In 1985 Nozières and Schmitt-Rink considered the behavior of the critical temperature using the Hubbard model with negative coupling constant [10]. The authors realized that the calculation of the Tc with the crossover equations is correct only in the BCS side, while in the BEC side

Thermodynamics of Fermi Gases Across the BCS-BEC Crossover

3

it is not. The mean field approximation is therefore only suitable for a qualitative description at T = 0, while its extension at finite temperature completely fails the prediction of the critical temperature in the strong coupling, BEC, side. These results are also valid in a continuous system, as shown in [11] and reviewed in [1, 12]. The main features emerging from these results are the following • the change in sign of the chemical potential μ near the unitary regime, in the BEC side; • the difference in the interpretation of the gap parameter Δ in the two limits; • the presence of non-condensed bosons: the dissociation temperature is in fact much greater than the critical temperature in the BEC side; • around the unitary limit, the critical temperature can be the same order of magnitude as (or at least not significantly smaller than) the Fermi temperature. In the next sections, following the scheme of the presentation in [13], we review how to describe BCS-BEC crossover in the mean field approximation and how to include Gaussian fluctuation to go beyond mean field results.

2 BCS-BEC Crossover in the Mean Field Approximation A system composed of two species of fermionic particles with interparticle attractive interactions can show a different behavior depending on their interaction, passing continuously from a BEC to a set of Cooper pairs of weakly interacting fermions. This process is known as the BEC-BCS crossover and some historical remarks on it were presented in the previous section. An experimental realization of the phenomenon was possible in ultracold atoms [2], although it plays a role in several other systems. What allows the transition from a BCS state to a condensate of bosons is the presence of the Fano-Feshbach resonance which varies the strength of the interaction between the fermions depending on the external field applied. The interest of those who study this model is mainly aimed at the unitary regime, that is of the crossover between the BCS state and the BEC. The crossover region is defined by the values of the parameter 1/kF as approximately in the region −1  1/kF as  1. In particular the unitary limit occurs when kF1as = 0 and the scattering length in s-wave, as , diverges. The radius of the potential also tends to zero r0 → 0. Since there is no longer a reference scale of length, the system shows universal behaviors, independent of the form of the interparticle potential. In the ground state it remains, as the only reference quantity, the Fermi energy F , as in the case of free fermions, despite the fact that we are dealing with a strongly interacting system [14]. In this section and in the following we will analyze the properties of the crossover in the mean field approximation both at T = 0 and at finite temperature, in order to explain its thermodynamic features. The quantities that describing the crossover at T = 0 are the chemical potential μ and the gap parameter Δ. At a finite temperature, the critical temperature Tc has to be determined. In the following we will assume h¯ = kB = 1 and set β = 1/T .

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M. Iori et al.

2.1 Mean Field Results at T = 0 Consider the fermionic Hamiltonian in three spatial dimensions with an attractive delta-shape potential H =



dxψσ† (x)

σ

   ∇2 − μ ψσ (x) − g dxψ↑† (x)ψ↓† (x)ψ↓ (x)ψ↑ (x), − 2m (1)

where σ labels the fermionic species. At T = 0, at the mean field level, the BEC-BCS crossover is described by the following equations: 1 1  1 = , g V 2Ek k

  1  ξk n= 1− , V Ek k

2



k where ξk = 2m −μ, Ek = ξk2 + Δ2 and n is the particle density (V is the volume). The contact potential leads to a divergence that must be regularized. In the BCS theory a cut-off is used (the Debye frequency ωD ), however this regularization is no longer possible in the present context and we introduce the regularization through the scattering wave length as , using the two-body transfer matrix, obtaining:

1  1 m 1 =− + . 4πas g V 2k k

The mean field equations at T = 0 then read   ξk 1  1− , n= V Ek k   m 1  1 1 − = − . 4πas V 2Ek 2k k

(2)

Thermodynamics of Fermi Gases Across the BCS-BEC Crossover

5

In three dimensions it is convenient to define the following dimensionless quantities [15], k k2 = , 2m Δ ξk ξx = = x 2 − x0 , Δ

μ , Δ  Ek = ξx2 + 1, Ex = Δ

x2 =

x0 =

where x0 is the crossover parameter that varies between −∞ (BEC) to ∞ (BCS). We then have 3

(2mΔ) 2 n= I2 (x0 ), 2π 2

(3)

1

−

1 2(2mΔ) 2 = I1 (x0 ), as π

where, following [15], we defined the integrals 



∞

I1 (x0 ) =

dxx

2

0

 I2 (x0 ) =

∞

0

1 1 − 2 Ex x

 ,

  ξx , dxx 2 1 − Ex

To make the equations dimensionless it is necessary to introduce the Fermi 1 energy F . The Fermi wavevector is kF = (3π 2 n) 3 , so that F = kF2 /2m = 2

(3π 2 n) 3 /2m. Substituting the density in the number equation (3), one has: F =

 2 2 3 3 Δ [I2 (x0 )] 3 . 2 1

kF = (2mΔ) 2



3 I2 (x0 ) 2

1 3

.

Finally we obtain  2 3 Δ 2 1 = ; F 3 I2 (x0 ) 1 2 =− as kF π



2 1 3 I2 (x0 )

(4)

1 3

I1 (x0 ).

(5)

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M. Iori et al. 2

1,5

Δ EF

1

0,5

0 –3

–2

–1

1

0

1

2

kF aS Fig. 1

Δ F

at T = 0 in the mean field approach

μ Inverting (5) one can find an expression for x0 = Δ as a function of as1kF to get μ Δ Δ F and F = x0 F . The two limiting cases, BCS and BEC, can be obtained by expanding the integrals I1 (x0 ) and I2 (x0 ) as a function of the elliptic integrals of the first and second species as in [15]. The numerical solution of the system of Eqs. (4) and (5) is shown in Figs. 1 and 2. The following values of the variables involved are obtained at the unitary limit: μ = 0.59F , Δ = 0.68F . From Fig. 2 we can see how the chemical potential decreases starting from the BCS limit, until it becomes negative shortly after the unitary regime so that bound states can begin to form. Figure 3 more clearly represents the behavior of μ, as it was chosen a different normalization (based on the Fermi energy and the bound state energy 0 = 1/mas2) depending on the sign of the chemical potential itself.

Thermodynamics of Fermi Gases Across the BCS-BEC Crossover

7

1

0

m EF

–1

–2

–3 –3

–2

–1

1

0

1

2

k F aS Fig. 2

μ F

at T = 0 in the mean field approach

1

m ϵF 0

m ϵ0 /2 –1 –1

–2

0

1

1

k F aS Fig. 3

μ F

for μ > 0 and

μ 0 /2

for μ < 0, at T = 0, in the mean field approach

2

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M. Iori et al.

3 BCS-BEC Crossover at Finite Temperature The study of the BEC-BCS crossover at finite temperature allows to find the transition temperature for 1/kF as varying, in particular that in the unitary regime of the crossover. At T = 0 the mean field allowed to find the typical parameters of the two limits BEC and BCS. It will be reminded instead in this section that the mean field fails to describe the critical temperature of the strong coupling limit. Typical finite temperature results can be obtained using different equivalent formalisms, e.g. • extension of the BCS theory to the limit of strong coupling; • generalization of self-consistent equations (Hartree-Fock method); • path integral. The method that will be used here in the presentation is the path integral formalism. It is convenient to define bosonic operators related to the creation and destruction of pairs of fermions: φ † (x) = ψ↑† (x) ψ↓† (x) .

φ (x) = ψ↓ (x) ψ↑ (x) ,

We can then write the interacting term in the Hamiltonian (1) as Hint = −g dxφ † (x) φ (x). To simplify the treatment the transformation of Hubbard-Stratonovich is applied (see e.g. [16]), which maps a system of interacting fermions in a system of noninteracting fermions in an effective field. Let us perform the transformation: −gφ † (x) φ (x)

−→

Δ¯ (x) Δ (x) + φ † (x) Δ (x) + Δ¯ (x) φ (x) , g

where the field Δ can be thought of as the bosonic field that averages the original interaction, and is related to the order parameter of the theory and the gap parameter. This field has to satisfy the relation:  

 ¯ D Δ, Δ exp − dx



0

β

Δ¯ (x) Δ (x) dτ g

= 1,

where      d Δ¯ xj , τ dΔ xj , τ D Δ, Δ¯ = , N



τ,j

with

N =

2πig . dxdτ

Thermodynamics of Fermi Gases Across the BCS-BEC Crossover

9

This identity can then be inserted into the partition function: 

 

¯ D ψσ , ψ¯ σ e−S ψσ ,ψσ =

Z= 









D ψσ , ψ¯ σ

=



D ψσ , ψ¯ σ

=

 2 ∇ − μ ψσ (x) exp − dτ dx ψ¯ σ (x) ∂τ − 2m 0 σ   † − g dxφ (x) φ (x) =





β

  ¯ D Δ, Δ exp −









β



dτ H0 +

0

 Hint



,

with  H0 =

dx

 σ

 Hint

 =



 ∇2 − μ ψσ (x) , ψ¯ σ (x) ∂τ − 2m



 Δ¯ (x) Δ (x) † − gφ (x) φ (x) . dx g

Details on this derivation are given in Appendix. Performing the transformation Δ (x)

−→

Δ (x) + gφ (x) ,

Δ¯ (x)

−→

Δ¯ (x) + gφ † (x) ,

one gets:  Hint



  Δ¯ (x) + gφ † (x) (Δ (x) + gφ (x)) † − gφ (x) φ (x) = = dx g

 Δ¯ (x) Δ (x) + Δ¯ (x) φ (x) + φ † (x) Δ (x) . = dx g 

One can rewrite the partition function as  Z=



D ψσ , ψ¯ σ





 D Δ, Δ¯   ∇2 − μ ψσ (x) + ψ¯ σ (x) ∂τ − 2m 0 σ    Δ¯ (x) Δ (x) + Δ¯ (x) φ (x) + φ † (x) Δ (x) . + dx g

 exp −



β

dτ

dx



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M. Iori et al.

One can also write the partition function Z in terms of an effective Hamiltonian  Z=

¯

 − dx Δ(x)Δ(x) g D Δ, Δ¯ e



 ˜

D ψσ , ψ¯ σ e−S ,

where S˜ =





β

dτ

dx

0





ψ¯ σ (x) ∂τ ψσ (x) + Heff Δ, Δ¯



 ,

σ

with Heff

 Δ, Δ¯ =

 dx  +

 σ



 ∇2 − μ ψσ (x) + ψ¯ σ (x) ∂τ − 2m

  dx Δ¯ (x) φ (x) + φ † (x) Δ (x) .

˜ The

action S is quadratic in the fermionic operators and can be integrated to give D ψσ , ψ¯ σ using the formula of the Gaussian integrals with Grassmann variables (see Appendix), obtaining 





 ˜ D ψ, ψ¯ e−S = det ∂τ + heff Δ, Δ¯ ,

where heff is the matrix representation of Heff . Indeed one has   β   1 ˜ ¯ = ¯ dτ ΔΔ S ψ, ψ + dx g 0       β ∇2 1 ¯ ¯ ¯ ¯ − μ ψσ − g ψ↑ ψ↓ ψ↓ ψ↑ + ΔΔ . dτ ψσ ∂τ − = dx 2m g 0 This can be conveniently expressed by introducing the Nambu spinors   ψ Ψ = ¯↑ , ψ↓

  Ψ † = ψ¯ ↑ ψ↓ .

Thermodynamics of Fermi Gases Across the BCS-BEC Crossover

11

One then has   β   1 ¯ = dτ ΔΔ S˜ ψ, ψ¯ + dx g 0       β ∇2 ¯ ΔΔ 1  Δ † −∂τ + 2m + μ Ψ+ + dτ Ψ ξk , = dx ∇2 g V −μ Δ¯ −∂τ − 2m 0 k since ψ¯ σ ∂τ ψσ = ψσ ∂τ ψ¯ σ . In this way we can rewrite the Lagrangian as L =

1  |Δ|2 − Ψ + G−1 Ψ + ξk , g V k

where G−1 is the inverse of the fermionic Nambu propagator: G

−1



 ∇2   +μ Δ −∂τ + 2m δ x, x  . x, x = ∇2 ¯ Δ −∂τ − 2m − μ 





It is therefore possible to rewrite the partition function in terms of an effective action  Z=



 ¯ D Δ, Δ¯ e−Seff Δ,Δ ,

with



Seff Δ, Δ¯ =





β

dτ 0



  ¯

 1 ΔΔ − ln det G−1 Δ, Δ¯ + dx ξk . g V k

The Fredholm determinant formula is now used ln det A = Tr ln A, obtaining the effective action    β  

 |Δ (τ, x) |2 1 Seff Δ, Δ¯ = dτ dx ξk − Tr ln G−1 [Δ (τ, x)] + g V 0 k

(6) (trace is made on space, time and Nambu indices). −1 The propagator can be written as: G−1 = G−1 0 + M, where G0 is the inverse of the Nambu free propagator:  G

−1

=

−∂τ +

∇2 2m

0



+μ

0 −∂τ −

∇2 2m

−μ

 ed M =

 0 Δ . Δ¯ 0

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M. Iori et al.

Then one can rewrite Tr ln G−1 using another Fredholm determinant formula: Tr [ln (11 + A)] = −Tr

∞   (−1)n An n=1

n

,

for which      −1 + M = ln det G + ln [det (11 + MG0 )] = ln det G−1 0 0 ∞     G0 M n −1 −1 . = Tr ln G0 + Tr ln (11 + G0 M) = Tr ln G0 − Tr − n n=1

This expansion can be done near the critical temperature, where the order parameter Δ is small.

3.1 Effective Action Let us now introduce the spatial Fourier transform of the wave function ψσ (τ, x) =

 e−ikx √ ψσ (τ, k), V k  eikx √ ψ¯ σ (τ, k). V k

ψ¯ σ (τ, x) =

It is therefore possible to rewrite the effective action as



Seff Δ, Δ¯ =



β 0

     V −∂ − ξ Δ τ k ¯ − ln dτ ΔΔ det ξk , + Δ¯ −∂τ + ξk g k

2

k where ξk = k − μ and k = 2m . We also introduce the Fourier transform also in imaginary time

ψk (τ ) =

 e−iωn τ √ ψk,n , β n

ψ¯ k (τ ) =

 eiωn τ √ ψ¯ k,n , β n

k

Thermodynamics of Fermi Gases Across the BCS-BEC Crossover

13

where ωn are the frequencies of Matsubara fermionics defined as ωn = Ignoring the fluctuations of Δ in imaginary time, one has

(2n+1)π . β

⎤ ⎡     Δ ¯ = V β ΔΔ ¯ − ln ⎣ det iωn − ξk ⎦+β Seff [Δ, Δ] ξk . Δ¯ iωn + ξk g k,n

Since ln



k,n det Ak,n

det

=

k,n ln[det Ak,n ],

k

it is necessary to evaluate

    Δ iωn − ξk 2 2 2 . + ξ + |Δ| = − ω n k Δ¯ iωn + ξk

The following expression is obtained for the effective action: ¯ = Seff [Δ, Δ]

    Vβ ¯ ln ωn2 + ξk2 + |Δ|2 + β ξk . ΔΔ − g k,n

(7)

k

By introducing the analytical continuation in the complex plane of the FermiDirac distribution it is possible to sum on n of a function F (ωn ) depending on Matsubara frequencies, as discussed in textbooks [17]. We remind the identity for which, given a function F (ω), we can write !  ! 1 F (ωn ) = f (z)F (−iz)!! − β n z=iωn n  1  +∞ dxf (x + iyb ) [F (x + iyb + i) − F (x + iyb − i)] , 2πi y −∞ b

(8) where f (z) = (1 + eβz )−1 and where the imaginary poles of analytic continuation are the frequencies of Matsubara and yb are the branch points, considering  → 0+ . " dz Equation (8) can be obtained by developing the integral I = f (z)F (z) 2πi on the complex plane. If we consider the integral on the red (outer) paths of Fig. 4 one obtains: 

 I = f (zp )Res F (zp ) , zp

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M. Iori et al.

Fig. 4 Contour for the sum over the Matsubara frequencies.

Re(z)

while on the blue (inner) paths one has: I =−



Res (f (iωn )) F (ωn )+

n

 1  +∞ + dxf (x + iyb ) [F (x + iyb + i) − F (x + iyb − i)] 2πi y −∞ b

and equating the two expressions one has (8).

3.2 Saddle Point Approximation The mean field approach is equivalent to perform the saddle point approximation, ¯ in the expression for the partition function where the whole integral is on D[Δ, Δ] it is estimated by setting the order parameter to the value that minimizes the action. We then have ! ¯ !! −Seff [Δ,Δ] Z e . ! Δ=Δ0

Using this approximation it is possible to estimate the transition temperature of the system, in fact the trivial saddle point Δ = 0, stable for all couplings at sufficiently large temperatures, it becomes unstable below a certain temperature Tc . To estimate this temperature it is necessary to derive two equations, the equation for the energy gap and the equation for the number of particles. The first equation is

Thermodynamics of Fermi Gases Across the BCS-BEC Crossover

15

obtained by imposing the saddle point condition: ! δSeff !! = 0. δΔ !Δ=Δ0

(9)

The second one can be obtained via the thermodynamic relation N =−

∂Ω , ∂μ

where Ω is the thermodynamic potential defined by the partition function as Z = e−βΩ . Within the saddle point approximation one has then ! Seff !! Ω= β !

(10)

. Δ=Δ0

The equation for the energy gap is obtained by using (9) and we have ! ! ! δSeff !! Δ V βΔ  ! − = , ! ! 2 2 2 ¯ g δ Δ Δ=0 ω + ξ + |Δ| Δ=0 n k k,n from which it follows 1 1  Δ = . 2 g βV ω + ξk2 + |Δ|2 k,n n Defining Ek2 = ξk2 + |Δ|2 , one can perform the sum over Matsubara frequencies using (8), where in this case the function F (ω) is F (ωn ) = 2 21 2 . This ωn +ξk +|Δ|

function has two poles iωn = ±Ek . Summing in this way on the Matsubara frequencies, we obtain the equation   1  f (Ek ) f (−Ek ) 1 =− , − g V 2Ek 2Ek k

(f is the Fermi-Dirac equation). Since f (Ek ) − f (−Ek ) = − tanh   Ek β tanh  2 1 1 = . g V 2Ek k



Ek β 2

 , then

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M. Iori et al.

In the limit Δ → 0 one has the required equation:   ξk β 1  tanh 2 1 = . g V 2ξk k

As expected, a divergence is present and it has to be regularized. It is therefore necessary to introduce the s-wave scattering length as that can regularize the divergence by means of (2). The following equation is obtained for the energy gap   ⎡ ⎤ ξk β tanh  2 1 1 m ⎣ ⎦. = − − 4πas V 2ξk 2k

(11)

k

In order to derive the equation for the number of particles it is necessary to introduce the thermodynamic potential Ω, defined by the effective action in the mean field approximation Seff =

    Vβ ¯ − ΔΔ ln ωn2 + Ek2 + β ξk = g k,n

=

Vβ ¯ ΔΔ − g



k

[ln(iωn + Ek ) + ln(−iωn + Ek )] + β

k,n



ξk .

k

We sum over Matsubara frequencies using (8): 1 ln(−iωn + Ek ) = β n  ∞ 1 =− dxf (x) {ln [−x + Ek − i] − ln [−x + Ek + i]} , 2πi −∞ where one can extend the integral up to −∞. By writing the Fermi-Dirac distribud tion function in the form f (x) = − β1 dx ln(1 + e−βx ), inserting this expression in the previous equation and integrating by parts, we obtain  1 1  ln(−iωn + Ek ) = ln 1 + e−βEk . β n β The effective action in the mean field approximation is therefore Seff =

      Vβ ¯ − ln 1 + e−βEk + ln 1 + eβEk + βξk . ΔΔ g k

(12)

Thermodynamics of Fermi Gases Across the BCS-BEC Crossover

17

Thus, the thermodynamic potential, given by (10), is Ω=

    1   Vβ ¯ − ΔΔ ln 1 + e−βEk + ln 1 + eβEk + βξk . gβ β k

We thus derive the equation for the density     1  βEk ξk 1 ∂Ω = 1 + tanh . n=− V ∂μ V 2 Ek k

Then, in the limit Δ → 0, the equation for the number of particles becomes n=

   ξk β 1  . 1 − tanh V 2

(13)

k

By denoting with T0 the mean field critical temperature, the determination of T0 and μ operation can be performed analytically in the two limits of weak and strong coupling. In the weak coupling limit (g → 0) one can recover to the BCS theory, where μ T0 . We use the particle number equation to fix the chemical potential. In the BCS limit the Cooper pairs are mainly found around the Fermi wavevector kF . For  2 2 3π n

this reason the contribution to n is maximum if  = F = 2m and this happens if F μ. The chemical potential in the weak coupling regime is therefore μ F . The critical temperature is obtained by using the equation for the gap (11), which can be rewritten in terms of density of states g() as:

−

m = 4πas



∞

⎡ dg() ⎣

 tanh

−F 2T0



2( − F )

0

⎤ −

1 ⎦. 2

The density of states of a Fermi gas in three dimensions is √  . g() = (m) √ 2π 2 3 2

Then the left-hand side of Eq. (11) can be rewritten as m g(F )π = . 4π|as | 2kF |as |

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M. Iori et al.

Inserting it into the equation for the gap, after defining the variable x =

 F



g(F )π = g(F ) − 2kF |as |

⎡

∞

√ dx x ⎣

 tanh

F 2T0 (x

− 1)

2(x − 1)

0



, one has

⎤ 1 ⎦ − . 2x

The value of this integral is 

∞ 0

⎡





⎤  γE  1 8e ⎣ ⎦ − dzz , = ln 2(z − 1) 2z πe2 t 1 2

tanh

where t = T0 /F and γE = − previous equation we get

z−1 2t

∞ 0

dxe−x ln x is Euler’s constant. Using the

8eγE −2 − π F e 2kF |as | . π

T0 =

(14)

This is the critical temperature in mean field approximation in the weak coupling limit, i.e. it is valid in the case where as1kF → −∞. Inserting the values of the constants we recover the BCS result T0 0.61F e

π F |as |

− 2k

.

Corrections to this formula [18–20] have been extensively discussed in literature and will be not considered in the present review. We now consider the strong coupling limit (g → ∞), for which kF1as → ∞, with μ < 0. In this case we use the equation for the gap (11) in order to find the chemical potential. We expect to find strongly bound fermion pairs (diatomic molecules) with binding energy 0 = ma1 2 and non-degenerate fermions for which it holds: μ T0 .  s  +|μ| 1 then Eq. (11) reads

1, tanh Since |μ| T0 2T0 m − = 4πas Defining z =

 |μ| ,

∞ 0



 1 1 − dg() . 2( + |μ|) 2

one has 3

−



m2 m = √ 4πas 2π 2



∞

dz 0

  √ √ 1 |μ| z |μ| 1 − . |μ| 2(z + 1) 2z

Since 

∞ 0

√ dz z



 1 π 1 − =− , 2(z + 1) 2z 2

Thermodynamics of Fermi Gases Across the BCS-BEC Crossover

19

it follows 3√ m m 2 |μ| . − =− √ 4πas 2 2π

Then, in the strong coupling limit the chemical potential becomes |μ| =

1 0 = . 2 2mas 2

Let use the equation for density a in number in order to calculate the critical temperature: 

∞

n= 0





 + |μ| dg() 1 − tanh 2T0

 .

In terms of the polylogarithmic function Lis (z) 3

n=−

4(T0 m) 2 1 4 π2

#

  π Li 3 −e|μ|/T0 . 4 2

It is e−|μ|/T0  1 since |μ| T0 . Therefore Li 3 (x) x and 2

3

n

Recalling that n =

(T0 m) 2 −|μ|/T0 e . 3√ π2 2

3

(2mF ) 2 3π 2

, then one gets 0   3  .

T0 2 ln

0 F

(15)

2

This temperature is to be interpreted as Tdiss of breaking of the pairs, in fact it is much larger than the critical temperature Tc in the BEC regime of a diluted boson gas. It therefore appears that T0 grows to infinity, but this is an artifact of the approximation used. We conclude that the mean field approximation correctly describes the BCS limit, while it is not able to describe the BEC limit. As mentioned above, the analysis of the mean field behavior of the chemical potential and the critical temperature can be done numerically. The graphs of the chemical potential calculated at T = Tc and the critical temperature in the mean field approximation are shown in Figs. 5 and 6.

20

M. Iori et al.

1

0

–1

m EF

–2

–3

–4 –2

–1

0

1

2

1

kF aS Fig. 5 Chemical potential in the crossover as obtained numerically from the equations for the energy gap (11) and for the density (13). In the BCS limit ( as1kF → −∞) μ F , in the BEC limit ( as1kF → +∞) μ − 20

4 Gaussian Fluctuations A quantitative theory of the BEC-BCS crossover at finite temperature requires the inclusion of the effects beyond the mean field [10, 11]. This theory is expected to interpolate between the known results in both limits, BEC and BCS. An issue arises from the fact that the critical temperature obtained from the mean field solution corresponds to the formation temperature of the pairs and not at the temperature of their condensation. The two temperatures coincide only in the BCS limit, while in the BEC regime the formation of pairs occur at a much larger temperature than the one at which pairs condense. There are several approaches to address the problem to the continuum: • diagrammatic approach: this method consists in using the transfer matrix T; • path integral; • quantum Monte Carlo simulations performed with finite size systems. Using the functional integral formalism, we now present the calculation of the fluctuations on top of the mean field following the lines of [11], continuing with the approach of the previous section. The general idea is to develop the action to the successive orders of the order parameter.

Thermodynamics of Fermi Gases Across the BCS-BEC Crossover

21

0,4 Tc TF 0,2

0 –2

–1

0

1

1

kF aS Fig. 6 Mean field critical temperature in the crossover as obtained numerically from the equations for the energy gap (11) and the density (13). We remind that in the main text the mean field critical temperature is denoted by T0 . In the BCS limit ( as1kF → −∞) T0 TBCS , and in the BEC limit ( as1kF → +∞) T0 diverges

For this purpose, consider the effective action (6) with G−1 = G−1 0 + M the inverse of the Nambu propagator. M can be written in matrix form as  Mk,q =

 0 Δ(q − k) , ¯ − k) Δ(q 0

where we explicitly denoted the momentum dependence. The first term of correction to the effective action is Tr G0 M and easily seen to be vanishing. To properly handle fluctuations on the second order it is convenient to rewrite the product between the matrices G0 and M as:   ¯ − , G0 M = G0 Δσ + + Δσ where σ ± = 12 (σx ± iσy ) and where σx and σy are Pauli matrices: σx =

  01 , 10

σy =

  0 −i . i 0

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M. Iori et al.

Then, at the second order, we need to calculate the value of the following trace 1 − Tr(G0 σ + G0 σ + ΔΔ + G0 σ + G0 σ − ΔΔ¯  + 2 ¯  + G0 σ − G0 σ − Δ¯ Δ¯  ), + G0 σ − G0 σ + ΔΔ where Δ = Δ(q) and Δ = Δ (−q). Since Δ is even in q, then ΔΔ¯  = |Δ(q)|2. Some terms vanish, so the correction to the action to be calculated is therefore 1 ¯  ). − Tr(G0 σ + G0 σ − ΔΔ¯  + G0 σ − G0 σ + ΔΔ 2 G0 is a function of k = (ω, k), whereas G0 in k + q where q = (ν, q), since the Fourier transform of the order parameter is used, as done previously in the case of the mean field. The function Δ is a bosonic field, so we have iωn =

2(n + 12 )π : Matsubara fermionic frequency, β

iνm =

2mπ : Matsubara bosonic frequency. β

Using the Fourier transform Δ=

 e−iνn τ −iqx √ √ Δq n , V β q,n

one gets Tr|Δ|2 G0 σ + G0 σ − =

1  (iω − ξk )−1 (iω − iν + ξk+q )−1 |Δ(q)|2. V q,ν,k,ω

Summing on the the Matsubara frequencies using the relation (8), one gets f (−ξk+q ) f (ξk ) 1 (iω − ξk )−1 (iω − iν + ξk+q )−1 = + , β ω ξk − iν + ξk+q iν − ξk+q − ξk if f (ξ ) = f (ξ + iν). One then has the property f (−z) − 1 = −f (z). The trace is then: Tr|Δ|2 G0 σ + G0 σ − = −

1  1 − ξk − ξk+q |Δ(q)|2, V iν − ξk+q − ξk k,q,ν

Thermodynamics of Fermi Gases Across the BCS-BEC Crossover

(ξk = k − μ and k = one can show that

k2 2m ).

23

Calculating the other terms of the trace (G0 σ − G0 σ + ),

Tr|Δ(q)|2 G0 σ − G0 σ + = Tr|Δ(q)|2G0 σ + G0 σ − . We can now replace the fluctuations in the effective action in order to obtain ¯ = Seff [0] + SGauss [Δ, Δ]

1  −1 Γ (q, iν)|Δ(q, iν)|2 , V q,ν

(16)

where Γ −1 (q, iν) is found to be Γ −1 (q, iν) =

  m 1  1 − f (ξk ) − f (ξk+q ) 1 − + V iν − ξk − ξk+q 2k 4πas

(17)

k

using (2).

4.1 Equations for the Energy Gap and for the Number of Particles We can now analyze the equation for the gap by proceeding analogously to what was done for the mean field, imposing that the saddle point condition around the trivial saddle point Δ0 : ! δSGauss !! = 0. ! ¯ δ Δ(q) Δ=Δ 0

One finds ⎡

m 1 ⎣ − = 4πas V

 tanh

k

ξk β 2

2ξk



⎤ 1 ⎦ − . 2k

We can now treat the equation for the number of particles. Consider therefore the thermodynamic potential, defined by Ω = − β1 ln Z, from which we can derive the density by the relation n=−

1 ∂Ω . V ∂μ

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M. Iori et al.

The partition function is then rewritten with Gaussian fluctuations such as  Z = Z (0)

¯ −βV DΔD Δe

q,k

−1 (q,iν)Δ(q) ¯ Δ(q)Γ

,

¯ −S[0] . As Δ, Δ¯ are bosonic variables and we have a DΔD Δe with Z (0) = Gaussian integral, we can simplify the partition function. One finds Ω = Ω (0) + Ω (2): Ω = Ω (0) −

1 ln Γ −1 . β q

The first term is that of the mean field, from which it derives the density n0 given by (13):    ξk β 1  n0 = . 1 − tanh V 2 k

One can therefore write n = n0 + δn, where δn denotes the correction to the mean field   1 ∂Ω (2) 1 ∂  −1 . δn = − = ln Γ V ∂μ βV ∂μ q The bosonic frequencies of Matsubara are thus left to be summed. The function χ(q, iν) =

1  1 − f (ξk ) − f (ξq+k ) , V iν − ξk − ξk+q q,ν,k

computed in ν = 0 is real and it has to be 1 + gχ(q, 0) > 0. One defines the phase shift δ(q, ω) which moves the function above and below the real axis (the cut): Γ −1 (q, ω ± i0+ ) =

1 + χ(q, ω ± i0+ ) = |Γ −1 (q, ω)|e∓iδ(q,ω) . g

We can now perform the sum over the frequencies  1 1 +∞ ln Γ −1 (q, ω) = dωb(ω)δ(q, ω), β ω π −∞ −1  is the Bose-Einstein distribution function and where b(ω) = eβω − 1     ln |Γ |e−iδ − ln |Γ |eiδ = −δ. 2i

Thermodynamics of Fermi Gases Across the BCS-BEC Crossover

25

The equation for density is therefore n = n0 +

 ∂δ(q, ω) 1  +∞ dω b(ω) . V q −∞ π ∂μ

(18)

The critical temperature and the chemical potential, considering the Gaussian fluctuations, are therefore given by the solution of the following system   ⎡ ⎤ ξk m 1  ⎣ tanh 2Tc 1 ⎦ − = − ; 4πas V 2ξk 2ξk k

n = n0 +

 ∂δ(q, ω) 1  +∞ dω b(ω) . V q −∞ π ∂μ

This can be solved analytically only in the BCS and BEC limits, while in the general case it must be solved numerically. In the weak coupling limit the phase shift δ(q, ω) is small because, for g → 0, the term g1 dominates. For this reason n n0 and we recover the mean field results Tc T0 TBCS . In the limit of strong coupling instead the function Γ (q, ω) has a pole at a given frequency η, in fact Γ is dominated by χ(q, ω). The discrete pole on the real axis then corresponds to a bound state. For g → ∞ the energy of the bound state becomes very large and the density in number is dominated by the density of the pairs n0 and the mean field term can be neglected. In this regime T  T0 , we can then approximately write the function Γ (q, ω) as Γ (q, iν)

R(q) , iν − ωB (q) + 2μ

where ωB is the energy of the bound and the state ωB −0 +

|q|2 . 4m

26

M. Iori et al.

By introducing the bosonic field φ(q) = √Δ(q) , we can rewrite the partition R(q) function as  ¯ (0) B (q)+2μ)φ(q,iν) ¯ − ν,q φ(q,iν)(iν−ω Z Z . DφD φe Summing up the bosonic Matsubara frequencies, using the (8), we find in this approximation Ω = Ω (0) +



  ln 1 − eβ(ωB (q)−2μ) .

q

The bosons are formed as bound states of two fermions, so the bosonic density will be nb = n2 . The critical temperature is therefore found to be  Tc = 2π

1 6π 2 ζ (3/2)

2 3

F 0.218F .

The value found is the one expected for the condensation of bosons. Therefore, the introduction of Gaussian fluctuations seems to be sufficient to qualitatively find the correct critical temperature throughout the crossover.

4.2 Results for the Critical Temperature and the Bertsch Parameter The calculation of the critical temperature along the crossover including the Gaussian fluctuations involve the determination of multiple integrals, that can be nevertheless simplified significantly by the use of contour integration (see more details on [13]). We then numerically integrated the equations for all μ and Tc pairs for various values of the parameter 1/(as kF ) that satisfy the gap equation. These pairs are then inserted into the equation for the density. In this way it is possible to obtain the critical temperature throughout the BEC-BCS crossover [11]. The results of our calculations are summarized in Figs. 7 and 8. The values obtained at the unitary limit are μ = 0.35F

and Tc = 0.22F .

With quantum Monte Carlo simulations the critical temperature value is around Tc = 0.171F (see [1] and references therein).

Thermodynamics of Fermi Gases Across the BCS-BEC Crossover

27

0,5

0

m EF

–0,5

–1

–1,5

–2 –2

–1

0

1

2

1

kF aS Fig. 7 Chemical potential obtained by including Gaussian fluctuations

Coming to the Bertsch parameter ξ , the value ξ = 0.376(4) was obtained in [21] and ξ = 0.370(9) in [22]. An upper bound was found in [23] giving ξ ≤ 0.383(1), and quantum Monte Carlo simulations gave the value ξ = 0.372(5) [24]. We refer to [1, 25, 26] for a comparison of different values of ξ found by quantum Monte Carlo and other methods. For our purposes, we comment that the value of ξ found by mean field is significantly larger (0.59) and that Gaussian fluctuations improve it. As an example, at the critical temperature one has μ/F = 0.54 in mean field and μ/F = 0.35 adding the Gaussian fluctuations.

5 Conclusions In this chapter we reviewed the problem of the BCS-BEC crossover for a system of spin 1/2 fermions. We approached the problem both at zero and finite temperature using the path-integral formalism. All along the review we highlighted results for the unitary limit where the scattering length diverges. We commented about the Bertsch parameter and we discussed about the importance of adding the Gaussian fluctuations on top of the mean field approach. Finally, we comment that the study of the relation between rigorous few-body treatments and many-body findings is a very deserving subject of investigation, and that many interesting results are yet to come in this direction.

28

M. Iori et al.

0,4 Tc TF 0,2

0 –2

–1

1

0

1

kF a S Fig. 8 Critical temperature along the BEC-BCS crossover obtained in the mean field approximation (dotted line) and with Gaussian fluctuations (continuous line)

Acknowledgements Discussions during the years with N. Defenu, L. Dell’Anna, G. Dell’Antonio, S. Fantoni, G. Gori, M. Iazzi, A. Michelangeli, G. Panati and C. Sa De Mélo are very gratefully acknowledged. T.M. acknowledges CNPq for support through Bolsa de produtividade em Pesquisa n.311079/2015-6. This work was supported by the Serrapilheira Institute (grant number Serra-1812-27802), CAPES-NUFFIC project number 88887.156521/2017-00. T.M. and A.T. thanks the Physics Department of the University of L’Aquila for the hospitality where part of the work was done.

Appendix In this appendix we review the definition of the partition function for a fermionic system with the use of Grassmann variables. Consider a fermionic system described by fermionic operators which are associated with Grassmann variables [27]: Ψ |ψ = ψ |ψ

and

$ ! † $ ! ¯ ψ¯ ! Ψ = ψ¯ ! ψ,

Thermodynamics of Fermi Gases Across the BCS-BEC Crossover

29

$ ! where |ψ and ψ¯ ! indicate coherent states, which can be decomposed as: |ψ = |0 − ψ |1 . The states |0 and |1 are the only ones possible and indicate the states with zero and one particle respectively. The fermion operators anti-commutate and therefore we have: % & % & {Ψ, Ψ } = Ψ † , Ψ † = 0, Ψ, Ψ † = 1. The following properties of the Grassmann variables are obtained: ψ 2 = ψ¯ 2 = 0; $ ' $ ' ¯ ¯ ψ|ψ = 0|0 + 1|ψ¯ ψ|1 = eψψ ; 

 ψ dψ = −

dψ ψ = 1;

 1 dψ = 0. In general  xjn . . . xj1 dxi1 . . . dxin = 

  j1 . . . jn i1 . . . in

and is different from zero only if (j1 . . . jn ) coincide with (i1 . . . in ). 

  1 with an even number of permutations j1 . . . jn = i1 . . . in −1 with an odd number of permutations

We then have the following relations: • Gaussian integrals: 

 ¯ ¯ ψ = det M; e−ψMψ D ψ,

• completeness relation:  I=

$ ! ¯  ¯ ψ ; |ψ ψ¯ ! e−ψψ D ψ,

30

M. Iori et al.

• trace of an operator:  TrΩ =

$

!

 ¯ ¯ ψ ; −ψ¯ ! Ω |ψ e−ψψ D ψ,



¯ ψ we mean the functional integral. where with D ψ, The completeness relation is obtained developing coherent states and the exponential:    

 ¯ ¯ ψ = D ψ, (|0 + |1 ψ) 0| − ψ¯ 1| 1 − ψψ 

 

 ¯ + |1 1| ψ ψ¯ )D ψ, ¯ ψ = |0 0| + |1 1| = 11, − |0 0| ψψ

while an operator trace is obtained from:  TrΩ = 





   

¯ ψ = ¯ 0| + 1| ψ¯ Ω (|0 − ψ |1) 1 − ψψ D ψ,

 

¯ ψ = 0| Ω |0 + 1| Ω |1 = TrΩ. ¯ − 1| Ω |1 ψψ ¯ D ψ, − 0| Ω |0 ψψ

The partition function of a canonical ensemble is defined as: Z=



e−βEn = Tre−βEn .

n

It can be written using the path integral approach, developing the exponential trace, using e

−βH

N       βH βH βH ... 1 − = lim exp − = 1− , N→∞ N N N ( )* + N times

for large N. Then     ! 

βH βH ¯ |ψ0  e−ψ0 ψ0 D ψ¯ 0 , ψ0 = ... 1 − −ψ¯ 0 ! 1 − N N      ! ! $ $ βH βH |ψN−1  ψ¯ N−1 ! 1 − |ψN−2  . . . = −ψ¯ 0 ! 1 − N N   N−1  $ ! βH ¯ ! ¯ |ψ0  . . . ψ1 1 − e−ψi ψi d ψ¯ i ψi , N 

Z=

$

i=0

where N − 1 completeness relations were inserted.

Thermodynamics of Fermi Gases Across the BCS-BEC Crossover

31

Since we are dealing with fermionic systems, it is necessary to impose antiperiodic boundary conditions −ψ¯ 0 = ψ¯ N

and

− ψ0 = ψN .

Then     N−1  $ ! βH −ψ¯ i ψi ¯ ! ¯ |ψi  e ψi+1 1 − d ψi dψi , Z= N i=0

from which Z=

 N−1 

exp

i=0

β N



   ψ¯ i+1 − ψ¯ i ψi − H ψ¯ i+1 , ψi d ψ¯ i dψi . β/N

Define now τ as the imaginary time (the inverse of the temperature) that will vary ¯ −ψ¯ i as to the derivative with between 0 and β. We can then (non-rigorously) see ψi+1 β/N respect to this time. Analogously we can perform the approximation       β , ψ (τ ) . H ψ¯ i+1 , ψi = H ψ¯ τ + N Introducing the action S=

 N    β  ψ¯ i (ψi − ψi−1 ) + H ψ¯ i , ψi−1 , N i=0

in the limit  =

β N

→ 0 and then for N → ∞ one has 

β

S=

  dτ ψ¯ (τ ) ∂τ ψ (τ ) + H ψ¯ (τ ) , ψ (τ ) .

0

In conclusion it is possible to rewrite the system partition function as  Z=





 ¯ ¯ ψ e−S ψ,ψ , D ψ,

or, more explicitly  Z=

 ¯ ψ exp D ψ,



β 0



 ∂ − H ψ (τ ) dτ . ψ¯ (τ ) − ∂τ

32

M. Iori et al.

In the grand canonical ensemble, we can introduce the number operator Nˆ 

N=

† ckσ ckσ ,

k,σ =↑↓ † with ckσ e ckσ are defined from

ψσ (x) =

 eirk √ ckσ , V k

ψσ† (x) =

 e−irk † √ ckσ . V k

Therefore, in a fermionic system, with anti-periodic boundary conditions in imaginary time: and ψ¯ σ (β, x) = −ψ¯ σ (0, x) ,

ψσ (β, x) = −ψσ (0, x) the action of the system becomes 

S ψ¯ σ , ψσ =





β

dτ 0

dx

 σ

ψ¯ σ (x)



 ∂ − μ ψσ (x) + H (x) , ∂τ

with x = (τ, x).

References 1. Zwerger, W. (ed.): The BCS-BEC Crossover and Unitary Fermi Gas. Springer , Heidelberg (2012) 2. Giorgini, S., Pitaevskii, L.P., Stringari, S.: Theory of ultracold atomic Fermi gases. Rev. Mod. Phys. 80, 1215 (2008) 3. Ogg, R.A.: Bose-Einstein condensation of trapped electron pairs. Phase separation and superconductivity of metal-ammonia solutions. Phys. Rev. Lett. 69, 243 (1946) 4. Schafroth, M.R.: Theory of superconductivity. Phys. Rev. 96, 1442 (1954) 5. Blatt, J.M.: Theory of Superconductivity. Academic, New York (1974) 6. Bardeen, J., Cooper, L.N., Schrieffer, J.R.: Theory of superconductivity. Phys. Rev. 108, 1175 (1957) 7. Yang, C.N.: Concept of off-diagonal long-range order and the quantum phases of liquid He and of superconductors. Rev. Mod. Phys. 34, 694 (1962) 8. Eagles, D.M.: Possible pairing without superconductivity at low carrier concentrations in bulk and thin-film superconducting semiconductors. Phys. Rev. 186, 456 (1969) 9. Leggett, A.J.: Diatomic molecules and Cooper pairs. In: Modern Trends in the Theory of Condensed Matter. Lecture Notes in Physics, vol. 115, p. 13. Springer, Berlin, Heidelberg (1980); Cooper pairing in spin-polarized Fermi systems. J. Phys. (Paris) Colloq. 41, C7-19 (1980) 10. P. Nozières, S. Schmitt-Rink, Bose condensation in an attractive fermion gas: from weak to strong coupling superconductivity. J. Low Temp. Phys. 59, 195 (1985)

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11. Sa de Melo, C.A.R., Randeria, M., Engelbrecht, J.R.: Crossover from BCS to Bose superconductivity - Transition temparetaure and time-dependent Ginzburg-Landau theory. Phys. Rev. Lett. 71, 3202 (1993) 12. Leggett, A.J.: Quantum Liquids: Bose Condensation and Cooper Pairing in Condensed-Matter Systems. Oxford University Press, Oxford (2006) 13. Iori, M.: Proprietà termodinamiche di fermioni ultrafreddi con accoppiamento spin-orbita. Graduation Thesis. Università degli Studi di Torino (2013) 14. Ho, T.L.: Universal thermodynamics of degenerate quantum gases in the unitarity limit. Phys. Rev. Lett. 92, 090402 (2004) 15. Marini, M., Pistolesi, F., Strinati, G.C.: Evolution from BCS superconductivity to Bose condensation: analytic results for the crossover in three dimensions. Eur. Phys. J. 1, 151 (1998) 16. Altland, A., Simons, B.D.: Condensed Matter Field Theory. Cambridge University Press, Cambrdige (2006) 17. Fetter, A.L., Walecka, J.D.: Quantum Theory of Many-Particle Systems. McGraw-Hill, New York (1971) 18. Gorkov, L.P., Melik-Barkhudarov, T.M.: Contribution to the theory of superfluidity in an imperfect Fermi gas. Sov. Phys.: J. Exp. Theor. Phys. 13, 1018 (1961) [Zh. Eksp. Teor. Fiz. 40, 1452 (1961)] 19. Pethick, C., Smith, H.: Bose-Einstein Condensation in Dilute Gases. Cambridge University Press, Cambrdige (2002) 20. Pisani, L., Perali, A., Pieri, P., Strinati, G.C.: Entanglement between pairing and screening in the Gorkov-Melik-Barkhudarov correction to the critical temperature throughout the BCS-BEC crossover. Phys. Rev. B 97, 014528 (2018) 21. Ku, M.J.H., Sommer, A.T., Cheuk, L.W., Zwierlein, M.W.: Revealing the superfluid lambda transition in the universal thermodynamics of a unitary Fermi gas. Science 335, 563 (2012) 22. Zürn, G., Lompe, T., Wenz, A.N., Jochim, S., Julienne, P.S., Hutson, J.M.: Precise characterization of 6Li feshbach resonances using trap-sideband-resolved RF spectroscopy of weakly bound molecules. Phys. Rev. Lett. 110, 135301 (2013) 23. Forbes, M.M., Gandolfi, S., Gezerlis, A.: Resonantly interacting Fermions in a box. Phys. Rev. Lett. 106, 235303 (2011) 24. Carlson, J., Gandolfi, S., Schmidt, K.E., Zhang, S.: Auxiliary-field quantum Monte Carlo method for strongly paired fermions. Phys. Rev. A 84, 061602(R) (2011) 25. Carlson, J., Gandolfi, S., Gezerlis, A.: Quantum Monte Carlo approaches to nuclear and atomic physics. Prog. Theor. Exp. Phys. 2012, 01A209 (2012) 26. Pessoa, R., Gandolfi, S., Vitiello, S.A., Schmidt, K.E.: Contact interaction in an unitary ultracold Fermi gas. Phys. Rev. A 92, 063625 (2015) 27. Shankar, R.: Principles of Quantum Mechanics. Plenum, New York (1994)

Scattering Theory for Delta-Potentials Supported by Locally Deformed Planes Claudio Cacciapuoti, Davide Fermi, and Andrea Posilicano

Abstract We give a self-contained description of the main results from the paper (Cacciapuoti et al., J Math Anal Appl 473(1):215–257, 2019). We focus on the fundamental concepts and on the chief achievements, omitting some auxiliary results and a number of technical details given in the original paper. We discuss the scattering problem for a quantum particle in dimension three in the presence of a semitransparent unbounded obstacle, modelled by a δ-interaction supported on a surface obtained through a local, Lipschitz continuous deformation of a flat plane. We discuss existence and asymptotic completeness of the wave operators with respect to a suitable reference dynamics. Additionally, we provide an explicit expression for the related scattering matrix and show that it converges to the identity as the deformation goes to zero (giving a quantitative estimates on the rate of convergence). Keywords Scattering theory · Point interactions supported by unbounded hypersurfaces · Kre˘ın’s resolvent formulae MSC 2010 35P25, 47A40, 35J25

1 Introduction In this brief report we give a self-contained description of the main results from the paper [12]. We will only outline the strategy of the approach and refer to the original paper for the proofs and several technical details.

C. Cacciapuoti · A. Posilicano DiSAT, Sezione di Matematica, Università dell’Insubria, Como, Italy e-mail: [email protected]; [email protected] D. Fermi () Classe di Scienze, Scuola Normale Superiore, Pisa, Italy © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. Michelangeli (ed.), Mathematical Challenges of Zero-Range Physics, Springer INdAM Series 42, https://doi.org/10.1007/978-3-030-60453-0_2

35

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We study the scattering problem for a quantum particle in dimension three in the presence of a semitransparent unbounded obstacle. The obstacle is modeled by a δ-interaction supported on the surface πF := {x ≡ (x 1 , x ) ∈ R3 | x 1 = F (x )},

(1)

where F : R2 → R has compact support and is Lipschitz continuous, i.e., F ∈ C00,1 (R2 , R) .

(2)

The generator of the dynamics we are interested in is the self-adjoint operator formally written as AF = −Δ + α δπF ,

α > 0,

(3)

where δπF is the Dirac δ-distribution supported on the surface πF and α is a parameter associated to the opacity of the obstacle. We remark that the operator AF can be rigorously defined as a self-adjoint extension of the symmetric operator S := −Δ with domain Dom(S) := C0∞ (R3 \πF ) (the space of smooth functions with compact support outside πF ). This is done, rigorously, in Sect. 3, through the approach to self-adjoint extension theory of symmetric operators developed by one of the authors in [32]. Here we just notice that functions in the domain of AF belong to H 2 (R3 \πF ), are continuous on πF (their trace on πF , denoted by f πF , is well defined), and satisfy the semitransparent boundary condition [∂n f ]πF = α f πF , where [∂n f ]πF denotes the jump of the normal derivative across the surface πF . Moreover, AF f = −Δf for f ∈ C0∞ (R3 \πF ). We also remark that α = +∞ corresponds to total reflection. More precisely, for α = +∞ the operator AF coincides with the Friedrichs extension of S; i.e., the self-adjoint realization of the (positive) Laplacian defined on the set of functions in H 2 (R3 \πF ) which satisfy the Dirichlet boundary condition f πF = 0. Rigorous mathematical analysis of models with surface supported δ-interactions dates back to at least the late nineteen-eighties, see [3], where δ-interactions supported on the surface of a sphere were intended to model effective interactions among nucleons, within the so called “Surface Delta Interaction” model, see [23, 30]. In quantum mechanics, operators of the form (3) could also be understood as approximations for peaked potentials localized in a neighborhood of πF , in the same spirit as in [5]. More recently, similar models appeared in the analysis of propagation of acoustic and electromagnetic waves, see, e.g., [39].

Scattering Theory for Delta-Potentials Supported by Locally Deformed Planes

37

The scattering problem, in relation with operators of the form (3), has been studied first in [16], see also [22], under the assumption that α = α(x ) goes to zero as x → ∞. We remark that in this setting, since the obstacle is essentially transparent at infinity, it is reasonable to choose as reference dynamics the free one. In particular, it is possible to prove existence and completeness of the wave operators with respect to the scattering couple (AF , A∅ ), where A∅ := −Δ ,

Dom(A∅ ) := H 2 (R3 ).

(4)

In our analysis, instead, we keep α constant, so that AF models a genuinely unbounded semitransparent obstacle. For this reason, one cannot expect existence of the wave operators for the scattering couple (AF , A∅ ), while it is reasonable to consider the scattering couple (AF , A0 ), where A0 := AF =0 . We remark that the operator A0 enjoys a factorized structure (see Remark 2 () below) which allows to write it as the sum of a trivial term (A0 , the “free” (1) Laplacian in the directions parallel to π0 := πF =0 ) and a simple one (A0 , the one dimensional Laplacian perturbed by a δ-interaction). It is not difficult to convince oneself that relevant quantities related to A0 , such as the integral kernel of its resolvent and the generalized eigenfunctions, can be explicitly computed, see Eq. (22). An extensive literature exists on the scattering by unbounded obstacles modeled through a Dirichlet boundary condition, see, e.g., the monograph [33], also in the presence of rough surfaces, see, e.g., [13, 14]. Much less is known on the scattering by a semitransparent, unbounded obstacle such as the one described by (3). In the case α < 0, corresponding to an attractive (singular) potential term, one main question is whether there are isolated eigenvalues in the spectrum of the operator AF . This problem has been studied in several papers, see, e.g., [15, 18] and references therein for the 2D case, [21] for the 3D case, and the survey [17]. To the best of our knowledge, the scattering problem is, instead, for the most open. In the 2D case existence and completeness of the wave operators (with respect to the scattering couple (AF , A0 )) have been provided in [19]. For α < 0 the essential spectrum of AF (as well as A0 ) coincides with [−α 2 /4, +∞), and in [19] the authors also provide a formula for the scattering matrix for the negative part of the spectrum. We remark that in this case the scattering problem is essentially one-dimensional in the sense that it is described by a 2 × 2 matrix of reflection and transmission amplitudes. We consider the more involved three-dimensional problem. For α > 0 we provide a comprehensive scattering analysis. In particular: we obtain the Limiting Absorption Principle (LAP) both for the operators A0 and AF , see Proposition 1 and Theorem 2 below; we prove existence and completeness of the wave operators for the couple (AF , A0 ), see Theorem 3; and obtain an explicit formula for the corresponding scattering matrix SF (λ), for any energy λ ∈ (0, +∞)\σp+ (AF ), where σp+ (AF ) is the (possibly empty) discrete set of embedded eigenvalues of AF , see Corollary 1. We conjecture that our assumptions on F imply that the set σp+ (AF ) is empty.

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The latter formula allows us to prove that the scattering matrix converges (in operator norm) to the identity as F goes to zero, more precisely 

 SF (λ) − 12B(L2 (S2 )) = O

R2

dx |F (x )|γ

,

0 < γ < 1,

see Theorem 5. In what follows we give an outline of the strategy of the approach used in [12] and of the structure of this review. We use the approach developed in [32] to obtain Kre˘ın-type formulae for the resolvents of AF and A0 , see Eq.s (13) and (14) respectively. With these formulae we write the resolvents RF (z) and R0 (z) in terms of operators defined by means of R∅ (z) (the resolvent of the free Laplacian A∅ ) and of the trace operator τF (resp. τ0 ) on the surface πF (resp. π0 ). Afterwords, by simple algebraic manipulations, we obtain a formula for the difference RF (z) − R0 (z), see Eq.s (17) and (18). Immediate consequences of the Kre˘ın-type formulae are the rigorous definitions of the operators AF and A0 and the precise characterization of their domains (the same results could also be achieved through quadratic form methods, see, e.g., [4, 9, 20]). After this preparatory analysis we focus on the proof of the LAPs. We prove first the LAP for the operator A0 , see Proposition 1. This is achieved by exploiting the factorization of A0 (recall Remark 2). More precisely, we use an abstract result from Ben-Artzi and Devinatz [6], which allows to infer the LAP for an operator written as a sum of tensor products, once that certain assumptions on the single terms in the sum are verified. Among these assumptions are the LAPs () (1) for the operators A(1) 0 and A0 in Eq. (16). The LAP for A0 can be obtained by a straightforward, although lengthy, analysis which exploits an explicit formula for the resolvent of A(1) 0 , for the details we refer to Appendix A in [12]. The LAP for () A0 , instead, is a classical result, see, e.g., [1]. To prove the LAP for the operator AF , see Theorem 2, we use a perturbation result from Renger [34] and [35]. The perturbation method is based on the LAP for A0 and several bounds on the resolvents RF (z) and R0 (z), and their difference. The proof is rather technical and we refer to Sect. 4 below and to the main paper [12, Section 4.2] for the details. Here we just remark that the formula for the difference RF (z) − R0 (z) (Eqs. (17) and (18)) is used to prove the bound (21). The next step in our analysis is the proof of the existence and completeness of the wave operators for the couple (AF , A0 ), Theorem 3. To achieve this goal we follow the same strategy used in [26]. Due to the unboundedness of the obstacle, which does not allow us to choose as reference dynamics the trivial one generated by A∅ , several non trivial adjustments are needed. The first step is to obtain a different representation formula for the resolvent difference, see Eq. (28). This is again a Kre˘ın-type formula.

Scattering Theory for Delta-Potentials Supported by Locally Deformed Planes

39

A first consequence of this second representation, is identity (30). The latter allows us to use several general results from Schechter (see, in particular, [36, Section 4.2]), which, in turn, are the starting point to set up the approach developed in [26] to prove existence and completeness of the wave operators for the couple (AF , A0 ). Formula (28), is also a fundamental tool in the study of the scattering matrix, for which we follow the same kind of reasoning as in [26, Sec. 4] (see also [27]). The main observation is that the resolvent difference is written in the factorized form RF (μ) − R0 (μ) = g˘Σ (μ)∗ ΛF,Σ (μ) g˘Σ (μ)

μ < 0.

The latter identity allows us to use some general result from the monograph [40] (see, in particular, [40, Section V.5.3]) to show that the wave operators for the couple (RF (μ), R0 (μ)) exist and are complete, and obtain an expression for the corresponding scattering matrix S μ (λ), see Theorem 4 below. Once that the scattering problem for the couple (RF (μ), R0 (μ)) is settled, one can use the Birman-Kato invariance principle, in particular formula (35), to obtain the scattering matrix for the couple (AF , A0 ) (see Corollary 1). To conclude, we use the explicit formula for the scattering matrix for the couple (AF , A0 ), to obtain a bound in terms of the deformation F , see Theorem 5. We point out another relevant feature of formula (28). All the quantities on the right-hand side of the identity involve, instead of the trace operators τF and τ0 , the trace operator τΣ (see Eq. (24)) on the compact support of the deformation F (Σ denotes the support of F ). In this sense, formula (28) somehow exploits the fact that the operators AF and A0 coincide on functions which are zero inside a certain compact region. Formula (28) resembles the one used in [19] (see, especially, Th. 2.5 and Eq. (2.14) therein). The paper is structured as follows. In Sect. 2 we discuss several preliminary results, concerning the boundedness properties of the free resolvent R∅ (z) and of the trace operators between (possibly weighted) Sobolev spaces. In Sect. 3, we present the Kre˘ın-type resolvent formulae for RF (z) and R0 (z) in terms of the free resolvent R∅ (z), and obtain the first formula for the resolvent difference, see Eqs. (17)–(18). In Sect. 4 we discuss LAP for the operators A0 and AF . In Sect. 5 we present the second Kre˘ın-type resolvent formula for the difference RF (z) − R0 (z), see Eq. (28). In Sect. 6, we discuss existence and completeness of the wave operators for the couple (AF , A0 ). In Sect. 7 we present the explicit formula for the scattering matrix for the couple (AF , A0 ) and the bound on SF (λ) − 1 as F → 0.

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2 Preliminaries For k ∈ N, let D(Rk ) be the topological vector space of compactly supported, smooth functions f : Rk → C and let D  (Rk ) be its dual, i.e., the space of Schwartz distributions on Rk . Most of our considerations involve Sobolev spaces of L2 -type, that we indicate with H r (Rk ) for r ∈ R. For n ∈ {0, 1, 2, ...}, H n (Rk ) is the usual Hilbert space of distributions which are square integrable along with all their derivatives up to order n (of course, H 0 (Rk ) ≡ L2 (Rk )); for θ ∈ (0, 1), H n+θ (Rk ) can be characterized via complex interpolation setting H n+θ (Rk ) := [H n (Rk ), H n+1 (Rk )]θ ; finally, for any r  0, H −r (Rk ) := (H r (Rk )) is the topological dual of H r (Rk ). As well known, D(Rk ) is a dense subset of H r (Rk ) for any r ∈ R and we have the continuous  embedding H r (Rk ) → H r (Rk ) for r  r  . Let us also remark that the distributional  Fourier transform F : D (Rk ) → D  (Rk ) allows to define the equivalent norm f → (1 + |k|2 )r/2 Ff L2 (Rk ) on H r (Rk ) for any r ∈ R. Our considerations refer to the 3-dimensional setting described hereafter. The free Laplacian is the self-adjoint operator A∅ defined in Eq. (4); this has purely absolutely continuous spectrum σ (A∅ ) = σac (A∅ ) = [0, +∞) and the corresponding free resolvent operator is R∅ (z) := (A∅ − z)−1 : L2 (R3 ) → H 2 (R3 ) ,

z ∈ C\[0, +∞) .

Notice that R∅ (z) ∈ B(H r (R3 ), H r+2 (R3 )) for all r ∈ R.1 Recall that F : R2 → R is any compactly supported and Lipschitz continuous function, see Eq. (2). Correspondingly, we consider the surface πF defined in Eq. (1). This is referred to as the “deformed” plane, since it is regarded as a compact deformation of the “flat” reference plane corresponding to F ≡ 0, namely π0 := {x ∈ R3 | x 1 = 0} . Note that the change of coordinates (x 1 , x ) → (x 1 − F (x ), x ) on R3 allows to identify πF with R2 ; the induced map f (x 1 , x ) → f (x 1 + F (x ), x ) defines an automorphism of H r (R3 ) for any |r|  1 (see, e.g., [24, Sec. 1.3.3] or [29, Ch.3]). The evaluation map τF associated to πF , acting on continuous functions f ∈ C 0 (R3 ) as τF f := f (F (x ), x ), can be uniquely extended to the bounded and surjective trace operator (see, e.g., [29, Th. 3.37]) τF ∈ B(H r+1/2 (R3 ), H r (R2 ))

for r ∈ (0, 1/2] .

(5)

Note that the map τ0 related to π0 fulfills τ0 ∈ B(H r+1/2 (R3 ), H r (R2 )) for all r > 0. 1 For any pair of Banach spaces X and Y , the Banach space of bounded linear operators from X to Y and the corresponding two-sided ideal of compact operators are denoted, respectively, with B(X, Y ) (B(X) ≡ B(X, X)) and S∞ (X, Y ) (S∞ (X) ≡ S∞ (X, X)).

Scattering Theory for Delta-Potentials Supported by Locally Deformed Planes

41

On the other hand, consider the C 0,1 open domains ΩF± := {x ∈ R3 | ± x 1 > ±F (x )} and the associated lateral traces τF± ∈ B(H r+1/2(ΩF± ), H r (R2 )) for r ∈ (0, 1), defined as the unique bounded extensions of the evaluation maps (τF± f± )(x ) := limε↓0 f± (F (x )±ε, x ) for f± ∈ C 0 (ΩF± ); using these elements, τF can be uniquely extended by continuity setting, for r ∈ (0, 1), τF : H r+1/2(R3 \πF ) ≡ H r+1/2(ΩF− ) ⊕ H r+1/2(ΩF+ ) → H r (R2 ) , τF (f− ⊕f+ ) :=

1 − (τ f− + τF+ f+ ) . 2 F

For later notational convenience, we introduce the notations f π ± := τF± f , F

f πF := τF f ,

[f ]πF := τF+ f − τF− f .

The limiting absorption principles reported in Sect. 4 naturally reside in the framework of weighted Sobolev spaces. For k ∈ N, let w ∈ L1loc (Rk, [0, +∞)) be such that 1/w ∈ L1loc (Rk, [0, +∞)) : for n ∈ {0, 1, 2, ...}, we set Hwn (Rk ) := {f ∈ D  (Rk ) | |β|n Rk dx w(x) |(∂ βf )(x)|2 < +∞} and Hw0 (Rk ) ≡ L2w (Rk ); for θ ∈ (0, 1), by complex interpolation we define Hwn+θ (Rk ) := [Hwn (Rk ), Hwn+1 (Rk )]θ ; for r  0, using the standard duality induced by the L2 scalar product we put r (Rk )) . Hw−r (Rk ) := (H1/w In our analysis we consider the weights ws1(x 1 ) := (1 + |x 1 |2 )s1 on R, ws(x ) := (1 + |x |2 )s on R2 and ws1,s(x 1, x ) := ws1(x 1 ) ws(x ) on R3 , for suitable s1, s ∈ R. Correspondingly, for r ∈ R we set Hsr1(R) ≡ Hwr s (R) , 1

Hsr(R2 ) ≡ Hwr s (R2 ) ,

Hsr1,s(R3 ) ≡ Hwr s ,s (R3 ) .



1 

−r Notice that (Hsr1(R)) = H−s (R). Besides, for any s1 ∈ R and all r ∈ R, the map 1 √  f1 → ws1 f1 in D (R) induces the isomorphism of Banach spaces Hsr1(R) → H r (R) (see, e.g., [38, §6.1]); in particular, this yields the continuous embeddings  Hsr1(R) → Hsr1 (R) for r  r  . Analogous results hold also for the spaces Hsr(R2 ) r

min(r1,r)

and Hsr1,s(R3 ). To say more, we have Hsr11(R) ⊗ Hs(R2 ) → Hs1,s r1, r  0. Regarding the free resolvent operator R∅ (z), there holds

(R3 ) for all

(R3 )) for z ∈ C\[0, +∞), s1, s ∈ R, r ∈ R . R∅ (z) ∈ B(Hsr1,s(R3 ), Hsr+2 1,s (6)

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On the other hand, for F ∈ C00,1 (R2 , R) as before, the trace operator τF possesses a unique bounded extension r+1/2

τF ∈ B(Hs1,s (R3 ), Hsr(R2 ))

for s1, s ∈ R, r ∈ (0, 1/2] ;

(7)

in particular, when F ≡ 0 a claim analogous to Eq. (7) holds true for all r > 0. Our main results regarding the scattering matrix (see Sect. 7) are stated in terms of Sobolev spaces on domains, defined as follows for any open set Ω ⊆ Rk with Lipschitz continuous boundary and for all r ∈ R: H r (Ω) := {f ∈ D  (Ω) | ∃ f˜ ∈ H r (Rk ) such that f˜  Ω = f } ,

(8)

HΩr (Rk ) := {f ∈ H r (Rk ) | suppf ⊆ Ω} . Notice that the spaces H r (Ω) share the interpolation property (see, e.g., [37]); besides, by [29, Th.s 3.14 and 3.29] we have the duality isomorphic identification (H r (Ω)) HΩ−r (Rk )

for any r ∈ R.

(9)

Before proceeding, let us mention a few basic facts underlying the derivation of a number of results mentioned in this work. On the one hand, the algebraic tensor product D(R) ⊗ D(R2 ) is a dense subset of D(R3 ) (see, e.g., [8, Prop. 6.1]); this allows to extend to larger functional spaces some properties which can be readily proved for factorized test functions of the form f = f1 ⊗f . On the other hand, since all types of Sobolev spaces introduced above enjoy both the interpolation property and a specific duality relation, one can often work solely with spaces of positive integer order and then extend the related conclusions to the whole Sobolev scale by standard arguments of interpolation theory (see [7, Cor. 4.5.2] and [25, Th. 2.6]).

3 Delta-Type Perturbations of A∅ Supported on the Planes πF , π0 For F ∈ C00,1 (R2 , R) as before and z ∈ C\[0, +∞), we consider the operator ˘ F (z) := τF R∅ (z) : L2 (R3 ) → H 1/2 (R2 ) G and the corresponding adjoint with conjugate spectral parameter, i.e., the single layer operator ˘ F (¯z)∗ : H −1/2 (R2 ) → L2 (R3 ) . GF (z) := G

Scattering Theory for Delta-Potentials Supported by Locally Deformed Planes

43

These operators possess uniquely determined bounded extensions, for r ∈ (0, 1/2]: ˘ F (z) ∈ B(H r−3/2(R3 ), H r (R2 )) , G

(10)

GF (z) ∈ B(H −r (R2 ), H 3/2−r (R3 )) .

(11)

On account of Eqs. (5) and (11), we further define MF (z) := τF GF (z) ∈ B(H −1/2(R2 ), H 1/2 (R2 )) .

(12)

Next, let us fix α > 0 and introduce the operator ΓF (z) := 1 + α MF (z) . Notice that ΓF (z)∗ = ΓF (¯z), by construction. Besides, ΓF (z) admits a bounded inverse whenever z is far enough from the real positive semiaxis; more precisely, indicating with dz := infλ∈[0,+∞) |λ − z| the distance of z ∈ C from [0, +∞), we have that (see [12, Lem. 3.3]) ∃ z0 ∈ C\[0, +∞) s.t. ΓF (z)−1∈ B(H r (R2 )) for |r| < 1/2, z ∈ C with dz > dz0 . Taking into account the facts mentioned above, by [32, Th. 2.1] and [11, Th. 2.19] we obtain the following Theorem 1 There holds ΓF (z)−1 ∈ B(H r (R2 ))

for |r| < 1/2 and z ∈ C\[0, +∞) ,

and the bounded linear operator ˘ F (z) RF (z) := R∅ (z) − α GF (z) ΓF (z)−1 G

with z ∈ C\[0, +∞)

(13)

is the resolvent of the self-adjoint operator AF defined by Dom(AF ) := {f ∈ L2 (R3 ) | f = fz − α GF (z) ΓF (z)−1 τF fz , fz ∈ H 2 (R3 )} , (AF − z)f := (A∅ − z)fz . Indicating with n the unit vector normal to πF pointing to the right and with ∂n the related normal derivative, the operator AF can be equivalently characterized as Dom(AF ) = {f ∈ H 2 (R3 \πF ) | f π + = f π − = f πF , [∂n f ]πF = α f πF } , F

AF f = −Δf

3

F

in R \πF , for f ∈ Dom(AF ) .

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This shows that AF corresponds to the singular perturbation of the free Laplacian formally written as AF = −Δ + α δπF , including a delta-type potential of strength α supported on the plane πF . Remark 1 Theorem 1 grants, amongst else, that RF (z) ∈ B(L2 (R3 )) for all z ∈ C\[0, +∞); equivalently, we have σ (AF ) ⊆ [0, +∞). In this connection, let us anticipate that the subsequent Remark 3 actually yields the identity σ (AF ) = [0, +∞). All facts mentioned before certainly hold true, in particular, for F = 0; let R0 (z) := RF =0 (z), then ˘ 0 (z) R0 (z) = R∅ (z) − α G0 (z) Γ0 (z)−1 G

(14)

is bounded in L2 (R3 ) for all z ∈ C\[0, +∞) and it is the resolvent of the self-adjoint operator (cf. Theorem 1 and the related comments) Dom(A0 ) = {f ∈ H 2 (R3 \π0 ) | f π + = f π − = f π0 , [∂n f ]π0 = α f π0 } , 0

A0 f = −Δf

0

in R \π0 , for f ∈ Dom(A0 ) . 3

Indeed, in the case F = 0 more can be said profiting from the smoothness and factorization properties of the flat plane configuration. ˘ 0 (z) and G0 (z) fulfill analogous versions of Eqs. (10) and On the one hand, G (11) for any r > 0. On the other hand, indicating with −Δ the free Laplacian on R2 we have (see [10] and cf. Eq. (12)) i (z+Δ )−1/2 ∈ B(H r−1/2 (R2 ), H r+1/2 (R2 )) for z ∈ C\[0, +∞), r ∈ R , 2 √ with the determination of the square root Im z > 0 for all z ∈ C\[0, +∞). From −1 r 2 here it follows that Γ0 (z) ∈ B(H (R )) for any z ∈ C\[0, +∞) and r ∈ R (see [12, Rem. 3.8]). In particular, setting r = 1/2 in Eqs. (10) and (11) (to be understood ˘ 0 (z) ∈ B(H −1 (R3 ), H 1 (R3 )), hence, with F = 0) one infers G0 (z) Γ0 (z)−1 G M0 (z) =

R0 (z) ∈ B(H −1 (R3 ), H 1 (R3 ))

(15)

for any z ∈ C\[0, +∞). Remark 2 A0 can be equivalently represented as the sum of tensor products (1)

()

A0 = A0 ⊗ 1 + 11 ⊗ A0 ,

(16)

where 11 , 1 are the identity operators on L2 (R) and L2 (R2 ), A(1) := 0 (1) (1) (− ∂x 1 x 1 ) : Dom A0 ⊂ L2 (R) → L2 (R) with Dom A0 := {u ∈ H 1 (R) ∩ () H 2 (R\{0}) | u (0+ ) − u (0− ) = α u(0)} and A0 := (− Δ ) : H 2 (R2 ) → L2 (R2 ).

Scattering Theory for Delta-Potentials Supported by Locally Deformed Planes

45

The above representation (16) and [2, Ch. I.3, Th.s 3.1.1 and 3.1.4] prove that A0 has purely absolutely continuous spectrum σ (A0 ) = σac (A0 ) = [0, +∞). Before proceeding let us mention the following basic identities, on which rely some of the subsequent considerations: RF (z) − R0 (z)

  ˘ 0 (z) ˘ F (z) − G = − α G0 (z) Γ0 (z)−1 G      ˘ F (z) ; ˘ F (z) + G0 (z) ΓF (z)−1 − Γ0 (z)−1 G + GF (z) − G0 (z) ΓF (z)−1 G (17) and ΓF (z)−1 − Γ0 (z)−1

  = − α Γ0 (z)−1 (τF − τ0 ) GF (z) + τ0 GF (z) − G0 (z) ΓF (z)−1 .

(18)

4 The Limiting Absorption Principle The analysis of the scattering theory for the couple (AF , A0 ) to be reported in the forthcoming Sects. 6 and 7 fundamentally relies on the use of Limiting Absorption Principles (LAPs) for the resolvent operators R0 (z) and RF (z). Essentially, these LAPs characterize the limits ε ↓ 0 of R0 (λ ± iε) and RF (λ ± iε) for λ ∈ (0, +∞) as bounded operators on weighted Sobolev spaces, mapping functions decreasing polynomially at infinity to functions increasing with an inverse rate in the same limit. () (1) Let us first consider the factorized representation A0 = A0 ⊗ 1 + 11 ⊗ A0 mentioned in Remark 2 and introduce, for z ∈ C\[0, +∞), the resolvent operators R0 (z) := (A0 − z)−1 : L2 (R) → Dom A0 , (1)

(1)

(1)

R0 (z) := (A0 − z)−1 : L2 (R2 ) → H 2 (R2 ) . ()

()

For any λ ∈ (0, +∞), s1 > 1/2 and θ ∈ (0, 1/2), the limits limε↓0 R0(1) (λ ± iε) 1+θ (R)) and the convergence is uniform in any compact exist in B(L2s1(R), H−s 1 subset K ⊂ (0, +∞). Moreover, from [1, Sec. 4] (see also [6, Prop. 5.1]) we have () 2 (R2 )) for all λ ∈ (0, +∞) and s > 1/2. limε↓0 R0 (λ ± iε) ∈ B(L2s(R2 ), H−s   Due to a fundamental result by Ben-Artzi and Devinatz on operators which are sums of tensor products (see [6, Th. 3.8]), the facts mentioned above suffice to infer the following LAP for A0 (see [12, Prop. 4.2]).

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Proposition 1 Let θ ∈ (0, 1/2) and s1, s > 1/2 . Then, for any λ ∈ (0, +∞), the limits R0± (λ) := lim R0 (λ ± iε)

(19)

ε↓0

1+θ (R3 )) and the convergence is uniform in any compact exist in B(L2s1,s(R3 ), H−s 1,−s subset K ⊂ (0, +∞); in particular,

R0± (λ) ∈ B(L2s1,s(R3 ), L2−s1,−s(R3 )) . An analogous LAP can be derived for RF (z) by means of an abstract perturbation method of Renger [34], starting from the above Proposition 1. This requires to check preliminarily that A0 and AF do satisfy Hypotheses 1, 8, and 9 of [34]. On account of the mapping properties (6) and (7) for the free resolvent R∅ (z) 3 r 2 ˘ F (z) ∈ B(Hsr−3/2 and for the trace operator τF , we have G 1,s (R ), Hs(R )), GF (z) ∈ 

B(Hs−r (R2 ), Hs1,s (R3 )) and ΓF (z)−1 ∈ B(Hsr (R2 )) for any z ∈ C\[0, +∞),  r ∈ (0, 1/2], |r  | < 1/2 and all s1, s ∈ R; notably, for F ≡ 0 analogous results hold true under the weaker assumptions r > 0, r  ∈ R. Then, recalling the explicit Kre˘ın-type representation (13) of RF (z) we have 3/2−r

RF (z) ∈ B(L2s1,s(R3 ))

for z ∈ C\[0, +∞), s1, s ∈ R .

(20)

Next, notice that the trace difference τF − τ0 defines a compact operator on suitable weighted Sobolev spaces; more precisely (see [12, Lem. 4.7]), r+1/2

τF − τ0 ∈ S∞ (Hs1,s (R3 ), Htr−ε (R2 )) 

for r ∈ (0, 1/2), ε ∈ (0, r], s1, s, t ∈ R .

Under the same assumptions on r, ε, s1, s, t, for all z ∈ C\[0, +∞) we r−ε 3 ˘ F (z) − G ˘ 0 (z) ∈ S∞ (Hsr−3/2 have G (R2 )) and GF (z) − G0 (z) ∈ 1,s (R ), Ht 3/2−r

S∞ (Htε−r (R2 ), Hs1,s (R3 )) (see [12, Lem. 4.8]). From here, recalling the basic  identities (17) and (18) we obtain (see [12, Prop. 4.9 and Rem. 4.10]) RF (z)−R0 (z) ∈ S∞ (L2 (R3 ), L2s1,s(R3 ))

for z ∈ C\[0, +∞), s1, s ∈ R .

(21)

Finally, consider the complete set of generalized eigenfunctions for A0 given by (cf. [2, p. 85, Eq. (3.4.1)]) eik1 x iα ei |k1 | |x | (1) ϕk1 (x 1 ) := √ √ − , 2|k1 | + i α 2π 2π 1

(1)

()

ϕk := ϕk1 ⊗ ϕk ,

for k = (k1 , k ) ∈ R×R2 .

1

()

ϕk (x ) :=

eik ·x 2π

(22)

Scattering Theory for Delta-Potentials Supported by Locally Deformed Planes

47

For any s1 > 1/2, s > 1 and u ∈ L2s1,s(R3 ), the map R3  k → ϕk |f  belongs to the class C 0,η (R3 ) of uniformly Hölder continuous functions, with exponent η ∈ (0, 1) such that η  min(s1 − 1/2, s − 1); besides, regarding the operators R0± (λ) defined in Eq. (19), for any compact subset K ⊂ (0, +∞) there holds the implication R0+ (λ)f = R0− (λ)f for all λ ∈ K

⇒

ϕk |f  = 0 for all k ∈ R3 with |k|2 ∈ K .

In view of the basic identity R0 (z)f 2L2 (R3 ) = R3 dk |ϕk |f |2 / | |k|2 − z|2 (see, e.g., [31, p. 111, Th. 2]), the facts mentioned above imply that, for any compact subset K ⊂ (0, +∞) and for all f ∈ L2s1,s(R3 ) such that R0+ (λ)f = R0− (λ)f , there exits a constant c > 0 such that (see [12, Prop. 4.12]) R0± (λ) f L2 (R3 )  c f L2s ,s

1 

(R3 )

for all λ ∈ K, s1 > 1, s > 3/2 .

(23)

Summing up, Eqs. (20), (21) and (23) prove, respectively, that Hypotheses 1, 9 and 8 of [34] are fulfilled in our setting, for suitable choices of the weights indexes; thus, by [34, Th.7] we have the following theorem (see [12, Th. 4.13] for more details): Theorem 2 Let s1 > 1/2, s > 3/4. Then, the (possibly empty) set σp+ (AF ) of embedded eigenvalues of AF is discrete; moreover, for λ ∈ (0, +∞)\σp+ (AF ), the limits RF± (λ) := lim RF (λ ± iε) ε↓0

exist in B(L2s1,s(R3 ), L2−s1,−s(R3 )) and the convergence is uniform in any compact subset K ⊂ (0, +∞)\σp+ (AF ) . Remark 3 As a direct consequence of LAP for RF (z), we infer that the singular continuous spectrum of AF is empty, namely σsc (AF ) = ∅ (see, e.g., [28, Cor. 4.7]). Thus, we have σac (AF ) = σac (A0 ) = [0, +∞) . We conjecture that there are no embedded eigenvalues for AF ; if this is the case, then AF has purely absolutely continuous spectrum σ (AF ) = σac (AF ).

5 A Convenient Representation of RF (z) − R0 (z) To examine the scattering theory for the couple (AF , A0 ), it is convenient to consider a representation of the resolvent difference RF (z) − R0 (z) which depends only on the restriction of the traces τF and τ0 to the support of the function F . In

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this connection, it should be noticed that τF and τ0 (together with the free resolvent R∅ (z)) completely characterize all the operators appearing in the definitions (13) and (14) of RF (z), R0 (z); on top of that, their restrictions to the complement of the support of F do indeed coincide. Let us indicate the support of F with Σ ≡ supp(F ) . Recall Eqs. (5) and (8), and consider the trace operator defined as follows: τΣ ∈ B(H 1 (R3 ), H 1/2 (Σ) ⊕ H 1/2(Σ)) ,

f → τΣ f := (τF f ) Σ ⊕ (τ0 f ) Σ . (24)

Moreover, recall that R0 (z) ∈ B(H −1 (R3 ), H 1 (R3 )), see Eq. (15). Hence, g˘ Σ (z) := τΣ R0 (z) ∈ B(H −1 (R3 ), H 1/2(Σ) ⊕ H 1/2 (Σ)). Let gΣ (z) := g˘Σ (¯z)∗ . Taking into account the duality relation (9), it follows that −1/2

gΣ (z) ∈ B(HΣ

−1/2

(R2 ) ⊕ HΣ

−1/2

which in turn implies τΣ gΣ (z) ∈ B(HΣ H 1/2(Σ)). So that, in particular,

(R2 ), H 1 (R3 )) , −1/2

(R2 ) ⊕ HΣ

(R2 ), H 1/2(Σ) ⊕

g˘Σ (z) =τΣ R0 (z) ∈ B(L2 (R3 ), L2 (Σ) ⊕ L2 (Σ))

(25)

gΣ (z) =g˘Σ (¯z)∗ ∈ B(L2 (Σ) ⊕ L2 (Σ), L2 (R3 )) ,

(26)

and τΣ gΣ (z) ∈ B(L2 (Σ) ⊕ L2 (Σ)) , where we have used the identification L2Σ (R2 ) L2 (Σ). We also recall that, see the proof of Th. 5.1 in [12], ΛF,Σ (z) := − α (1 + α J τΣ gΣ (z))−1 J ∈ B(L2 (Σ) ⊕ L2 (Σ))

for z ∈ C\[0, +∞) , (27)

where J is the involution in L2 (Σ) ⊕ L2 (Σ) defined by J (u ⊕ v) := u ⊕ (−v). Taking into account the above definitions, by a number of purely algebraic manipulations we obtain the alternative representation (see [12, Th. 5.1]) RF (z) − R0 (z) = gΣ (z) ΛF,Σ (z) g˘ Σ (z) for all z ∈ C\[0, +∞) .

(28)

Scattering Theory for Delta-Potentials Supported by Locally Deformed Planes

49

Let μ < 0. By formula (28) it is clear that every function v ∈ Dom(AF ) can be written as v = v0 + gΣ (μ)q ,

(29)

where v0 ∈ Dom(A0 ) is given by R0 (μ)f for some f ∈ L2 (R3 ), and q = ΛF,Σ (μ) g˘Σ (μ)f ∈ L2 (Σ) ⊕ L2 (Σ) . Moreover, see [32, Th. 2.1], (AF − μ)v = (A0 − μ)v0 , and the decomposition (29) is unique, so that one can define the map ρΣ : Dom(AF ) → L2 (Σ) ⊕ L2 (Σ)

such that ρΣ v = q for any v ∈ Dom(AF ).

Remark 4 We claim that for any v ∈ Dom(AF ) and u ∈ Dom(A0 ), it holds (u, AF v)L2 (R3 ) − (A0 u, v0 )L2 (R3 ) = −(τΣ u, ρΣ v)L2 (Σ)⊕L2 (Σ) .

(30)

To prove that this is indeed the case, note the chain of identities (u, (AF − μ)v)L2 (R3 ) =(u, (A0 − μ)v0 )L2 (R3 ) = ((A0 − μ)u, v0 )L2 (R3 ) =((A0 − μ)u, v)L2 (R3 ) − ((A0 − μ)u, gΣ (μ)q)L2 (R3 ) . Since, gΣ (μ)∗ (A0 − μ)u = τΣ u, for any u ∈ Dom(A0 ), and q = ρΣ v identity (30) follows.

6 The Wave Operators for the Couple AF , A0 In view of the results reported in the previous sections, existence and completeness of the wave operators associated to the scattering couple (AF , A0 ) can be inferred from [26, Th. 2.8]. More precisely, consider the representation (28) of the resolvent difference RF (z) − R0 (z) in terms of the operators gΣ (z), ΛF,Σ (z), defined according to Eqs. (25), (26), and (27). Recalling that σ (A0 ) = σac (A0 ) = [0, +∞) (see Remark 2) and building on LAPs for RF (z), R0 (z) (see Proposition 1 and Theorem 2), by

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computations similar to those performed in [26] (see, especially, Eqs. (3.16) and (4.2) therein) we get sup

(λ,ε)∈I ×(0,1)

√ ε gΣ (λ ± iε)B(L2 (Σ)⊕L2 (Σ),L2 (R3 )) < +∞

(31)

ΛF,Σ (λ ± iε)B(L2 (Σ)⊕L2 (Σ)) < +∞ ,

(32)

sup

(λ,ε)∈I ×(0,1)

for any open and bounded interval I such that I ⊂ R\({0} ∪ σp+ (AF )). Equations (31) and (32) prove the hypotheses of [26, Th. 2.8] (see also Rem. 2.1 therein); then, recalling as well that σsc (AF ) = ∅ (see Remark 3), we obtain the following (see [10, Th. 6.1] for more details) Theorem 3 Let Pac (AF ) be the orthogonal projector on the absolutely continuous subspace relative to AF . Then, the wave operators W± (AF , A0 ) := s-lim eit AF e−it A0 , t →±∞

W± (A0 , AF ) := s-lim eit A0 e−it AF Pac (AF ) , t →±∞

exist and are asymptotically complete.

7 The Scattering Matrix Consider the usual definition of the scattering operator for the couple (AF , A0 ) in terms of the wave operators W± (AF , A0 ), whose existence and (asymptotic) completeness was addressed previously in Theorem 3: SF := W+ (AF , A0 )∗ W− (AF , A0 ) : L2 (R3 ) → L2 (R3 ) . Indicating with F0 : L2 (R3 ) → L2 ((0, +∞); L2 (S2 )) (S2 ≡ sphere of unit radius in R3 ) the unitary operator which diagonalizes A0 and recalling that σp+ (AF ) is the (possibly empty) set of embedded eigenvalues of AF , we define the scattering matrix SF (λ) : L2 (S2 ) → L2 (S2 ) SF (λ) u(λ) = [F0 SF F0∗ u](λ)

for λ ∈ (0, +∞)\σp+ (AF ) , for u ∈ L2 ((0, +∞); L2 (S2 )) .

Using the Birman-Kato invariance principle, an explicit characterization of SF (λ) can be recovered from the analogous scattering matrix for the resolvent couple (RF (μ), R0 (μ)) with μ < 0 (the same strategy was employed in [26, Sec. 4]).

Scattering Theory for Delta-Potentials Supported by Locally Deformed Planes

51

To this purpose, consider again the representation of RF (z) − R0 (z) in terms of gΣ (z) and ΛF,Σ (z) (see Eqs. (25)–(28)). Note that the basic relations (RF/0 (μ) − 

z)−1 = − 1z

1 z

RF/0 (μ + 1z ) + 1 and LAPs for RF (z), R0 (z) established in Sect. 4

imply the existence of the following limits, for μ + 1/λ ∈ (0, +∞)\σp+ (AF ): lim g˘Σ (μ) (R0 (μ) − (λ ± iε))−1 , ε↓0

lim g˘Σ (μ) (RF (μ) − (λ ± iε))−1 , ε↓0

lim g˘Σ (μ) (RF (μ) − (λ ± iε))−1 gΣ (μ) . ε↓0

(33) On the other hand, we have that g˘ Σ (μ) is weakly-R 0 (μ) smooth; namely, by [40, p. 154, Lem. 2] there holds sup ε g˘Σ (μ)(R0 (μ) − (λ ± iε))−1 2B(L2 (Σ),L2 (Σ)⊕L2 (Σ)) < +∞ for a.e. λ .

0 0. Later, the renormalisation procedure was applied also to the massless case m = 0 and the properties of the Hamiltonians with and without cutoff were investigated, see e.g. [1, 4, 13, 16]. The result of Griesemer and Wünsch equally holds for the massless case. In the article [10], Jonas Lampart together with the author answered the question in the way it was suggested by Edward Nelson in the above quote. That is, to give a direct description of the operator H∞ and its domain, from which the answer to the second question can be read off. This is achieved by using abstract boundary conditions. More concretely, a dense domain D(H ) on Fock space is identified, which contains the domain of the free operator D(L). Elements of this domain are the sum of a regular part, which is an element of D(L), and a singular part. Then the action of L is extended to this larger domain in such a way that it encodes the action of the creation operator. In addition, also the action of the annihilation operator is extended to the domain D(H ) and it is shown that their sum defines a self-adjoint operator H , bounded from below. Afterwards it turns out, that this operator is in fact the limit of the sequence of cutoff operators HΛ , so it becomes clear that H is equal to the renormalised Hamiltonian H∞ . Characterising elements of D(H ) in this way can be viewed as imposing abstract boundary conditions on them. These boundary conditions, which are called interiorboundary conditions, are formulated in strong analogy with the theory of point interactions, see also [23] and [22] for earlier contributions. The main difference

The Massless Nelson Hamiltonian and Its Domain

59

being the fact that the boundary space or space of charges of the theory of point interactions is on each sector of Fock space identified with the sector with one boson less. In this way the boundary space can be identified with the Hilbert space H itself. The singular behavior of the wave function on one sector is determined by the wave function one sector below. The Skornyakov–Ter-Martyrosyan (STM) operator appears in this construction not as part of a boundary condition and it is therefore not used to label self-adjoint realisations, for the latter alternative see, e.g. [14]. Instead, the STM operator T is identified as the correct extension of the annihilation operator to the singular functions and is therefore part of the action of the Hamiltonian. Thus it is not necessary to study T as an operator on the space of charges, but as an operator on H . In [10], the case of nonrelativistic particles was considered. In [19], the construction was extended to treat also pseudorelativistic models with dispersion relations / Θ(p) = p2 + μ2 . If the renormalisation constant EΛ diverges too fast, the method of [10] has to be suitably modified. This was done for the first time in [8]. In [9] and [7], the enhanced method of the former article is applied to a Polaron-type model and the Bogoliubov-Fröhlich Hamiltonian, respectively. So far however, these results on interior-boundary conditions were concerned with the massive case: it was always assumed that the dispersion of the bosons is bounded from below by a positive constant. As a consequence, the free operator is bounded from below by the number (of bosons) operator, i.e. N ≤ L. Now naturally the question arises whether the construction using abstract interior boundary conditions can be extended also to the massless case. After all, within renormalisation schemes, there is no difficulty in treating these cases as well. In the present note, we will give a more detailed description of the domain D(H ) with or without mass. Roughly speaking, we will differentiate Nelson’s second question between the full free operator L and the part of it that only acts on the field degrees of freedom, dΓ (ω). In this way, we will prove self-adjointness of the Hamiltonian H with or without mass. Neither an ultraviolet nor an infrared cutoff is used in the construction, not even in an intermediate step. We will focus on a class of models in three space dimensions where one nonrelativistic particle interacts with the bosonic field. In [12], interior boundary conditions were used in a multi-time formulation for massless Dirac particles in one space dimension. There the number of particles is bounded. As we will explain in more detail below, the main problems with massless fields occur only if an arbitrary number of quanta is allowed. For physical aspects and more general discussions of the IBC approach, we refer the reader to [3, 6, 20] and [21]. Recently in [18], a generalised variant of interior boundary conditions has been embedded in the more abstract framework of [17].

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2 The Model In this section we will define the basic objects of our model. Then we will introduce a spectral parameter and justify its use by demonstrating that the domain and the extended annihilation operator are actually parameter-independent. Our model will be defined on the Hilbert space H :=

∞ 0

L2 (R3 ) ⊗ L2sym (R3n )

n=0

of the composite system of the particle and the field. We will formulate the model in Fourier representation where elements of the sectors of this Hilbert space are wavefunctions ψ (n) (p, k1 , . . . , kn ) , which are symmetric under exchange of either two of the k-variables. The operator that governs the dynamics of the nonrelativistic particle is given by the multiplication operator p2 . The dispersion relation of the field is given by a non-negative 3 function ω ∈ L∞ loc (R ). Its second quantisation will be denoted by Ω := dΓ (ω). We can now define the free operator L = p2 + Ω, which is self-adjoint and nonnegative with domain D(L) ⊂ H . Since Ω ≥ 0, the operator Ωμ := Ω + μ is invertible for any μ > 0 and so is Lμ := p2 + Ωμ . The interaction between the field and the particle is characterised by a coupling function v ∈ L2loc (R3 ), which is called the form factor. The formal expression for a Hamiltonian of the model is L + a(V ) + a ∗ (V ) , where the annihilation operator a(V ) acts sector-wise as (a(V )ψ)

(n)

√ (p, k1 , . . . , kn ) := n + 1

 R3

v(k)ψ (n+1) (p − k, k1 , . . . , kn , k) dk .

The creation operator a ∗ (V ) is the formal adjoint of a(V ), with action given by (a ∗ (V )ψ)(n) (p, k1 , . . . , kn ) := n−1/2

n 

v(kj )ψ (n−1) (p + kj , k1 , . . . , kˆj , . . . , kn ) .

j =1

As usual, kˆj means that the j -th variable is omitted. The operator a ∗ (V ) is a densely defined operator on H if and only if v ∈ L2 (Rd ). However, in all relevant examples, this is not the case. Often v is in L2loc (Rd ) but is not decaying fast enough at infinity such that v ∈ / L2 . This is what we will assume in the following.

The Massless Nelson Hamiltonian and Its Domain

61

If we wanted to start with a renormalisation procedure, we would now simply replace v by χΛ v where χΛ is the characteristic function of a ball of radius Λ in R3 . Instead, we proceed by defining an operator G∗μ := −a(V )L−1 μ . Later, we will make assumptions on v which guarantee that this operator is bounded. As a consequence, ! the symmetric operator L0,μ := Lμ !ker a(V ) is closed for any μ ≥ 0. Because v ∈ / 2 L , its domain ker a(V ) is also dense in H , see [10, Lem. 2.2]. Therefore the adjoint L∗0,μ is unique. Observe that the operator Gμ maps elements of H into ker L∗0,μ , because for all ψ ∈ ker a(V ) it holds by definition of Gμ that L∗0,μ Gμ ϕ, ψ = ϕ, G∗μ L0,μ ψ = −ϕ, a(V )ψ = 0 . We will now define a family of subspaces of the adjoint domain D(L∗0,μ ). In order to do so, we decompose elements of H in the same way as in the theory of point interactions into the sum of two terms: one is regular, i.e. in D(L), and one term is singular, that is, of the form Gμ ϕ. If we would like to define a sum of point interaction domains in H , we would introduce a boundary or charge space where ϕ lives. But because H is an infinite sum, there is another possibility, namely to take ψ itself as the charge. This is what we will do. Note that the decomposition ψ = (1 − Gμ )ψ + Gμ ψ holds for any ψ ∈ H and μ > 0. Then the family of domains is given by Dμ := {ψ ∈ H |(1 − Gμ )ψ ∈ D(L)} . For μ, λ > 0, the resolvent identity yields −1 −1 ∗ (Gμ − Gλ )∗ = −a(V )(λ − μ)L−1 μ Lλ = ((λ − μ)Lμ Gλ ) . −1 In particular it holds that 1 − Gμ = (1 − Gλ ) − (λ − μ)L−1 μ Gλ . Because Lμ Gλ maps into D(L), this shows that the domain Dμ is in fact independent of the chosen μ > 0. We will denote it by D from now on. In the next step we have to extend the action of a(V ) from D(L) to the enlarged domain D. The formal action of the annihilation operator on the range of Gμ would read

a(V )Gμ ψ (n) (p, k1 , . . . , kn ) |v(k )|2 = −ψ (n) (p, k1 , . . . , kn ) R3 Lμ (p,k1n+1 ,...,kn+1 ) dkn+1 n ψ (n) (p+kj −kn+1 ,k1 ,...,kˆj ,...,kn+1 ) − dkn+1 . R3 v(kn+1 )v(kj ) Lμ (p,k1 ,...,kn+1 )

(1)

j =1

Here Lμ (p, k1 , . . . , kn+1 ) denotes the functions to which the operator Lμ reduce to on one sector of H in the Fourier representation. The off-diagonal part of this sum, the second line of (1), constitutes an integral operator, which we will denote μ by Tod . The integral in the first line of (1) does in general not converge. In order to

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J. Schmidt

regularise this expression, we define the diagonal part of the T -operator μ

Td ψ(p, k1 , . . . , kn ) := −Iμ (p, k1 , . . . , kn ) · ψ (n) (p, k1 , . . . , kn ) , (2)  |v(kn+1 )|2 |v(kn+1 )|2 dkn+1 . where Iμ (p, k1 , . . . , kn ) := − 2 kn+1 + ω(kn+1 ) R3 Lμ (p, k1 , . . . , kn+1 )

(3) μ

μ

Now define the action of T μ ψ := Td ψ + Tod ψ on a (maximal) domain Dμ ⊂ H . At first, this definition seems to depend again on the choice of μ > 0. Note however that, because the second term of the integral Iμ in (3) is independent of the parameter μ > 0, it holds that ∗ T μ − T λ = a(V )(Gμ − Gλ ) = a(V )(λ − μ)L−1 μ Gλ = (μ − λ)Gμ Gλ .

(4)

Because the operators Gμ are continuous, this implies that ψ ∈ Dλ for any λ > 0 as soon as ψ ∈ Dμ for some μ > 0. Set D(T ) = Dμ . While the action of T μ does of course still depend on the chosen parameter, this operator gives rise to the desired extension of a(V ). We define the action of the full extension for all ψ ∈ D(T ) ∩ D as Aμ ψ := a(V )(1 − Gμ )ψ + T μ ψ .

(5)

As a consequence of (4), we have Aμ = a(V )(1 − Gλ ) + a(V )(Gλ − Gμ ) + T μ = a(V )(1 − Gλ ) + T λ = Aλ . Therefore we can define the operator (A, D ∩ D(T )) by choosing any μ > 0. Finally we may also define the action of our Hamiltonian manifestly independent of the spectral parameter: H := L∗0,0 + A . Using the definition of Gμ and T μ , we can rewrite it in a convenient form that contains the positive spectral parameter: H = (1 − Gμ )∗ Lμ (1 − Gμ ) + T μ − μ .

(6)

In [10], it was assumed that ω ≥ 1, and as a consequence of the resulting bound N ≤ L, it was possible to define G∗ := G∗0 = −a(V )L−1 without the need for a parameter. We would however like to make clear that the use of a spectral parameter was avoided only for convenience and better readability and is by no means the real benefit of the assumption ω ≥ 1.

The Massless Nelson Hamiltonian and Its Domain

63

In order to show self-adjointness of H , we will adopt the strategy of [10], where μ the representation (6) (for μ = 0) was used. At first, we have to show that H0 := ∗ (1 − Gμ ) Lμ (1 − Gμ ) is self-adjoint. In [10, Lem. 3.3] the estimate N ≤ L was envoked to show directly the continuous invertibility of (1 − G0 ), from which the self-adjointness of H00 follows. Since 1 1we can not use this estimate, we will show that there exists μ0 > such that 1Gμ 1 < 1 for all μ > μ0 . The main problem to overcome is however the inclusion D ⊂ D(T ) or, more precisely, the relative μ boundedness of T μ with respect to H0 . The proof of the relative bound for T 0 in [10] makes extensive use of the inequality N ≤ L and the resulting fact that (1 − G0 ) leaves D(N) invariant. For that reason, this strategy is not helpful in the massless case. In fact, because there is no relation between N and L, it will be necessary to use characterisations of the domains D(T ) and D that are independent of N altogether. We will illustrate the problems that occur with this strategy for the example of the Nelson model. While [10, Prop. 3.5] gives—for this specific model—an n-independent inclusion D(L1/2 ) ⊂ D(T ), the statement of [10, Lem. 3.2] yields that G0 maps H into D(Lη ) for any 0 ≤ η < 1/4. These exponents do not match together and this is the very problem we have to overcome if we want to define T μ . Differentiating between the diagonal and the off-diagonal part of T μ , we easily observe that, what is actually proven in [10] is that on the one hand D(Ω 1/2 ) ⊂ D(Tod ), but on the other hand D(Lε ) ⊂ D(Td ) for all ε > 0. Thus, at least in the Nelson model, the diagonal part of the operator T seems to pose no problems. The off-diagonal part could be dealt with, if the mapping properties of Gμ are such that D ⊂ D(Ω 1/2 ). This is exactly what we will prove in the following for a certain class of models under some assumptions on v and ω in three space dimensions.

3 Assumptions and Theorems Let the dimension of the physical space be equal to three and assume that there exist α ∈ [0, 3/2) and a constant c > 0 such that for v ∈ L2loc (R3 ) it holds that c(1 + |k|α )−1 ≤ |v(k)| ≤ |k|−α . Furthermore, there exists β ∈ (0, 2] and a constant β β 3 m ≥ 0 such that for ω ∈ L∞ loc (R ) it holds that |k| ≤ ω(k) ≤ |k| + m. Defining D := 1 − 2α we always assume that 0 ≤ D < β. Note that the Nelson model is contained in this class because v = ω−1/2 allows us to choose α√ = 1/2. Clearly β is equal to 1. The upper and lower bounds on ω hold because k 2 + m2 ≤ |k| + m. It will not be necessary to distinguish between the massive and the massless case, for the only important thing is the pair (β, D), which is equal to (1, 0) in the Nelson model. Our first result, Proposition 3.1, is concerned with regularity properties of a family of domains Dσ . Its proof can be found in Sect. 4.2. Proposition 3.1 Let β ∈ (0, 2], let 0 ≤ D < β/2 if β < 2 and 0 < D < 1 if β = 2. Let ψ #= 0 and κ, η ∈ [0, σ ] for some σ ∈ (0, 1].

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If ψ ∈ Dσ = {ψ ∈ H |(1 − Gμ )ψ ∈ D(Lσ ) for some μ > 0} , then ψ ∈ D(Lκ ) if and only if κ
β, which is in particular fulfilled for the Nelson model. For determining supersets of the operator domain D(H ) = D, the IBC method is the only tool available. For the Nelson model, massive or massless, Proposition 3.1 implies that D(H ) ⊂ D(Ω η ) for all η < 1 but D(H ) ∩ D(Ω) = {0}.

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4 Constructing the Hamiltonian In the main part of the article we will carry out the program that has been sketched in the introduction. The possibility to set up the operators G and T using positive parameters μ > 0 and the results about the parameter-independence of the domains Dσ and the operator A will not be repeated. They can be found in Sect. 2. We will discuss the mapping properties of Gμ and fit them together with those of T μ . In this way, we will prove self-adjointness of the Hamiltonian H (Theorem 3.3) and obtain the characterisation of the domains Dσ in terms of domains of powers of Ω and L (Proposition 3.1). We will from now on assume that the spectral parameter μ is greater than one, μ ≥ 1. When writing D(Lx ) without index for some x ∈ R \ {0} we mean the domain D(Lxμ ) for any μ ≥ 1. Note also that the assumption on μ guarantees y monotonicity in the exponent, i.e. Lxμ ≤ Lμ if x ≤ y. We will denote by K the collection of variables K := (k1 , . . . , kn ). Consequently Kˆ j := (k1 , . . . , kˆj , . . . , kn ) is the collection of variables with the j -th component omitted. We will use the symbols Lμ (p, K) = p2 + Ωμ (K) to denote the functions to which the operators reduce to on one sector of H in the Fourier representation. Powers of the self-adjoint operators Ω and L are self-adjoint on their respective domains D(Lκ ) etc., which are all continuously embedded in H . We will regard the domains as Banach spaces equipped with the norms ψD(Lκ ) = Lκ ψH + ψH . The intersection of two such subspaces is a Banach space with norm ψD(Lκ )∩D(Ω η ) := max(ψD(Lκ ) , ψD(Ω η ) ). We will mostly use the equivalent norm given by the sum, i.e. ψD(Lκ ) + ψD(Ω η ) .

4.1 Mapping Properties of Gμ Let us begin with a technical lemma that will be useful later on. It is concerned with certain properties of the affine function u(s) := (βs − D)/2. This function itself plays an important role in the following because many relations between the parameters can be expressed with its help. Lemma 4.1 Let β ∈ (0, 2], let 0 ≤ D < β if β < 2 and 0 < D < 2 if β = 2. Let ε0 > 0 be such that D + ε0 = β. Define for any 0 < ε < ε0 the function  θε (β, D) :=

2−D−ε 2−β

D>

max(1/β, 1)

D≤

3β−2 β 3β−2 β

−ε −ε.

(7)

Let the affine transformation u for all s ∈ [0, ∞) be defined as u(s) := (βs − D)/2. Then it holds that θε ≥ 1. Furthermore 1 + u(θε ) − θε ≥ ε and u(θε ) < 1. Proof If θε = 1, the hypothesis clearly implies that u(θε ) < 1. When θε = 1/β, then u(θε ) = (1 − D)/2 ≤ 1/2. If D > 3β−2 β − ε then, by definition of ε0 , it holds

The Massless Nelson Hamiltonian and Its Domain

67

that β 2 > 3β − 2. This implies that β ∈ (0, 1), in particular β/(2 − β) < 1 and therefore u(θε ) < (2 − D − ε − D)/2 < 1. In the upper case of (7), the equality 1 + u(θε ) − θε = ε holds by construction. Because 1 + u(s) − s is non-increasing, it remains to prove that 2−D−ε 2−β is an upper bound for θε . For 1/β this is the case if and only if D ≤ follows easily because by definition 2 − D − ε > 2 − β. The last step also proves that θε ≥ 1.

3β−2 β

− ε. If θε = 1, this % $

Now we will consider Gμ as an operator into D(Lκ ) under some conditions on κ. Later, when the target space will be enlarged to D(Ω η ), we will build on some of the formulas obtained here. Lemma 4.2 Let β ∈ (0, 2], let 0 ≤ D < β if β < 2 and 0 < D < 2 if β = 2. Then for any 0 ≤ κ < (2 − D)/4 and any μ ≥ 1 it holds that Gμ is continuous from D(Ω κ ) to D(Lκ ). There exists μ0 ≥ 1 such that the norm of Gμ is smaller than 1 for all μ > μ0 . 1 1 1 1 1 κ−(1+u(s)−s) 1 Proof We will show that 1Lκ Gμ ψ 1 ≤ C 1Ωμ ψ 1 for some constant C > 0 and any s ≥ 1. In view of Lemma 4.1, this proves the claim because 1 1 1 1 1 κ−ε/2 1 ψ 1 ≤ μ−ε/2 1Ωμκ ψ 1 ≤ μ−ε/2 ψD(Ωμκ ) . 1 Ωμ ! η !2 For later use, we will write Ξμ := Lμ at first. To estimate !Ξμ Gμ ψ ! , we multiply by ω(kj )s /ω(kj )s for s ≥ 1 and use the finite dimensional Cauchy-Schwarz inequality: ! !2 ! ! !v(kj )!2 Ξμ (p, K)2κ !!ψ (n) (p + kj , Kˆ j )!! n+1 n+1  ! !2  s ω(kν ) ! κ ! !Ξμ Gμ ψ (n) (p, K)! ≤ n+1 Lμ (p, K)2 ω(kj )s j =1 ν=1

! !2 ! ! !v(kj )!2 Ξμ (p, K)2κ !!ψ (n) (p + kj , Kˆ j )!! n+1 s s  ˆ ω(kj ) + Ω(Kj ) ≤ . n+1 Lλ (p, K)2 ω(kj )s j =1

In the second step, the fact that s ≥ 1 is essential. We now use the assumptions |v(k)| ≤ |k|−α and ω(k) ≥ |k|β . This yields for the translated expression ! ! κ !Ξ Gμ ψ (n) (p − kj , K)!2 the bound μ !2 ! ! !−2α−βs ˆ j )!! n+1 !!ψ (n) (p, K ! !2  Ξμ (p − kj , K)2κ !kj ! Ω(Kˆ j )s ! κ ! (n) !Ξμ Gμ ψ (p − kj , K)! ≤ n+1 Lμ (p − kj , K)2 j =1

! !2 ! !−2α ˆ j )!! n+1 !!ψ (n) (p, K  Ξμ (p − kj , K)2κ !kj ! + . n+1 Lμ (p − kj , K)2 j =1

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J. Schmidt

Now we use the symmetry of ψ, L and Ξ to note that we can bound the integral over these sums by the integral over the first term of the sums times n + 1. That is, we have a bound 1 !2 12  ! 1 κ ! ! (n) 1 Ξ G ψ 1 μ μ 1 = !Ξμκ Gμ ψ (n) (p − kj , K)! dKdp  ! !2  ! ! (n) ˆ ≤ !ψ (p, K1 )!

Ξμ

R3

γd

Ξ + γodμ dk1 dKˆ 1 dp

where Ξ

Ξ

γd μ (p, K) + γodμ (p, K) :=

Ξμ (p − k1 , K)2κ Ξμ (p − k1 , K)2κ Ω(Kˆ 1 )s + . 2α Lμ (p − k1 , K)2 |k1 | Lμ (p − k1 , K)2 |k1 |2α+βs (8)

We now specify to Ξμ = Lμ and estimate it from below by |p − k1 |2 + Ωμ (Kˆ 1 ). Recall that since D ≥ 0 we have by hypothesis κ < 1/2. So we can bound the integral over k1 of the off-diagonal part by 

L

R3



γodμ (p, K) dk1 ≤

R3

Ωμ (Kˆ 1 )s |k1 |−2α−βs dk1 . (|p − k1 |2 + Ωμ (Kˆ 1 ))2(1−κ)

If u(s) < 1 and 2κ < u(s) + 1, this integral is by scaling bounded by a constant times Ωμ (Kˆ 1 )s+2(κ−1)+

3−2α−βs 2

2 = Ωμ (Kˆ 1 )

  κ− 1+u(s)−s 2

.

If 2κ < u(0) + 1, we obtain similarly for some C > 0 a bound for the diagonal part: 

γodμ (p, K) dk1 ≤ CΩμ (Kˆ 1 )2(κ−1)+ L

R3

3−2α 2

= CΩμ (Kˆ 1 )

  2 κ− 1+u(0) 2

.

Because β > 0, the function u is increasing so the hypothesis 2κ < u(0) + 1 = (2 − D)/2 clearly implies 2κ < u(s) + 1. In addition β ≤ 2, so we can estimate 1 + u(s) − s ≤ 1 + u(0). % $ The next lemma deals with the most important step of the construction, namely the mapping properties of Gμ into D(Ω η ). It is only here (because more explicit computations are used) where the fact that the dimension is equal to three is relevant.

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69

Lemma 4.3 Let β ∈ (0, 2], let 0 ≤ D < β if β < 2 and 0 < D < 2 if β = 2. Assume that there exists εod > 0 small enough such that 0≤η≤

⎧ β(2−D−ε ) ⎨ 2−β od −D+1 2β

⎩ 2−D

D> D≤

2β

3β−2 β 3β−2 β

− εod , − εod .

Define for any εd ≥ 0 the map qεd (η) := max (0, η + εd − (β + 2 − 2D)/(4β)). Then for any μ ≥ 1 and any εd > 0 it holds that Gμ is continuous from D(Ω η ) ∩ D(Lqεd (η) ) to D(Ω η ) and there exists μ0 ≥ 1 such that the norm of Gμ as a map between these two spaces is smaller than 1 for all μ > μ0 . η

Proof To estimate the norm of Ωμ Gμ ψ (n) , we start directly with the expressions η Ω η as they have been defined in (8). Note that we have replaced the γdΩ and γod 1/2 1/2 exponent κ by η. By defining the rescaled variables p˜ := p/Ωλ and k˜ := k1 /Ωλ we can estimate 

η

R3



γdΩ (p, K) dk1 ≤



2η |k1 |−2α |k1 |β + m + Ωμ (Kˆ 1 )

dk1 (|p − k1 |2 + |k1 |β + Ωμ (Kˆ 1 ))2 2η ! ! ! ! ! ˜ !−2α ! ˜ !β  !k ! !k ! + m + 1 = Ωμ (Kˆ 1 )2η−(u(0)+1) 2 dk˜ . ! ! ! !2 β−2 ! !β R3 ! ! !p˜ − k˜ ! + Ωμ (Kˆ 1 ) 2 !k˜ ! + 1 R3

In the very same way we obtain for the integral over k1 of the off-diagonal part in (8) the upper bound

Ωμ (Kˆ 1 )2η−(1+u(s)−s)

! ! 2η ! ! ! ˜ !β ! ˜ !−2α−βs !k ! + m + 1 !k !

 R3

! 2 dk˜ . !2 ! ! β−2 ! !β ! ! !p˜ − k˜ ! + Ωμ (Kˆ 1 ) 2 !k˜ ! + 1

Abbreviate Ω := Ωμ (Kˆ 1 ), set M := m + 1 ∈ (0, ∞) and denote the remaining integral by ! ! 2η ! ! ! ˜ !β ! ˜ !−2α−βs !k ! + M !k !

 Υ (s, μ, p) ˜ :=

R3

! 2 dk˜ . ! !β !2 ! ! ! ! !p˜ − k˜ ! + Ω β/2−1 !k˜ ! + 1

(9)

and The integral Υ is clearly bounded for any p˜ ∈ R3 as long as η < 1+u(s) β u(s) < 1. If |p| ˜ ≤ 1, we therefore estimate it simply by a constant. So assume in

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J. Schmidt

the following that |p| ˜ > 1 and compute using spherical coordinates ∞ 1

 Υ (s, μ, p) ˜ = 2π 0

 = 2π 0

−1

∞

(r β + M)2η r 2−2α−βs (r 2 + p˜ 2 − 2r pσ ˜ + rβ Ω

β−2 2

+ 1)2

drdσ

(r β + M)2η r 2−2α−βs ((r − p) ˜ 2 + rβ Ω  ∞

≤ 2π(p˜ 2 )ηβ−(u(s)+1)

0

β−2 2

+ 1)((r + p) ˜ 2 + rβ Ω

β−2 2

+ 1)

dr

(x β + M)2η + x 2−2α−βs dx . ((x − 1)2 x β pβ−2 + p˜ −2 )((x + 1)2 + x β pβ−2 + p˜ −2 )

We have replaced M/p˜ β simply by M because |p| ˜ > 1. The integral from x = 2 to infinity is bounded by a constant, independent of p, ˜ for any η < 1+u(s) β . The same −1/β . Consider the integral from 21/β < 1 is true of the integral from zero to x = 2 to 2. On this interval, the numerator of the integral can be estimated by a constant that depends on M, the factor in the denominator that contains the (x + 1)2 -term is bounded from below by one. It remains to estimate the factor which has a pole at x = 1. This can be done by enlarging the domain and making use of fact that the antiderivative of (1 + x 2 )−1 is the arctan. So we have 

 2 1 1 dx ≤ dx 2 + x β pβ−2 + p˜ −2 ) 2 + 1/2pβ−2 + p˜ −2 ) −1/β −1/β ((x − 1) ((x − 1) 2 2  −1/2  1 β−2 −2 ≤ + p ˜ . dx = π 1/2p 2 β−2 + p˜ −2 ) R ((x − 1) + 1/2p 2

Recall that the other parts of this integral are bounded by a constant. So, because p˜ > 1 implies p > 1, we can bound as a whole:  −1/2   β−2 −2 χ{p>1} Υ (s, μ, p) ˜ ≤ χ{p>1} + p˜ C + 1/2p ˜ ˜ ≤ C  χ{p>1} (p ˜

2−β 2

)(1−t )(p) ˜ t.

(10)

Here we have introduced a parameter t ∈ [0, 1]. Now we have to distinguish between the diagonal term in (8), where we have s = 0 and choose t = 0 in (10), and the off-diagonal term where we choose t = 1 in (10) and observe that s ≥ 1 is required. The off-diagonal term hence can be bounded by  R3

  Ωη γod (p, K) dk1 ≤ CΩμ (Kˆ 1 )2η−(u(s)+1−s) χ{p≤1} + χ{p>1} p˜ 2ηβ−2(u(s)+1))+1 . ˜ ˜

We would like to have—for the off-diagonal term—a bound independent of p. To achieve this, we apply Lemma 4.1 and choose s = θεod for an εod > 0 admissible there. Then we can see that our upper bounds on η are such that the exponent of p˜

The Massless Nelson Hamiltonian and Its Domain

71

is non-positive. This is because for s = θεod the exponent becomes 2ηβ − 2(u(θεod ) + 1)) + 1 = 2βη − 2β

⎧ β(2−D)−βε od −D+1 ⎨ 2−β 2β ⎩ β max(1,1/β)−D+1 2β

D> D≤

3β−2 β 3β−2 β

− εod − εod ,

and obviously 1 ≤ β max(1, 1/β). These considerations imply that the norm of the 1 1 1 η 1 1 η−ε /2 12 off-diagonal term is bounded by 1Ωμ od ψ 1 ≤ μ−εod 1Ωμ ψ 1. We are not able to obtain a bound independent of p also for the diagonal term in η (8). Setting s = t = 0 in (10), yields for the integral R3 γdΩ (p, K) dk1 a bound of the form constant times   2−β + χ{p>1} Ωμ (Kˆ 1 )−ηβ+(u(0)+1)p2ηβ−2(u(0)+1)+ 2 Ωμ (Kˆ 1 )2η−(u(0)+1) χ{p≤1} ˜ ˜ + χ{p>1} Ωμ (Kˆ 1 )2η−ηβ p2β(η−(β+2−2D)/(4β)) . = Ωμ (Kˆ 1 )2η−(u(0)+1)χ{p≤1} ˜ ˜ Due to the fact that D < β ≤ 2, the first term here is bounded by μ−u(0)−1 Ωμ for all p˜ ∈ [0, ∞). To bound the second term, introduce an εd > 0, which yields 2η

χ{p>1} Ωμ (Kˆ 1 )2η−ηβ p2β(η−(β+2−2D)/(4β)) ˜ ≤ χ{p>1} Ωμ (Kˆ 1 )η(2−β) μ−εd β (p2 + μ)β(η+εd −(β+2−2D)/(4β)) ˜ ≤ Ωμ (Kˆ 1 )η(2−β) μ−εd β (p2 + μ)βqεd (η) We have used in particular that μ ≥ 1 to get rid of the characteristic function. Now we apply Young’s inequality with ν = 2/(2 − β) and ξ = 2/β, which leads to the upper bound   Cμ−εβ Ωμ (Kˆ 1 )2η + (p2 + μ)2qεd (η) . Because β > 0, the norm of this term goes to zero as μ → ∞. This proves the claim. $ % The Neumann series is a candidate for the inverse of the operator 1 − Gμ . On domains where the norm of Gμ is decreasing, the series will converge for large enough μ. Corollary 4.4 Let β ∈ (0, 2], let 0 ≤ D < β if β < 2 and 0 < D < 2 if β = 2. Let η, κ ≥ 0. Assume that for any ε > 0 small enough 0≤η
D≤

3β−2 β 3β−2 β

− ε, −ε

(11)

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J. Schmidt

and max(κ, q0 (η)) < 2−D 4 . Then there exists μ0 ≥ 1 such that 1 − Gμ is continuously invertible on D(Ω η ) ∩ D(Lmax(κ,qε (η)) ) for any μ > μ0 , possibly for a smaller ε > 0. Proof We make ε > 0 possibly smaller, such that also max(κ, qε (η)) < Lemma 4.2 implies that for any η ≥ 0 it holds that

2−D 4 .

Then

1 1 1Gμ ψ 1 max(κ,q (η)) ≤ c(μ)ψ ε D(Ω max(κ,qε (η)) ) ≤ c(μ)ψD(Ω η )∩D(Lmax(κ,qε (η)) ) D(L ) with c(μ) < 1 for μ larger than some μ0 ≥ 1. Due to the assumptions we have made on η, the Lemma 4.3 gives 1 1 1 Gμ ψ 1

D(Ω η )

≤ C(μ)ψD(Ω η )∩D(Lqε (η) ) ≤ C(μ)ψD(Ω η )∩D(Lmax(κ,qε (η)) )

with C(μ) < 1 if μ > μ0 for some μ0 ≥ 1. The last inequality simply holds because μ ≥ 1 and qε (η) ≤ max(κ, qε (η)). % $ We are now ready to prove that the “free” operator H0 := (1 −Gμ )∗ Lμ (1 −Gμ ) is self-adjoint. To prove self-adjointness of the whole operator H in Sect. 4.3, the μ operator T μ will be regarded as an operator perturbation of H0 . μ

Corollary 4.5 Let β ∈ (0, 2], let 0 ≤ D < β if β < 2 and 0 < D < 2 if β = 2. μ μ Then H0 is self-adjoint and positive on D(H0 ) = D = {ψ ∈ H |(1 − Gμ )ψ ∈ D(L) for some μ > 0}. Proof Apply Corollary 4.4 with η = κ = 0. This is possible because the upper bounds on η and κ are positive for D < β and in addition q0 (0) ≤ 0. That means μ that (1 − Gμ ) is invertible on H for μ ≥ 1 large enough, so D(H0 ) := D is dense μ in H . The operator H0 is clearly symmetric and positive and it is easy to see that μ μ ϕ ∈ D((H0 )∗ ) implies ϕ ∈ D(H0 ). % $

4.2 The Domain D: Proof of Proposition 3.1 In order to determine supersets for D, we can now build on the results of the previous section. The domain can be characterised as D = (1 − Gμ )−1 D(L) for any μ ≥ 1 admissible in Corollary 4.5. Therefore any subspace of the form (1 − Gμ )−1 S with D(L) → S ⊂ H is also a superset for D. If 1−Gμ is invertible on (S , ·S ), we have (1 − Gμ )−1 S = S , which then allows us to explicitly characterise this space. In this section, we will restrict the range of parameters to pairs where D < β/2 in contrast to β. In this way, the various conditions on η can be significantly simplified.

The Massless Nelson Hamiltonian and Its Domain

73

Proposition 4.6 Let β ∈ (0, 2], let 0 ≤ D < β/2 if β < 2 and 0 < D < 1 if β = 2. Define for any σ ∈ (0, 1] the subspace Dσ = {ψ ∈ H |(1 − Gμ )ψ ∈ D(Lσ ) for some μ > 0}. σ η qε (η) ) for • For any η ∈ [0, σ ] with η < 2−D 2β it holds that D ⊂ D(Ω ) ∩ D(L any ε > 0 small enough. σ κ • For any κ ∈ [0, σ ] with κ < 2−D 4 it holds that D ⊂ D(L ).

Proof The first task will be to perform the promised simplification of the conditions on η in Corollary 4.4. First, observe that η ≤ σ means of course also η ≤ 1. We will now prove that η ≤ 1 together with D < β/2 implies that, if ε > 0 can be arbitrarily small, then

η


3β − 2 −ε. β

To show this, observe that 3β−2 β < D + ε < β/2 + ε means that β has to fulfill the 2 inequality 2βε > 6β − β − 4. This can, for ε small enough, only be satisfied for β < 4/5. Using again D < β/2 we bound, possibly making ε > 0 smaller, β(2−D−ε) 2−β

−D+1

2β

−1> =

β(2 − β/2 − ε) − (β/2 − 1 + 2β)(2 − β) 2β(2 − β) 5−2 − (4/5)ε (1 − β)2 − βε > > 0. β(2 − β) 2(4/5)

To sum up, we have shown that if η ≤ 1 then the upper case of (11) is fulfilled. The lower case in this very condition is also satisfied by hypothesis. Our second step is to show that the assumptions η ≤ 1 and η < 2−D 2β are such that also q0 (η) < 2−D 4 . Note that the latter condition is equivalent to η < β+2(1−D) . Using D < β/2 we now bound from below 4β

2−D 4

+

2−D β + 2(1 − D) 4 − 4β − β 2 + −1> 4 4β 8β and

β + 2(1 − D) 2 − D 6β − β 2 − 4 2−D + − > . 4 4β 2β 8β

Observe that for any β at least one of these functions is positive. So if either η ≤ 1 2−D or η < 2−D 2β then also q0 (η) < 4 . The above considerations allow us to apply the Corollary 4.4 and proceed with the main part of the proof. For η, κ fulfilling the hypothesis, we define S1 := Ω η and S2 := Lmax(κ,qε (η)) and S = (D(S1 ) ∩ D(S2 ), ·D(S1 ) + ·D(S2 ) ). Recall that μ ≥ 1 implies D(Lσ ) → D(Lmin(η,κ) ). Therefore we may consider the chain of inclusions

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σ D(L) → D(L 1 ) → S . Furthermore Si ψH ≤ ψS and denoting Cμ := 1 1(1 − Gμ )−1 1 L (S ) we have

1 1 1 1 Si ψH ≤ ψS = 1(1 − Gμ )−1 (1 − Gμ )ψ 1 1 1 ≤ Cμ C  1(1 − Gμ )ψ 1D(Lσ )

S

1 1 ≤ Cμ 1(1 − Gμ )ψ 1S

(12) 1 1 1 1  = Cμ C  1Lσμ (1 − Gμ )ψ 1H + 1(1 − Gμ )ψ 1H .

Inserting 1 = (1 − Gμ)−∗ (1 − Gμ )∗ yields the desired bound. In order to obtain the first part of the statement, we can set κ = 0. For the second part we choose η = 0 which implies D(Ω η ) = H and qε (η) ≤ 0 for ε > 0 small enough. % $ Corollary 4.7 Let β ∈ (0, 2], let 0 ≤ D < β/2 if β < 2 and 0 < D < 1 if β = 2. • For any η ∈ [0, 1) with η < 2−D 2β there exists μ0 ≥ 1 such that for any μ > μ0 η μ the operator Ωμ is infinitesimally bounded with respect to H0 • For any κ ≥ 0 with κ < 2−D 4 there exists λ0 ≥ 1 such that for any λ > λ0 the operator Lκλ is infinitesimally bounded with respect to H0λ . Proof Because η < 1, by Young’s inequality, we have 1 1 1 η 1 1L ϕ 1 ≤ C(ε ˜ 1Lμ ϕ 1 + ε−η/(1−η) ϕ) . μ

(13)

for any ε > 0 and any ϕ ∈ D(L). In (12) we can set σ = η, and because ϕ = (1 − Gμ )ψ ∈ (1 − Gμ )Dη ⊂ D(L), we can use (13) such that  1 1 η 1 1 1 1  1Ω ψ 1 ≤ Cμ C  C˜ ε1Lμ (1 − Gμ )ψ 1 + (1 + ε−η/(1−η) )1(1 − Gμ )ψ 1 H H H . Using 1 = (1 − Gμ )−∗ (1 − Gμ )∗ , we prove infinitesimal boundedness of Ω η with μ respect to H0 if μ is large enough. The case of Lκ can be proved in exactly the same way. % $ Now we are well prepared to prove Proposition 3.1. Proof (Proof of Proposition 3.1) One of the implications is provided by Proposition 4.6. It remains to prove that 0 #= ψ ∈ Dσ implies that Lκ ψ or Ω η ψ are 2−D infinite if κ ≥ 2−D 4 or η ≥ 2β , respectively. For later use we write Ξμ to denote either Lμ or Ωμ . Decomposing Ξ κ ψ = Ξμκ (1 − Gμ )ψ + Ξμκ Gμ ψ we see that, because in any case κ, η ≤ σ , the norm of the first term is always finite. Recall that we have μ ≥ 1. Choose n ∈ N such that ψ (n) #= 0. For any r > 0 we define the set ! ! Ur := {(p, K) ∈ R3+3(n+1)||p| < r, !kj ! < r for all 2 ≤ j ≤ n + 1} .

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1 12 We will now show that we can choose r > 0 such that 1Ξμκ Gμ ψ (n) 1L2 (U ) is r infinite. To do so we will split the sum that constitutes Gμ and apply the inequality n t −1 2 2 2 2 t a − (t − 1)b ≤ |a + b| for t = 2. In addition we use that ( j =1 aj ) ≤ n 2 n j =1 aj . Taken together, this leads to the lower bound ! !2 ! ! !Lμ Gμ ψ (n) ! ≥

!2 ! ! ! |v(k1 )|2 Ξμ (p,K)2κ !ψ(p+k1 ,Kˆ 1 )! 2(n+1)Lμ (p,K)2

!2 ! 2 2κ ! ˆ ! n+1 |v(kj )| Ξμ (p,K) !ψ(p+kj ,Kj )! j =2 Lμ (p,K)2

−

(14) .

We proceed by showing that the integral over Ur of the n lower terms in (14), all coming with a minus, is finite, but the integral of the first term is not. We enlarge the domain of integration to all p ∈ R3 and perform a change of variables in p → p+kj to obtain an upper bound for the integral over one of these terms:  Ur

! !2 ! ! !v(kj )!2 Ξμ (p, K)2κ !!ψ(p + kj , Kˆ j )!!

dpdK Lμ (p, K)2  !2  ! Ξμ (p − kj , K)2κ ! ! ≤ !ψ(p, Kˆ j )! ! !2α dpdK . R×Brn |kj | 0 the expresμ sion Td given by (2) defines a symmetric operator on the domain D(Lmax(ε,D/2)) for any μ ≥ 1. Lemma 4.9 (Lemma 3.8 of [19]) Assume D ≥ 0. Then, for all s > 0 such that μ u(s) < 1 and 0 < u(u(s)), the operator Tod , defined in (1), is bounded from D(N max(0,1−s)Ω s−u(s)) to H and is symmetric on this domain for any μ ≥ 1. In order to apply the result of Lemma 4.9, we clearly have to restrict to s ≥ 1 as usual. Proof (Proof of Theorem 3.3) Decompose into diagonal and off-diagonal terms 1 1 μ 1 1 1 max(ε,D/2) 1 μ μ T μ = Td + Tod . Due to Lemma 4.8, we have a bound 1Td ψ 1 ≤ 1Lμ ψ 1. As long as μ is greater than some μ0 and D < 2/3, the second part of Corollary 4.7 μ implies that the diagonal part of the operator is infinitesimally bounded by H0 . To proceed analogously for the off-diagonal part we need that for s ≥ 1 Lemma 4.9 is applicable, so necessarily u(s) < 1

(16)

u(u(s)) > 0 .

(17)

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77

μ

s−u(s)

In this way we can bound the norm of Tod ψ by the norm of Ωμ ψ. Then we would like to conclude the infinitesimal boundedness by setting η = s − u(s) in Corollary 4.7. To do so, we have to make sure that s − u(s) < 1, s − u(s)
βs − 2 =: f16 (s) D 1 particles could be included in the analysis as well. This is because when bounding norms of Gμ ψ from above, the relevant estimates are the same for M = 1 and M > 1. For bounds from below, as in Sect. 4.2, one has to take care of some more cross-terms because the domain of integration is chosen to be not symmetric under exchange of particles. It is only the T -operator where a significant difference occurs. The off-diagonal part of T consists for M > 1 of additional terms, which are called θ -terms in [10]. They are however bounded on D(Lmax(ε,D/2)) for any ε > 0, exactly as the diagonal part of T , see [19, Lem. 3.7]. In the context of the above analysis, these θ -terms can therefore be μ put together with Td and pose almost no constraints on the allowed pairs (β, D).

References 1. Bachmann, S., Deckert, D.-A., Pizzo, A.: The mass shell of the Nelson model without cut-offs. J. Funct. Anal. 263(5), 1224–1282 (2012) 2. Correggi, M., Dell–Antonio, G., Finco, D., Michelangeli, A., Teta, A.: A class of Hamiltonians for a three-particle fermionic system at unitarity. Math. Phys. Anal. Geom. 18(1), 32 (2015) 3. Dür, D., Goldstein, S., Teufel, S., Tumulka, R., Zanghí, N.: Bohmian trajectories for hamiltonians with interior-boundary conditions (2018). https://doi.org/10.1007/s10955-01902335-y 4. Fröhlich, J.: Existence of dressed one electron states in a class of persistent models. Fortschr. Phys. 2, 159–198 (1974) 5. Griesemer, M., Wünsch, A.: On the domain of the Nelson Hamiltonian. J. Math. Phys. 59(4), 042111 (2018) 6. Keppeler, S., Sieber, M.: Particle creation and annihilation at interior boundaries: onedimensional models. J. Phys. A Math. Gen. 49(12), 125204 (2016) 7. Lampart, J.: The renormalised Bogoliubov-Fröhlich Hamiltonian (2018). https://doi.org/10. 1063/5.0014217 8. Lampart, J.: A nonrelativistic quantum field theory with point interactions in three dimensions. Ann. Henri Poincaré 20, 3509–3541 (2019) 9. J. Lampart, Ultraviolet properties of a Polaron model with point interactions and a number cutoff, in Mathematical Challenges of Zero-Range Physics: Models, Methods, Rigorous Results, Open Problems, ed. by A. Michelangeli. Springer INdAM Series, vol. 42 (Springer, Cham, 2020). https://doi.org/10.1007/978-3-030-60453-0_6 10. Lampart, J., Schmidt, J.: On Nelson-type Hamiltonians and abstract boundary conditions (2018). https://doi.org/10.1007/s00220-019-03294-x

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11. Lampart, J., Schmidt, J., Teufel, S., Tumulka, R.: Particle creation at a point source by means of interior-boundary conditions. Math. Phys. Anal. Geom. 21(2), 12 (2018) 12. Lienert, M., Nickel, L.: Multi-time formulation of particle creation and annihilation via interior-boundary conditions (2018). https://doi.org/10.1142/S0129055X2050004X 13. Matte, O., Møller, J.S.: Feynman-Kac formulas for the ultra-violet renormalized Nelson model (2017). Preprint. arXiv:1701.02600 14. Michelangeli, A., Ottolini, A.: On point interactions realised as Ter-Martyrosyan–Skornyakov operators. Rep. Math. Phys. 79(2), 215–260 (2017) 15. Nelson, E.: Interaction of nonrelativistic particles with a quantized scalar field. J. Math. Phys. 5(9), 1190–1197 (1964) 16. Pizzo, A.: One-particle (improper) states in Nelson’s massless model. Ann. Henri Poincaré 4(3), 439 (2003) 17. Posilicano, A.: Self-adjoint extensions of restrictions. Oper. Matrices 2(4), 483–506 (2008) 18. Posilicano, A.: On the self-adjointness of H + A∗ + A (2020). Preprint. arXiv:2003.05412 19. Schmidt, J.: On a direct description of pseudorelativistic Nelson Hamiltonians. J. Math. Phys. 60(10), 102303 (2019) 20. Schmidt, J., Tumulka, R.: Complex charges, time reversal asymmetry, and interior-boundary conditions in quantum field theory (2018). Preprint. arXiv:1810.02173 21. Teufel, S., Tumulka, R.: Avoiding ultraviolet divergence by means of interior–boundary conditions, in Quantum Mathematical Physics, ed. by F. Finster, J. Kleiner, C. Röken, and J. Tolksdorf (Birkhäuser, Cham, 2016), pp. 293–311 22. Thomas, L.E.: Multiparticle Schrödinger Hamiltonians with point interactions, Phys. Rev. D 30, 1233–1237 (1984) 23. Yafaev, D.R.: On a zero-range interaction of a quantum particle with the vacuum. J. Phys. A Math. Gen. 25(4), 963 (1992)

A Note on the Dirac Operator with Kirchoff-Type Vertex Conditions on Metric Graphs William Borrelli, Raffaele Carlone, and Lorenzo Tentarelli

Abstract In this note we present some properties of the Dirac operator on noncompact metric graphs with Kirchoff-type vertex conditions. In particular, we discuss its spectral features and describe the associated quadratic form. Keywords Dirac operator · Metric graphs · Spectral properties · Kirchoff-type conditions · Boundary triplets

1 Introduction The investigation of evolution equations on metric graphs (see, e.g., Fig. 1) has become very popular nowadays as they are assumed to represent effective models for the study of the dynamics of physical systems confined in branched spatial domains. A considerable attention has been devoted to the cubic Schrödinger equation, as it is supposed to well approximate the behavior of Bose-Einstein condensates in ramified traps (see, e.g., [33] and the references therein). This, naturally, has lead to the study of the graph versions of the laplacian, given by suitable vertex conditions and, especially, to the study of the standing waves of the associated NonLinear Schrödinger Equation (NLSE) (see, e.g., [1–5, 21, 22, 24–27, 46–48]). In particular, the most investigated sub case has been that of the

W. Borrelli () Scuola Normale Superiore, Centro De Giorgi, Pisa, Italy e-mail: [email protected] R. Carlone Università “Federico II” di Napoli, Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, Napoli, Italy e-mail: [email protected] L. Tentarelli Politecnico di Torino, Dipartimento di Scienze Matematiche “G.L. Lagrange”, Torino, Italy e-mail: [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. Michelangeli (ed.), Mathematical Challenges of Zero-Range Physics, Springer INdAM Series 42, https://doi.org/10.1007/978-3-030-60453-0_4

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Fig. 1 A general noncompact metric graph

Fig. 2 The infinite 3-star graph

Kirchhoff vertex conditions, which impose at each vertex v: (i) continuity of the function: ue (v) = uf (v), ∀e, f ( v, e (v) = 0, ∀v ∈ K, (ii) “balance” of the derivatives: e(v du dxe

∀v ∈ K,

where K denotes the compact core of the graph (i.e., the subgraph of the bounded e edges), e ( v means that the edge e is incident at the vertex v and du dxe (v) stands   for ue (0) or −ue (−#e ) according to the parametrization of the edge (for more see Sect. 2). The above conditions correspond to the free case, namely, where there is no interaction at the vertices which are then mere junctions between edges. As a further development, in the last years the study of the Dirac operator on metric graphs has also generated a growing interest (see, e.g., [7, 12, 19, 42]). Moreover, recently [45] proposed (although if in the prototypical case of the infinite 3-star graph depicted in Fig. 2) the study of the NonLinear Dirac Equation (NLDE) on networks, where the Dirac operator is given by D := −ıc

d ⊗ σ1 + mc2 ⊗ σ3 . dx

(1)

Here m > 0 and c > 0 are two parameters representing the mass of the generic particle of the system and the speed of light (respectively), and σ1 and σ3 are the so-called Pauli matrices, i.e.

A Note on the Dirac Operator with Kirchoff-Type Vertex Conditions on Metric Graphs

  01 σ1 := 10

and

  1 0 σ3 := . 0 −1

83

(2)

Precisely, as for the majority of the works on the NLSE, Sabirov et al. [45] suggests the study of the stationary solutions, that is those 2-spinors such that χ(t, x) = e−iωt ψ(x), with ω ∈ R and with ψ solving Dψ − |ψ|p−2 ψ = ω ψ. Remark 1 We observe that, strictly speaking, the parameter c in (1) corresponds to the speed of light only in truly relativistic models, whereas in other contexts it should be rather considered as a phenomenological parameter depending on the model under study. Nevertheless, for the sake of simplicity, we will refer to it as “speed of light” throughout the paper. The attention recently attracted by the linear and the nonlinear Dirac equations is due to their physical applications, as effective equations, both in solid state physics and in nonlinear optics (see [34, 35]). While initially the NLDE appeared as a field equation for relativistic interacting fermions (see [29, 38]), thereafter it was used in particle physics (to simulate features of quark confinement), in acoustics and in the context of Bose-Einstein Condensates (see [35]). Recently, it also appeared that certain properties of some physical models, as thin carbon structures, are effectively described by the NLDE (see [10, 13–15, 20, 30, 31, 40]). On the other hand, in the context of metric graphs NLDE may describe the constrained dynamics of genuine relativist particles, or be regarded as an effective model for solid state/nonlinear optics systems (as already remarked). In particular, it applies in the analysis of effective models of condensed matter physics and field theory (see [45]). Moreover, Dirac solitons in networks may be realized in optics, in atomic physics, etc. (see again [45] and the references therein). Concerning the existence of standing waves for the NLDE on metric graphs, to our knowledge, the first rigorous mathematical work on the subject is [17], where a nonlinearity localized on the compact core of the graph is considered (refer to [18] for a survey on the standing waves of NLDE and NLSE with localized nonlinearity). Recently, in [16] the authors also studied the Cauchy problem for nonlinear Dirac equations with extended nonlinearities (for general metric graphs) and the existence of standing waves on infinite N-star graphs. It is clear that (as for the NLSE), preliminarily to the study of the nonlinear case, it is necessary to find suitable vertex conditions for the operator D that make it selfadjoint. In this paper, we consider those conditions that converge to the Kirchhoff ones in the nonrelativistic limit (for details see [16, Appendix A]), and that we call Kirchhoff-type, which represents (as well as Kirchhoff for Schrödinger) the free case. Roughly speaking, these conditions “split” the requirements of Kirchhoff conditions: the continuity condition is imposed only on the first component of the spinor, while the second component (in place of the derivative) has to satisfy the “balance” condition (see (4) and (5) below).

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Precisely, here we discuss self-adjointness of this graph realization of D, its more relevant spectral properties and some question arising in the rigorous definition of the associated quadratic form. Although this could seem a slightly detached topic with respect to zero-range interactions, it actually is one of its natural evolutions. The attachment of two (or more) edges of a metric graph is usually modeled, indeed, by means of an interplay between a point interaction on each of the two (or more) edges. As a consequence, the techniques to investigate self-adjoint extensions of differential operators on graphs relies on the very same techniques developed for the study self-adjoint extensions of differential operators with point/contact interactions. The Kirchhoff-type case, even representing the free case in the graph context, can be managed simply as a particular case in this framework. We limit ourselves to the case of noncompact graphs with a finite number of edges since it is the most studied one in the nonlinear context. The paper is organized as follows. In Sect. 2 we recall some fundamental notions on metric graphs and we give the definition of the Dirac operator with Kirchhofftype vertex conditions. In Sect. 3, we give a justification of the self adjointness of the operator and of its spectral properties. A model case is presented in Sect. 4, in order to clarify the general arguments developed in Sect. 3. Finally, Sect. 5 accounts for the particular features of the quadratic form associated with D and its form domain.

2 Functional Setting Preliminarily, we recall some basic notions on metric graphs. More details can be found in [1, 11, 37] and references therein. Throughout, by metric graph G = (V, E) we mean a connected multigraph (i.e., possibly with multiple edges and self-loops) with a finite number of edges and vertices. Each edge is a finite or half-infinite segment of real line and the edges are glued together at their endpoints (the vertices of G) according to the topology of the graph (see, e.g., Fig. 1). Unbounded edges are identified with copies of R+ = [0, +∞) and are called half-lines, while bounded edges are identified with closed and bounded intervals Ie = [0, #e ], #e > 0. Each edge, bounded or unbounded, is endowed with a coordinate xe which possess an arbitrary orientation when the interval is bounded and the natural orientation in case of a half-line. As a consequence, the graph G is a locally compact metric space, the metric given by the shortest distance along the edges. Clearly, since we assume a finite number of edges and vertices, G is compact if and only if it does not contain any half-line. Consistently, a function u : G → C is actually a family of functions (ue ), where ue : Ie → C is the restriction of u to the edge e. The usual Lp spaces can be defined in the natural way, with norms p

uLp (G) :=

 e∈E

p

ue Lp (Ie ) ,

if p ∈ [1, ∞),

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85

and uL∞ (G) := max ue L∞ (Ie ) , e∈E

while H m (G) are the spaces of functions u = (ue ) such that ue ∈ H m (Ie ) for every edge e ∈ E, with norm u2H m (G) =

m 

u(i) 2L2 (G) .

i=0

Accordingly, a spinor ψ = (ψ 1 , ψ 2 )T : G → C2 is a family of 2-spinors  ψe =

ψe1 ψe2

 : Ie −→ C2 ,

∀e ∈ E,

and thus Lp (G, C2 ) :=

0

Lp (Ie , C2 ),

e∈E

endowed with the norms p

ψLp (G,C2 ) :=



p

ψe Lp (I

e ,C

2)

,

if p ∈ [1, ∞),

e∈E

and ψL∞ (G,C2 ) := max ψe L∞ (Ie ,C2 ) , e∈E

whereas H m (G, C2 ) :=

0

H m (Ie , C2 )

e∈E

endowed with the norm ψ2H m (G,C2 ) :=



ψe 2H m (I

e ,C

2)

e∈E

Remark 2 Usually, graph Sobolev spaces are not defined as before. They also contain some further requirement on the behavior of the functions at the vertices of the graph (in the case m = 1, for instance, global continuity is often required). However, in a Dirac context it is convenient to keep integrability requirements and conditions at the vertices separated.

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Now, we can define the Kirchhoff-type realization of the Dirac operator on graphs. Definition 1 Let G be a metric graph and let m, c > 0. We define the Dirac operator with Kirchhoff-type vertex conditions the operator D : L2 (G, C2 ) → L2 (G, C2 ) with action D|Ie ψ = De ψe := −ıc σ1 ψe + mc2 σ3 ψe ,

∀e ∈ E,

(3)

σ1 , σ3 being the matrices defined in (2), and domain % & dom(D) := ψ ∈ H 1 (G, C2 ) : ψ satisfies (4) and (5) , where ψe1 (v) = ψf1 (v), 

∀e, f ( v,

ψe2 (v)± = 0,

∀v ∈ K,

∀v ∈ K,

(4) (5)

e(v

ψe2 (v)± standing for ψe2 (0) or −ψe2 (#e ) according to whether xe is equal to 0 or #e at vertex v. An immediate, albeit informal, way to see why the previous one can be considered a Kirchhoff-type realization of the Dirac operator is the following. First, recall that the domain of the Kirchhoff laplacian consists of the H 2 (G)-functions that also satisfy conditions (i) and (ii) of Sect. 1. Therefore, since, roughly speaking, the laplacian is (a component of) the square of the Dirac operator (up to a zero order term), one can square D and check if the resulting operator is in fact the Kirchhoff laplacian (up to a zero order term). This is, indeed, the case, since D2 clearly acts as (−$ + m2c4 ) ⊗ IC2 and since, considering spinors of the type ψ = (ψ 1 , 0)T , if one imposes that ψ ∈ dom(D2 ) (namely, that ψ ∈ dom(D) and that Dψ ∈ dom(D)), then ψ 1 ∈ dom(−$).

3 Self-adjointness and Spectrum In this section, we prove the self-adjointness of the operator D and present its main spectral features. Preliminarily, we observe that the proof of the self-adjointness is not new (see, e.g. [8, 19, 39, 41]). In particular, [19] shows it for a wide class of vertex conditions, including the Kirchoff-type ones. Here we give an alternative justification using the theory of boundary triplets.

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The study the of D requires some introductory notions. Let A be a densely defined closed symmetric operator in a separable Hilbert space H with equal deficiency indices n± (A) := dim N±ı  ∞, where Nz := ker(A∗ − z) ,

z ∈ C\ R,

is the defect subspace. Definition 2 A triplet % = {H, &0 , &1 } is said a boundary triplet for the adjoint operator A∗ if and only if H is a Hilbert space and &0 , &1 : dom(A∗ ) → H are linear mappings such that A∗ f |g − f |A∗ g = &1 f, &0 gH − &0 f, &1 gH ,

f, g ∈ dom(A∗ ),

holds (with ·, ·H the scalar product in H) and the mapping & :=

  &0 &1

: dom(A∗ ) → H ⊕ H

is surjective. Definition 3 Let % = {H, &0 , &1 } be a boundary triplet for the adjoint operator A∗ , define the operator A0 := A∗ |ker &0 and denote by ρ(A0 ) its resolvent set. Then, the γ -field and Weyl function associated with % are the operator valued functions γ (·) : ρ(A0 ) → L(H, H) and M(·) : ρ(A0 ) → L(H), respectively, defined by −1  γ (z) := &0 |Nz

and

M(z) := &1 ◦ γ (z),

z ∈ ρ(A0 ).

(6)

3.1 Self-adjointness In order to apply the theory of boundary triplets to the definition of the Kirchhofftype Dirac operator, one has to study the operator on the single components of the graph (segments and half lines) imposing suitable boundary conditions. Then, one describes the effect of connecting these one-dimensional components, according to the topology of the graph, through the vertex conditions (4) and (5). First, observe that the set E of the edges of a metric graph G can be decomposed in two subsets, namely, the set of the bounded edges Es and set of the half-lines -e on Eh . Fix, then, e ∈ Es and consider the corresponding minimal operator D 2 2 1 He = L (Ie , C ), which has the same action of (3) but domain H0 (Ie , C2 ). As

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a consequence, the adjoint operator possesses the same action and domain -e∗ ) = H 1 (Ie , C2 ). dom(D On the other hand, a suitable choice of trace operators (introduced in [32]) is given e : H 1 (I , C2 ) → C2 , with by &0,1 e  1 ψe e

&0

ψe2

 =

ψe1 (0)



ıcψe2 (#e )

,

 1 ψe e

&1

ψe2

 =

ıcψe2 (0)



ψe1 (#e )

8 7 and hence, given the boundary triplet He , &0e , &1e , with He = C2 , one can compute -e∗ has defect indices the gamma field and the Weyl function using (6) and prove that D n± (De ) = 2. Moreover, note that in this way we can define an operator De with the -e∗ (and D -e ) and domain same action of D dom(De ) = ker &0e , which is self-adjoint by construction. 9 Analogously, fix now e ∈ Eh and consider the minimal operator D e defined on 2 2 1 He = L (R+ , C ), with the same action as before and domain H0 (R+ , C2 ). The domain of the adjoint reads ∗ ) = H 1 (R , C2 ) 9 dom(D + e 

e and the trace operators &0,1 : H 1 (R+ , C2 ) → C can be properly given by  &0e

 1 ψe ψe2

 =

ıcψe1 (0),

 &1e

ψe1



ψe2

= ψe2 (0).

Again, the gamma field and the Weyl function are provided by (6) (with respect   to the boundary triplet {He , &0e , &1e }, with He = C), while the defect indices are 9 n± (D e ) = 1. In addition, one can define again a self-adjoint (by construction) operator as ∗, 9 De := D e

dom(De ) := ker &0e .

Then, we can describe the Dirac operator introduced in Definition 1 using Boundary Triplets. Let D0 be the operator D0 :=

0 e∈Es

De ⊕

0 e ∈Eh

De ,

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: : defined on H = e∈Es He ⊕ e ∈Eh He , with domain given by the direct sum of the domains of the summands. Consider, also, the operator ; := D

0

0

-e ⊕ D

9 D e ,

e ∈Eh

e∈Es

and its adjoint -∗ := D

0

-∗e ⊕ D

e∈Es

0 e ∈Eh

∗ 9 D e

(with the natural definitions of the domains). Introduce, also, the trace operators &0,1 =

0 e∈Es

e &0,1 ⊕

0



e &0,1 .

e ∈Eh

One can check that {H, &0 , &1 }, with H = CM and M = 2|Es | + |Eh |, is a -∗ (and hence one can compute gamma-field and boundary triplet for the operator D Weyl function as before). On the other hand, note that boundary conditions (4)–(5) are “local”, in the sense that at each vertex they are expressed independently of the conditions on other vertices. As a consequence, they can be written using proper block diagonal matrices A, B ∈ CM×M , with AB ∗ = BA∗ , as A&0 ψ = B&1 ψ (see also the model case in Sect. 4). The sign convention of (5) is incorporated in the definition of the matrix B. Therefore the Dirac operator with Kirchoff-type conditions can be defined as -∗ , D := D

dom(D) := ker(A&0 − B&1 ),

and then, by construction, Theorem 1 The Dirac operator with Kirchhoff-type boundary conditions D, defined by Definition 1 is self-adjoint. Remark 3 The boundary triplets method provides an alternative way to prove the self-adjointness of D. More classical approaches à la Von Neumann can be found in [19].

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3.2 Essential Spectrum Now, we can focus on the essential spectrum of D. It can be studied adapting the strategy used for the Schrödinger case in [36]. Preliminarily, note that the spectrum of D0 is given by the union of the spectra of each summand, that is < < σ (D0 ) = σ (De ) ∪ σ (De ). e∈Es

e ∈Eh

Precisely, following [23], each segment Ie , e ∈ Es , contributes to the point spectrum of D0 with eigenvalues given by ⎧ = ⎫   ⎨ ⎬ 1 2 2mc2π 2 2 c4 , j ∈ N , σ (De ) = σp (De ) = ± j + + m ⎩ ⎭ 2 #2e

∀e ∈ Es ,

and each half-lines has a purely absolutely continuous spectrum σ (De ) = σac (De ) = (−∞, −mc2 ] ∪ [mc2 , +∞),

∀e ∈ Eh .

Now, one can check that a Krein-type formula for the resolvent operators holds, namely (D − z)−1 − (D0 − z)−1 = γ (z) (B M(z) − A)−1 Bγ ∗ (z),

∀z ∈ ρ(D) ∩ ρ(D0 )

(7) (with γ (·) and M(·) the gamma-field and the Weyl function, respectively, associated with D—see [23]). Hence, the resolvent of the operator D is as a perturbation of the resolvent of the operator D0 . Since one can prove that the operator at the right-hand side of (7) is of finite rank, Weyl’s Theorem [44, Thm XIII.14] gives Theorem 2 The essential spectrum of the operator D introduced by Definition 1 is given by σess (D) = σess (D0 ) = (−∞, −mc2] ∪ [mc2 , +∞).

3.3 Absence of Eigenvalues in the Spectral Gap A natural question raised by Theorem 2 concerns the existence of eigenvalues and their location.

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Unfortunately, there is no easy and general answer. In principle, eigenvalues can be located in different parts of the spectrum, depending to the topology and the metric of the graph. Indeed, they can be embedded in the essential spectrum or at the thresholds (see next sections for examples). However, they cannot be in the spectral gap, as shown by the following computation. Let λ ∈ σ (D) be an eigenvalue. As a consequence, there exists 0 #= ψ ∈ dom(D) such that Dψ = λψ, or equivalently, such that −ıc

dψ 2 = (λ − mc2 )ψ 1 , dx

(8)

−ıc

dψ 1 = (λ + mc2 )ψ 2 . dx

(9)

If |λ| #= m, then we can divide both sides of (9) by (λ + mc2 ) and plug the value of ψ 2 into (8), obtaining − c2

d 2ψ 1 = (λ2 − m2 c4 )ψ 1 . dx 2

(10)

Furthermore, using (4) and (5), we can prove that  dψ 1 e

e(v

dx

(v) = 0,

ψe1i (v) = ψe1j (v),

∀ei , ej ( v,

so that ψ 1 is eigenfunction of the Kirchhoff laplacian on G. Hence, multiplying (10) times ψ 1 and integrating, |λ| > mc2 , namely Proposition 1 If λ ∈ R is an eigenvalue of the operator D (defined by Definition 1) then |λ|  mc2 .

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Fig. 3 A 3-star graph with two segments

f1 f3 f2

3.4 Graphs with Eigenvalues at the Thresholds As already remarked, D may present eigenvalues at thresholds. This is the content of the following Proposition 2 Let G be a graph with two terminal edges incident at the same vertex v ∈ K. Then λ = ±mc2 are eigenvalue of the operator D (defined by Definition 1). Remark 4 For simplicity we prove the result for the case depicted in Fig. 3, which is the simplest one having the property stated above. The same proof applies to more general graphs provided that they present at least two terminal edges, simply considering spinors which vanish identically everywhere except on the two terminal edges. Proof (Proof of Proposition 2) First, identify the bounded edges of the graph in Fig. 3 with the compact intervals Ij = [0, #j ], j = 2, 3, and the common vertex with 0. Let λ = mc2 . Then, Eqs. (8) and (9) read dψ 2 = 0, dx dψ 1 = 2ımcψ 2 . dx Now, let ψf1 ≡ 0. Integrating the above equations on f2 , f3 yields ψf1j (x) = 2ımcAj x + Bj ,

for x ∈ [0, #j ], with j = 2, 3

and ψf2j (x) ≡ Aj ,

for x ∈ [0, #j ], with j = 2, 3

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where Aj , Bj ∈ C. Therefore, as (4) and (5) have to be satisfied at v 0, we find B2 = B3 = 0,

and

A3 = −A2

and thus λ = mc2 is an eigenvalue of D. Let us, now, turn to λ = −mc2 . In this case the eigenvalue equation becomes dψ 2 = −2ımcψ 1 , dx dψ 1 = 0. dx Setting again ψf1 ≡ 0, we have ψf1j (x) ≡ Ej ,

for x ∈ [0, #j ], with j = 2, 3

and ψf2j (x) = −2ımcEj x + Fj ,

for x ∈ [0, #j ], with j = 2, 3

where Ej , Fj ∈ C, and again by (4) and (5) E2 = E3 = 0,

and

F3 = −F2 .

Then, also λ = −mc2 is an eigenvalue of D and the proof is completed.

% $

3.5 Graphs with Embedded Eigenvalues As we mentioned before, it is also possible, properly tuning the topology and the metric of the graph, to give rise to eigenvalues embedded in the essential spectrum. Here we limit ourselves to show a simple case, which nevertheless displays all the most important features of the phenomenon: the tadpole graph (see, e.g., Fig. 4). e

Fig. 4 Tadpole graph

f

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Arguing as before, first one identifies the circle e of the graph in Fig. 4 with the compact interval I = [0, #], with x = 0 and x = # representing the same vertex v. Then, assume |λ| > mc2 in (8) and (9) and set ψf ≡ 0. In other words, we search for eigenvalues λ ∈ (−∞, −mc2 ) ∪ (mc2 , +∞) whose eigenfunctions are supported on the circle. Namely, we look for spinors ψe ∈ H 1 (I, C2 ), such that −ıc

dψe2 = (λ − mc2 )ψe1 , dx

in (0, #),

−ıc

dψe1 = (λ + mc2 )ψe2 , dx

in (0, #),

and ψe1 (0) = ψe1 (#)

ψe2 (0) = −ψe2 (#) .

,

This is clearly equivalent to finding ψe1 ∈ H 2 (I ) such that d 2 ψe1 m2 c4 − λ2 1 = ψe , dx 2 c2

in (0, #),

with ψe1 (0) = ψe1 (#)

and

dψe1 dx (0)

=−

dψe1 dx (#).

Now, it is easy to check that there are infinitely many values of λ ∈ (−∞, −mc2) ∪ (mc2 , +∞) for which the problem admits a solution, i.e. = λk := sgn(k)

m2 c 4 +

4π 2 c2 2 k , #2

k ∈ Z\{0},

and easy computations yield that solutions are of the form    2x +1 = A sin kπ #    2x −ı2πkcA ψe2 (x) = cos kπ + 1 #(λk + mc2 ) # ψe1 (x)

k ∈ Z\{0}

with A ∈ R. Hence, we found a sequence of eigenvalues (λk )k∈Z\{0} embedded in the essential spectrum, unbounded both from below and from above.

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4 A Model Case: The Triple Junction The aim of the present section is to clarify the main ideas explained before by means of an example. Consider a 3-star graph with one bounded edge and two half-lines, as depicted in Fig. 5. In this case the finite edge is identified with the interval I = [0, #e3 ] and 0 corresponds to the common vertex of the segment and the half-lines. Here trace operators can be defined as ⎛

⎞

ψe11 (0)

⎟ ⎟ ⎟ ⎟, ⎟ ⎠

⎜ ⎜ ψ 1 (0) e2 ⎜ &0 ψ = ⎜ ⎜ ψ 2 (#e ) ⎝ e3 3

⎛

ıcψe21 (0)

⎞

⎟ ⎜ ⎜ ıcψ 2 (0) ⎟ e2 ⎟ ⎜ &1 ψ = ⎜ ⎟, ⎜ ıcψ 2 (0) ⎟ e3 ⎠ ⎝

ıcψe13 (#e3 )

ψe13 (#e3 )

so that, again, Kirchoff-type conditions (4)-(5) can be written as A&0 ψ = B&1 ψ, with AB ∗ = BA∗ given by ⎛

−2 1

1 0

⎜ 2⎜ ⎜ 1 −2 1 A= ⎜ 3 ⎜ 1 1 −2 ⎝ 0

0

⎞

⎟ 0⎟ ⎟ ⎟, 0⎟ ⎠

0 a

⎛

1110

⎜ 2⎜ ⎜1 1 1 B = −ı ⎜ 3 ⎜1 1 1 ⎝

⎞

⎟ 0⎟ ⎟ ⎟ 0⎟ ⎠

000b

(where, properly choosing a, b ∈ C, one can fix the value of the spinor on the non-connected vertex). Fig. 5 A 3-star graph with a finite edge

e1 e3 e2

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Remark 5 Looking at the example above, and recalling that conditions (4) and (5) are defined independently on each vertex, one can easily see how to iterate the above construction for a more general graph structure, thus obtaining matrices A, B with a block structure, each block corresponding to a vertex. In order to investigate the spectral properties of D on the graph depicted by Fig. 5, we have to use Weyl’s Theorem and compute the singularities of the resolvent (see (7)). Hence, it is necessary to compute the gamma-field γ (z) and the Weyl function M(z). The structure of γ (z) and M(z) can be recovered using a block composition as in [23, 32]. In particular, the eigenvalues of the operator D are given by the zeroes of the determinant of (BM(z) − A), so that the computation of (BM(z) − A)−1 B, is needed. Remark 6 In order to simplify some notations in the following, we choose m = 1/2 in the definition of the operator (1), so that the thresholds of the spectrum become λ = ±c/2. Let us define k(z) :=

 1 2 z − (c2 /2)2 , c

z ∈ C,

and = ck(z) k1 (z) := = z + c2 /2

z − c2 /2 , z + c2 /2

z ∈ C.

√ where the branch of the multifunction · is selected such that k(x) > 0 for x > c2 /2. It this way k(·) is holomorphic in C with two cuts along the half-lines (−∞, −c2 /2] and [c2 /2, ∞). Then, the Weyl function reads ⎛

0 ick1(z) ⎜ 0 ick 1 (z) ⎜ M(z) = ⎜ ⎜ 0 0 ⎝ 0 0

0 0

0 0

ck1 (z) sin(#e3 k(z)) 1 cos(#e3 k(z)) cos(#e3 k(z)) sin(#e3 k(z)) 1 cos(#e3 k(z)) ck1 (z)cos(#e3 k(z))

⎞ ⎟ ⎟ ⎟, ⎟ ⎠

Assume also, for the sake of simplicity, that a = 0, b = 1 and fix #e3 = c = 1. After some calculations, one sees that the zeroes of the determinant of BM(z) − A are given by the zeroes of the following function:   /  /   /    1 1 3 8 4z2 − 1 + sin 4z2 − 1 + 2i sec4 4z2 − 1 . f (z) = − i sin 9 2 2 2 (11)

A Note on the Dirac Operator with Kirchoff-Type Vertex Conditions on Metric Graphs

Let us consider, initially, z ∈ (−1/2, 1/2). For such values then (11) rewrites

97

√ 4z2 − 1 ∈ ıR, and

  /    /    / 8 3 1 4 1 2 2 2 f (z) = 1 − 4z + sinh 1 − 4z + 2 sech 1 − 4z . sinh 9 2 2 2 which is a positive real function. For z ∈ / (−1/2, 1/2), on the contrary, R and so  /  16 4 1 2 sec *(f (z)) = 4z − 1 9 2

√ 4z2 − 1 ∈

and  /  /  8 4z2 − 1 , ,(f (z)) = − sin 2 4z2 − 1 cos 9 so that the real part of (11) cannot vanishes, while the imaginary part is periodic and with alternate sign. We can thus conclude that there are eigenvalues in the spectrum. This should be compared with the example presented in Sect. 3.4, where threshold eigenvalues do exist for a small change of the graph structure Fig. 6. Moreover, we remark that the above analysis does not exclude the presence of resonances. They may appear, when one imposes Kirchoff-type conditions, due to the eigenvalues of the operator given by the direct sum of the Dirac operators on the edge (without any boundary condition) that are embedded in the continuous part of the spectrum. However, although the explicit analysis of the spectrum for a Dirac graph can be complicated when the topological structure is complex, the spectral characterization and the analysis of resonances appear to be even more challenging and interesting problems.

2.5

2.0 1.5 1.0

0.5

–0.4

–0.2

0.2

0.4

Fig. 6 The real and imaginary part of f (z) for z ∈ [−1/2, 1/2], in blue and yellow, respectively

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5 The Form-Domain We conclude the paper with some remarks concerning the quadratic form associated with D. The reason for which this point deserves some particular attention can be easily explained. In the standard cases (Rd , with d = 1, 2, 3) the quadratic form associated with the Dirac operator can be easily defined using the Fourier transform (see, e.g., [28]). Unfortunately, in the framework of non compact metric graphs this tool is not available. In addition, also classical duality arguments seems to be prevented as, generally speaking, H −1/2 (G) is not the topological dual of H 1/2 (G), due to the presence of the compact core of the graph. Therefore, one has to resort to the spectral theorem, where the associated quadratic form QD and its domain dom(QD ) are defined as  dom(QD ) := ψ ∈ L2 (G, C2 ) :

σ (D)

|ν| dμD (ν) , ψ

 QD (ψ) :=

σ (D)

ν dμD ψ (ν),

with μD ψ the spectral measure associated with D and ψ. However, such a definition is very implicit and thus not useful in concrete cases (as, for instance, in [17]). On the other hand, a useful characterization of the form domain can be obtained arguing as follows, using real interpolation theory (see, e.g., [6, 9]). Define the space   Y := L2 (G, C2 ), dom(D) 1 ,

(12)

2

namely, the interpolated space of order 1/2 between L2 and the domain of the Dirac operator. First, note that Y → H 1/2 (G, C2 ) :=

0

  H 1/2(Ie , C2 ) = L2 (G, C2 ), H 1 (G, C2 ) 1 , 2

e∈E

where Y is endowed with the interpolation norm (14) and H 1/2(G, C2 ) with the natural norm. Therefore, by Sobolev embeddings, Y → Lp (G, C2 ),

∀p  2,

and, also, the embedding in Lp (K, C2 ) is compact, due to the compactness of K. Let us prove that indeed dom(QD ) = Y.

(13)

This characterization turns out to be particularly useful, for instance, in the nonlinear case where one studies the existence of standing waves by variational methods [17]. In order to prove (13), we exploit again the spectral theorem, but in a different form

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(see Theorem 3 below). In particular, it states that, roughly speaking, every selfadjoint operator on a Hilbert space is isometric to a multiplication operator on a suitable L2 -space. In this sense self-adjoint operators can be “diagonalized” in an abstract way. Theorem 3 ([43, thm. VIII.4]) Let H be a self-adjoint operator on a separable Hilbert space H with domain dom(H ). There exists a measure space (M, μ), with μ a finite measure, a unitary operator U : H −→ L2 (M, dμ) , and a real valued function f on M, a.e. finite, such that 2 1. ψ ∈ dom(H ) if and only if ψ)(·)  f (·)(U  ∈ L (M, dμ), −1 2. if ϕ ∈ U (dom(H )), then U H U ϕ (m) = f (m)ϕ(m),

∀m ∈ M.

The above theorem, in other words, states that H is isometric to the multiplication operator by f (still denoted by the same symbol) on the space L2 (M, dμ), whose domain is given by % & dom(f ) := ϕ ∈ L2 (M, dμ) : f (·)ϕ(·) ∈ L2 (M, dμ) , endowed with the norm  ϕ21 :=

(1 + f (m)2 )|ϕ(m)|2 dμ(m) M

The form domain of f has an obvious explicit definition, as f is a multiplication operator, that is % & / ϕ ∈ L2 (M, dμ) : |f (·)| ϕ(·) ∈ L2 (M, dμ) and we will prove in the sequel that it satisfies (13) (we follow the presentation given in [6, 9]). Consider the Hilbert spaces H0 := L2 (M, dμ) with the norm x0 := xL2 (dμ), and H1 := dom(f ), so that H1 ⊂ H0 . The squared norm x21 is a densely defined quadratic form on H0 , represented by x21 = (1 + f 2 (·))x, x0 , where ·, ·0 is the scalar product of H0 . Define, in addition, the following quadratic version of Peetre’s K-functional % & K(t, x) := inf x0 20 + tx1 21 : x = x0 + x1 , x0 ∈ H0 , x1 ∈ H1 .

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By standard arguments (see  or [6, Ch. 7] and references therein) the

e.g. [9] intermediate spaces Hθ := H0 , H1 θ ⊂ H0 , 0 < θ < 1, are given by the elements x ∈ H0 such that the following quantity is finite:  x2θ =

∞

t −θ K(t, x)

 dt t

0

< ∞.

(14)

Remark 7 In the interpolation procedure we endow H1 with the H 1 -norm, as a direct computation shows that it is equivalent to the graph-norm of D. Now we can prove the following equivalence. Proposition 3 For every θ ∈ (0, 1), one has x2θ = (1 + f 2 (·))θ x, x0 ,

for x ∈ Hθ .

(15)

Proof Preliminarily, for ease of notation, set A := (1 + f 2 (·)). The operator A is positive and densely defined on H0 and also its (positive) square root A1/2 has a domain A dense in H0 . Now, let us divide the proof in two step. Step 1. There holds G K(t, x) =

tA x, x 1 + tA

H ,

t > 0,

x ∈ H0 .

(16)

0

First, observe that the bounded operator in (16) is defined via the functional calculus for A. Then, take x ∈ A and fix t > 0. By a standard convexity argument one gets the existence of a unique decomposition x = x0,t + x1,t , such that K(t, x) = x0,t 20 + tx1,t 21

(17)

(note also that xj,t ∈ A, j = 0, 1). Then, for all y ∈ A, using the minimality requirement in the definition of K, one gets  d  x0,t + sy20 + tx1,t − sy21 = 0, |s=0 ds and then, recalling that x21 = A1/2 x20 , we have A−1/2 x0,t − tA1/2 x1,t , A1/2 y0 = 0.

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Since the above inequality must be true for all y in the dense subset A ∈ H0 we conclude that A−1/2 x0,t = tA1/2 x1,t , so that we obtain x0,t =

tA x, 1 + tA

x1,t =

1 x 1 + tA

(18)

Combining (17) and (18) we get the claim. Step 2. Proof of (15). By Step 1 and exploiting the functional calculus for A, we get  x2θ =

∞

t −θ K(t, x)

0



dt = t

∞

t −θ

G

0

A x, x 1 + tA

G  = A

∞ 0

H dt 0

 H dt x, x . t θ (1 + tA) 0

(19)

Consider, then, the differentiable function 

∞

f (a) :=

dt t θ (1 + ta)

0

,

a > 0.

Integrating by parts, one easily gets 

∞ 0

a dt = t θ (1 + ta) 1−θ



∞ 0

tdt a f  (a). =− t θ (1 + ta)2 1−θ

Then f fulfills f  (a) =

(θ − 1) f (a), a

a > 0,

and integrating f (a) = a θ−1 . Note that we have set the integration constant equal to zero in order to get the correct formula as θ → 1− . Combining the above observations, one sees that (19) reads x2θ = Aθ x, x0 , thus proving the claim.

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Remark 8 Observe that, in the proof above, the bounded operators tA , 1 + tA

1 , 1 + tA

t  0,

are defined using the functional calculus for the self-adjoint operator A. Finally, in view of the previous proposition, if one sets for θ = 12 , then one recovers the form domain of the operator f and, hence, setting H = D and H = L2 (G, C2 ), one has that (12) is exactly the form domain of D, with Y = U −1 H 1 . 2 Consequently, (13) is satisfied and, summing up, we have shown the following Theorem 4 The form domain of D (defined by Definition 1) satisfies   dom(QD ) = L2 (G, C2 ), dom(D) 1 , 2

namely, is equal to the interpolated space of order 1/2 between L2 and the operator domain.

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38. Lee, S.Y., Kuo, T.K., Gavrielides, A.: Exact localized solutions of two-dimensional field theories of massive fermions with Fermi interactions. Phys. Rev. D 12(8), 2249–2253 (1975) 39. Noja, D.: Nonlinear Schrödinger equation on graphs: recent results and open problems. Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 372(2007), 20 pp., article number 20130002 (2014) 40. Pelinovsky, D.E.: Survey on global existence in the nonlinear Dirac equations in one spatial dimension, in Harmonic Analysis and Nonlinear Partial Differential Equations (RIMS Kôkyûroku Bessatsu, B26, Res. Inst. Math. Sci. (RIMS), Kyoto, 2011), pp. 37–50 41. Posilicano, A.: Self-adjoint extensions of restrictions. Oper. Matrices 2(4), 483–506 (2008) 42. Post, O.: Equilateral quantum graphs and boundary triples, in Analysis on Graphs and its Applications. Proceedings of Symposia in Pure Mathematics, vol. 77 (AMS, Providence, 2008), pp. 469–490 43. Reed, M., Simon, B.: Methods of Modern Mathematical Physics. I. Functional Analysis (Academic Press, New York, 1972) 44. Reed, M., Simon, B.: Methods of Modern Mathematical Physics. IV. Analysis of Operators (Academic Press, New York, 1978) 45. Sabirov, K.K., Babajanov, D.B., Matrasulov, D.U., Kevrekidis, P.G.: Dynamics of Dirac solitons in networks. J. Phys. A 51(43), 13 pp., article number 435203 (2018) 46. Serra, E., Tentarelli, L.: Bound states of the NLS equation on metric graphs with localized nonlinearities. J. Differential Equations 260(7), 5627–5644 (2016) 47. Serra, E., Tentarelli, L.: On the lack of bound states for certain NLS equations on metric graphs. Nonlinear Anal. 145, 68–82 (2016) 48. Tentarelli, L.: NLS ground states on metric graphs with localized nonlinearities. J. Math. Anal. Appl. 433(1), 291–304 (2016)

Dirac Operators and Shell Interactions: A Survey Thomas Ourmières-Bonafos and Fabio Pizzichillo

Abstract In this survey we gather recent results on Dirac operators coupled with δ-shell interactions. We start by discussing recent advances regarding the question of self-adjointness for these operators. Afterwards we switch to an approximation question: can these operators be recovered as limits of Dirac operators coupled with squeezing potentials? We also discuss spectral features of these models. Namely, we recall the main spectral consequences of a resolvent formula and conclude the survey by commenting a result of asymptotic nature for the eigenvalues in the gap of a Dirac operator coupled with a Lorentz-scalar interaction. Keywords Dirac operator · δ-shell interaction · Klein’s paradox · Self-adjoint operator · Spectral theory

1 Introduction 1.1 Singular Interactions in Non-relativistic Quantum Mechanics Some non-relativistic quantum systems are efficiently described by Schrödinger operators with singular δ-type potentials supported on a zero Lebesgue measure set.

T. Ourmières-Bonafos () Aix Marseille Univ, CNRS, Centrale Marseille, Marseille, France e-mail: [email protected] F. Pizzichillo CNRS & CEREMADE, Université Paris-Dauphine, PSL Research University, Paris, France e-mail: [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. Michelangeli (ed.), Mathematical Challenges of Zero-Range Physics, Springer INdAM Series 42, https://doi.org/10.1007/978-3-030-60453-0_5

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For example, such Hamiltonians arise as approximations of atomic systems in strong homogeneous magnetic fields [16] or when investigating photonic crystals with high contrast [25]. In this survey, we focus on the particular case of a bounded Lipschitz surface without boundary ' ⊂ R3 which splits the euclidean space R3 into two domains *± ⊂ R3 , that is R3 = *+ ∪ *− ∪ '. Such a surface ' is called a shell and we consider a Hamiltonian acting in L2 (R3 ) which formally writes Sτ := −$ − τ δ' ,

(1)

where τ ∈ R is a coupling parameter and δ' is the distribution defined for all ϕ ∈ C0∞ (R3 ) as  ϕ ds. (2) δ' , ϕD(R3 ),D (R3 ) := '

Here, ds denotes the two-dimensional Haussdorff measure on '. Definition and Self-Adjointness In order to investigate rigorously the operator Sτ given in (1), one has to answer first the following two preliminary questions. (Q1): How can the Schrödinger operator Sτ be defined rigorously? (Q2): Is the Schrödinger operator Sτ self-adjoint? These questions are answered in [9] to which we refer for a rigorous and detailed approach. Nevertheless, for further purpose, we recall here the usual strategy. Start by considering the bilinear form   sτ [u, v] := ∇u, ∇vR3 dx − τ uv ds, u, v ∈ H 1 (R3 ). R3

'

It is well known that this bilinear form is symmetric, densely defined, closed and semi-bounded below in L2 (R3 ) (see, for instance, [15, Section 2]). In particular, the Schrödinger operator Sτ can be defined as the self-adjoint operator associated with the bilinear form sτ thanks to Kato’s first representation theorem [28, Chapter VI, Theorem 2.1]. In particular, it implies D(Sτ ) ⊂ H 1 (R3 ). Nevertheless, one could argue that such an implicit definition of D(Sτ ) does not fully answer Question (Q1) in the sense that we do not know neither the action of the Schrödinger operator Sτ nor have described its domain. Actually, it can be proved that ⎧ D(Sτ ) = {u = u+ ⊕ u− ∈ D(sτ ) : $u± ∈ L2 (*± ), ⎨ (3) u+ |' = u− |' , ∂n u|' = τ u|' }, ⎩ Sτ (u+ ⊕ u− ) = (−$u+ ) ⊕ (−$u− ),

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where we have identified L2 (R3 ) with L2 (*+ ) ⊕ L2 (*− ) and where ∂n denotes the usual jump of the normal derivatives of u+ and u− through the surface '. Here, of course, as D(Sτ ) ⊂ H 1 (R3 ) one can easily give a sense to the traces 1 u± |' ∈ H 2 (') (see, for instance, [30, Theorem 3.37]). However the jump of the traces of the normal derivatives has to be understood in a weak sense, which is possible because $u± ∈ L2 (*± ). Such jump conditions can be recovered using a naive approach. Indeed, take for instance u ∈ C ∞ (R3 \ ') ∩ L2 (R3 ) and apply the expression of the Schrödinger operator Sτ given in (1) in the sense of distributions. Using the jump formula, it gives: (−$ − τ δ' )u = −

3 

(∂j2 u+ )1*+ + (∂j2 u− )1*−



j =1

+ (∂n− u− |' + ∂n+ u+ |' )δ' − τ u|' δ' ,

(4)

where ∂n± u± |' are the Neumann traces of u± . For the right-hand side to belong to L2 (R3 ) we need the following equality to hold: ∂n u|' := ∂n− u− |' + ∂n+ u+ |' = τ u|' . It is exactly the jump condition given in (3). In particular the operator Sτ defined in (3) acts as expected in (1). Approximations of δ-Shell Potentials From a physical point of view, a Schrödinger operator with a δ-shell potential is an idealized Hamiltonian for a quantum particle submitted to an electric potential localized in a thin tubular neighborhood of the shell. To justify this modelling, pick a function V ∈ C0∞ (R3 ) and let V be a sequence of mollifiers such that   V → V (x)dx δ' ,  → 0. R3

One can wonder if the family of Hamiltonians (−$ − V )>0 has a limit when  → 0. These operators are self-adjoint on the domain H 2 (R3 ) and for the model to be physically consistent, we would like to obtain a connection between the spectrum Sp(−$−V ) of −$−V and the spectrum of its limit operator. Mathematically, this can be answered investigating the following question (see [36, Section VIII.7]). (Q3): For some operator topology, does the following convergence hold −$ − V −→ SτV , →0

for some τV ∈ R depending on the potential V ?

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Finally, one would like to know whether or not this limiting procedure allows to recover the whole range of all possible coupling constants. (Q4): Given an operator Sτ , can it be realized as an operator SτV , for some V ∈ C0∞ (R3 )? Both Questions (Q3) and (Q4) are well investigated in the literature. Let us mention the one-dimensional case studied in [1] and [10] where the case of higher dimensions is dealt with for singular perturbations on general smooth hypersurfaces. This question is also discussed in [24, Section 10.1]. The main result is a norm resolvent convergence of the family of operators (−$− V )>0 to SτV where τV = R3 V (x)dx, answering both questions (Q3) and (Q4). Spectral Theory The structure of the spectrum of the operator Sτ attracted a lot of attention and is well understood, in particular as ' is compact we get Spess (Sτ ) = [0, +∞), see [8, Theorem 2.1] and if τ > 0, the interaction becomes attractive and bound states can appear below the threshold of the essential spectrum. The existence of such bound states, as well as their behaviour in the strong coupling regime τ → +∞ has been intensively investigated and we refer to [24, Chapter 10] and references therein for results in this direction.

1.2 Singular Interactions in Relativistic Quantum Mechanics The aim of this survey is to know until what extent questions (Q1)–(Q4) have been investigated for relativistic quantum particles. In this case, the Schrödinger operator (1) is replaced by the Dirac operator that acts in L2 (R3 , C4 ) as Dm := D := −i

3 

 αj ∂j + mβ = −iα · ∇ + mβ

(5)

j =1

where m ∈ R is the mass of the considered particle and α1 , α2 , α3 , β ∈ C4×4 are the Dirac matrices     0 σj 1 0 αj := . , β := 2 0 −12 σj 0

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Here σ1 , σ2 , σ3 ∈ C2×2 are the usual Pauli matrices   01 σ1 := , 10

  0 −i σ2 := , i 0

  1 0 σ3 := . 0 −1

It is well known (see [38, Section 1.4]) that D is essentially self-adjoint on C0∞ (R3 , C4 ). We denote by Dfree its self-adjoint extension and Dfree is called the free Dirac operator. It is defined on the domain D(Dfree ) := H 1 (R3 , C4 ) and Sp(Dfree ) = Spess (Dfree ) = (−∞, −|m|] ∪ [|m|, +∞).

(6)

The aim of this survey is to gather recent advances in the study of the Dirac operator coupled with a δ-shell interaction. This operator formally acts as Dτ,η := D + (τ 14 + ηβ)δ' ,

(7)

where τ, η ∈ R are coupling constants. From a physical point of view, this operator arises when one aims to study relativistic properties of spin-1/2 particles (such as electrons) coupled with an electrostatic potential of interaction strength τ and a Lorentz-scalar potential of interaction strength η, both localized on a shell '. Now, one could ask whether the relativistic counterpart of Question (Q1) has a natural answer as in the non-relativistic case evoked in Sect. 1.1 and it turns out the question is actually more involved, due to the relativistic nature of the model. Indeed, the energy functional of the Dirac operator is neither bounded below nor above as it can be seen by looking at the spectrum of the free-Dirac operator (6). Consequently, the approach involving a quadratic form is not available anymore and one needs to think about another rigorous strategy. When trying to apply the program (Q1)–(Q4) to relativistic particles we can see that the answer is not as straightforward as in the non-relativistic setting and this survey aims to illustrate the state of the art regarding these questions.

1.3 Structure of the Survey Section 2 is devoted to the rigorous definition of the Dirac operator coupled with both electrostatic and scalar interactions supported on a shell ', answering Questions (Q1) and (Q2) in the relativistic setting. Section 3 aims to justify that the Dirac operator coupled with either an electrostatic or a scalar interaction supported on a shell ' can be approached by a sequence of squeezing potentials, answering Questions (Q3) and (Q4). Finally, Sect. 4 deals with various properties of the spectrum of this operator that can be deduced from the previous definitions and results of Sect. 2. Namely, a

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resolvent formula is given and spectral asymptotics are obtained in the large mass limit for a pure Lorentz-scalar potential.

2 Definition of Relativistic Shell Interactions and Self-Adjointness In this section we discuss the various approaches used in the past few years to define the Dirac operator with a shell interaction. Sections 2.1–2.3 follow the chronological order of publications in order to emphasize on the key evolutions. Namely, we discuss the question of self-adjointness as dealt with in [2, 7, 11, 22, 34]. A reader only interested in the present state of the art can skip directly to Sect. 2.4 where we sum up the main results. We also state two open problems related to the question of self-adjointness of the Dirac operator with a shell interaction.

2.1 The Spherically Symmetric δ-Shell The first definition of the operator Dτ,η is given in [22] where the special case ' = S2 is considered. The authors look for a definition of the operator Dτ,η which preserves the spherical symmetry of the problem. To do this, they decompose the ambient Hilbert space L2 (R3 , C4 ) in partial wave subspaces associated to the Dirac operator. Namely, they are reduced to investigate the self-adjoint extensions of a countable family of operators of the form 

  ∞ 2 D(d) = C  0 (0, R) ∪ (R, +∞), C  , d du = − iσ2 dr + mσ3 + χr σ1 u,

  where these operators act in L2 (0, +∞), C2 and where χ ∈ Z \ {0} (see [22, Section 3] for details). Then, they suggest an extension dτ,η of d defined as ⎧   ⎪ ⎪ D(dτ,η ) = u = u+ ⊕ u− ∈ AC(I− , C2 ) ⊕ AC(I+ , C2 ) ∩ L2 (I+ , C2 ) : ⎪ ⎪ ⎪ ⎪   ⎪ ⎪ d ⎨ − iσ2 dr + χr σ1 u± ∈ L2 (I± , C2 ),

     ⎪ 1 ⎪ + ησ (1) + u (1) = iσ (1) − u (1) τ 1 u u ⎪ 2 3 + − 2 + − ⎪ 2 ⎪ ⎪     ⎪ ⎪ χ χ ⎩ dτ,η u = − iσ2 d + mσ3 + σ1 u+ ⊕ − iσ2 d + mσ3 + σ1 u− , dr r dr r

where we have set I+ = (0, 1), I− = (1, +∞).

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Such a choice for the jump condition at r = 1 is justified as follows. The distribution δS2 on the shell S2 can be understood in the partial wave decomposition as δ{r=1} , the Dirac distribution at r = 1. However, for a function u ∈ AC(I− ) ⊕ AC(I+ ), the expression (uδ{r=1} ) does not make any sense a priori. Hence, in [22, Equation (4.4)] they choose to define a distribution (uδ{r=1} ) as: uδ{r=1} =

 1 u+ (1) + u− (1) δ{r=1} . 2

(8)

We emphasize on the fact that this is a choice. The δ-shell is said to be symmetric because each boundary term u± (1) is considered with a coefficient 12 and although they may not be physically interesting, asymmetric δ-shell can also be considered (see [22, Appendix]). ˇ Thus, the strategy of Dittricht, Exner and Seba gives a natural answer to (Q1) in the relativistic setting. The operator Dτ,η is defined via a fibre decomposition in partial wave subspace and Question (Q2) about self-adjointness is answered thanks to the following proposition. Proposition 1 (Proposition 4.1 in [22]) The operator dτ,η is self-adjoint.

% $

Remark that this strategy defines the domain D(Dτ,η ) as the direct sum of the domains of the (countable) partial wave operators D(dτ,η ) and every information on the Sobolev regularity of functions in D(Dτ,η ) is implicitly hidden in this decomposition.

2.2 General Shells Let us describe the approach of [2], where the authors study the case of a general Lipschitz shell '. Consider the minimal Dirac operator

D(Dmin ) = C0∞ (R3 \ ', C4 ), Dmin u = (−iα · ∇ + mβ)u.

This operator is symmetric and we consider its adjoint, the maximal operator Dmax , defined as D(Dmax ) = {u ∈ L2 (R3 , C4 ) : (α · ∇)(u± ) ∈ L2 (*± , C4 )}     Dmax u = (−iα · ∇ + mβ)u+ ⊕ (−iα · ∇ + mβ)u− , where once again we have set u± := u1*± and identified L2 (*+ , C4 ) ⊕ L2 (*− , C4 ) with L2 (R3 , C4 ). Their main idea is to define the operator Dτ,η on a subdomain of D(Dmax ) providing extra jump conditions through the shell '.

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First Step They try to give an accurate description of the domain D(Dmax ) using various integral operators involving a fundamental solution φ of the free Dirac operator Dfree . Proposition 2 (Lemma 3.1 in [2]) Let m > 0. A fundamental solution of the free Dirac operator Dfree is given by φ(x) =

x  e−m|x|  mβ + (1 + m|x|)iα · 2 , 4π|x| |x|

for x ∈ R3 \ {0}.

% $

Then, they construct a linear and bounded operator + : L2 (', C4 ) → L2 (R3 , C4 ) defined as  +(g)(x) := φ(x − y)g(y)ds(y), for x ∈ R3 \ ', (9) '

see [2, Corollary 2.3]. Remark that the operator + is constructed in order to have for all g ∈ L2 (', C4 ) D+(g) = 0 as a distribution in D (*± ). In particular, +(g) is harmonic for the Dirac operator in the domains *± and +(g) ∈ D(Dmax ). Then, instead of working with the space D(Dmax ), they focus on its subspace E defined as E := {u + +(g) : u ∈ H 1 (R3 , C3 ), g ∈ L2 (', C4 )} ⊂ D(Dmax ).

(10)

Second Step They prove that functions in the vector space E have non-tangential traces in L2 (', C4 ). More precisely, one can define two linear bounded operators C± : L2 (', C4 ) → L2 (', C4 ) defined as C± (g)(x) :=

lim

nt

+(g)(y),

*± y →x

and they are related via a Plemelj-Sokhotski jump formula to the linear and bounded operator Cs : L2 (', C4 ) → L2 (', C4 ) defined for x ∈ L2 (', C4 ) as  Cs (g)(x) = lim

ε→0 '∩{|x−y|>ε}

φ(x−y)g(y)ds(y),

i C± = ∓ (α·n)+Cs , 2

(11)

where n denotes the outward pointing normal to *+ , see [2, Lemma 3.3]. Third Step In order to define the operator one needs to give a meaning to the expression (uδ' ) for u = v + +(g) ∈ E. By analogy with (8) for a spherical δ-shell interaction, one

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can define this expression as the distribution uδ' :=

  1 (u+ |' + u− |' )δ' = v|' + Cs (g) δ' . 2

(12)

With this definition and using the jump formula, we compute Dτ,η u in the sense of distributions and obtain:  Dτ,η u =

 − i(α · ∇u+ )1*+ − i(α · ∇u− )1*−

1 − i(α · n)(u− |' − u+ |' )δ' + (τ 14 + ηβ)(u+ |' + u− |' )δ' . 2

(13)

Thus, a natural jump condition through ' for u ∈ E is 1 (τ 14 + ηβ)(u+ |' + u− |' ) = i(α · n)(u− |' − u+ |' ). 2

(14)

Taking the Plemelj-Sokhotski jump formula (11) into account, it rewrites   (τ 14 + ηβ)v|' = − 1 + (τ + ηβ)Cs g. It leads to the following definition of the operator Dτ,η , that can be found in [2, Theorem 3.8] (for the pure electrostatic case η = 0). Definition 3 The operator Dτ,η is defined as ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

% D(Dτ,η ) = v + +(g) : v ∈ H 1 (R3 , C4 ), g ∈ L2 (', C4 ),  &  (τ 14 + ηβ)v|' = − 14 + (τ 14 + ηβ)Cs g , Dτ,η (v + +(g)) = (Dv+ ) ⊕ (Dv− ),

where we have identified L2 (R3 , C4 ) and L2 (*+ , C4 ) ⊕ L2 (*− , C4 ).

% $

Contrary to the strategy developed for spherical shells (see Sect. 2.1), Definition 3 describes the functions in the domain of D(Dτ,η ) as functions of the ambient Hilbert space L2 (R3 , C4 ) and precise their Sobolev regularity (actually, this can be made more precise as we will see thereafter in Sect. 2.3). The main result concerning self-adjointness reads as follows and concerns the pure electrostatic case (i.e. η = 0). Theorem 1 (Theorem 3.8 in [2]) Let ' be of class C 2 . As long as τ #= ±2 the operator Dτ,0 introduced in Definition 3 is self-adjoint. $ % Remark that we needed to impose two restrictions. The first one is that ' has to be sufficiently smooth but, the most surprising one is the existence of two critical

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strengths for the coupling constants τ = ±2. This last observation attracted a lot of attention in the past few years as we will see thereafter in Sect. 2.3. Remark 4 An analogue of Theorem 1 has been obtained in [3, Section 5.1] (more recently in [12]) for the general operator Dτ,η and reads as follows. Let ' be of class C 2 . As long as τ 2 − η2 #= 4 the operator Dτ,η defined in Definition 3 is self-adjoint. % $ Later on, Definition 3 attracted the attention of specialists in self-adjoint extensions of symmetric operators acquainted with the theory of quasi boundary triples, a slight modification of the general theory of boundary triples. (see [17] and references therein for an introduction to boundary triples and [5] for an introduction to quasi boundary triples). The main advantage of this theory is that it gives a systematic framework to define the operator, study its self-adjointness and spectral properties. Following this path, in [11], the authors propose a definition of Dτ,0 which coincides with the one given in Definition 3 (see [11, Definition 4.1]) and Theorem 1 is obtained as a consequence of the general theory of quasi boundary triples (see [11, Theorem 4.4]). The key argument in these two works lies in a link they establish between properties about the range and the kernel of an integral operator on the shell ' and the question of self-adjointness for Dτ,0 (see [2, Theorem 2.11] and [11, Theorem 2.4]). It turns out that in this study, the anticommutator {Cs , iα · n} plays a fundamental role. It is defined as   K := {Cs , iα · n} := i Cs (α · n) + (α · n)Cs

(15)

and as long as the shell ' is of class C 2 , K is a compact operator from L2 (', C4 ) onto itself and the problem is solved by an adequate application of Fredholm alternative. Remark that the hypothesis on the smoothness of the shell ' plays a fundamental role here: there is a priori no reason for this operator to be compact for less regular shells.

2.3 How to Handle the Critical Strengths? The critical strengths that appear in Theorem 1 motivated the simultaneous works [7, 34] where the authors wonder until which extent the operator Dτ,0 is self adjoint for the critical strengths τ = ±2. First, both works start with a different definition of the domain of the operator Dτ,0 .

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Definition 5 The Dirac operator with an electrostatic shell interaction of strength τ ∈ R is denoted Dτ and defined as ⎧ ⎨ D(Dτ ) = {u = u+ ⊕ u− ∈ H 1 (*+ , C4 ) ⊕ H 1 (*− , C4 )) : τ 2 (u+ |' + u− |' ) = i(α · n)(u− |' − u+ |' )}, ⎩ Dτ u = (Du+ ) ⊕ (Du− ).

% $

The jump condition is the one obtained in (14) and the main result reads as follows (see [34, Theorem 4.3] and [7, Theorem 1.1 & Theorem 1.2]). Theorem 2 Let τ ∈ R and let Dτ be the operator of Definition 5. The following alternative holds. (i) If τ #= ±2, Dτ is self-adjoint and coincides with the operator Dτ,0 of Definition 3. (ii) If τ = ±2, Dτ is essentially self-adjoint and there holds D(Dτ )  D(Dτ ) := {u = u+ ⊕ u− ∈ L2 (R3 , C4 ) : (α · ∇)u± ∈ L2 (*± , C4 ), τ 2 (u+ |' + u− |' ) = iα · n(u− |' − u+ |' )}, 1

where the transmission condition holds in H − 2 (', C4 ).

% $

Because the difference of the resolvents of Dτ and Dfree is a compact operator for the non-critical cases (i), an elemental spectral consequence of Theorem 2 is that the essential spectrum of Dτ is given by  

 Spess (Dτ ) = Spess (Dfree ) = − ∞, −|m| ∪ |m|, +∞ . Remark that this is not necessarily true for the critical cases, which may also prevent the functions in the domain D(Dτ ) to have any Sobolev regularity. Namely, in [7, Theorem 5.9] the authors prove the following theorem. Theorem 3 Let τ = ±2. If an open subset of ' is contained in a plane, there holds: 0 ∈ Spess (Dτ ). In particular, for all s > 0, D(Dτ ) can not be included in the Sobolev space H s (*+ , C4 ) ⊕ H s (*− , C4 ). % $ We briefly outline the strategy used to prove Theorem 2 in [34]. First Step In order to prove Theorem 2, one needs to understand what is missing in the space E in order to have an equality instead of an inclusion in (10). To do so, remark that a duality argument implies that functions in D(Dmax ) have 1 weak traces in H − 2 (', C4 ).

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Then, one proves that the operator + introduced in (9) extends into a linear 1 bounded operator from H − 2 (', C4 ) to D(Dmax ) (see [34, Theorem 2.2]). Finally, remark that the operators C± of (11) also extend as bounded operators 1 from H − 2 (', C4 ) onto itself. The Plemelj-Sokhotski jump formula (11) leads to introduce the bounded 1 projectors in H − 2 (', C4 ) defined as C± := ±C± (iα · n). They satisfy C+ + C− = I d and C2± = C± . They allow to describe accurately the maximal domain D(Dmax ). For this purpose, we are lead to introduce the spaces H01 (*± ) := {u ∈ H 1 (*± , C4 ) : C± (u|' ) = 0}. We have the following lemma. Lemma 6 The following direct sum of vector spaces holds.   · Hα (*± ) = H01 (*± ) + {+ (α · n)f : f ∈ ran C± }, where Hα (*± ) := {u ∈ L2 (*± , C4 ) : (α · ∇)u ∈ L2 (*± , C4 )}.

% $

Proof It is clear that the set in the right-hand side is included in Hα (*± ). Now, pick u± ∈ Hα (*± ). We have     u± = u± ∓ i+ (α · n)C± (u± |' ) ± i+ (α · n)C± (u± |' ) . )* + ( :=v±

Remark that v± ∈ Hα (*± ) and v± |' = u± |' − C± (u± ) = C∓ (u± |' ) ∈ 1 H 2 (', C4 ) by [34, Proposition 2.7]. Thus, by elliptic regularity (see [34, Proposition 2.16]), v± ∈ H 1 (*± , C4 ) and as v± |' = C∓ (u± |' ) we get C± (v± |' ) = 0 and v± ∈ H01 (*± ). Setting f = ±iC± (u|' ) we obtain that   Hα (*± ) = H01 (*± )+{+ (α · n)f : f ∈ ran C± } It remains to prove that the sum is direct. Assume that   u ∈ H01 (*± ) ∩ {+ (α · n)f : f ∈ ran C± }.   Hence, u = ∓i+ (α · n)f for some f ∈ ran C± . As u ∈ H01 (*± ), we obtain 0 = C± (u|' ) = f and u = 0. % $   Remark 7 The spaces {+ (α · n)f : f ∈ ran C± } can be seen as analogues of Bergman spaces for the Dirac operator, similarly as the usual Bergman space

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defined as the space of square integrable holomorphic functions in a domain of R2 . Moreover, remark that the space of traces of Dirac-harmonic functions in *± is ran C± . These projectors can be understood as the natural counterpart of Szegö projectors on the boundary ' for Dirac operators. % $ As the maximal domain D(Dmax ) satisfies D(Dmax ) = Hα (*+ ) ⊕ Hα (*− ), Lemma 6 provides an accurate description of D(Dmax ). Second Step Now, consider Dτ∗ , the adjoint of the operator Dτ introduced in Definition 5. One can prove that % & τ D(Dτ∗ ) = u = u+ ⊕ u− ∈ D(Dmax ), (u+ |' + u− |' ) = iα · n(u− |' − u+ |' ) , 2 where the transmission condition holds in H − 2 (', C4 ). If the traces u± |' ∈ 1 H 2 (', C4 ), by elliptic regularity (see [34, Proposition 2.16]), the non-critical case (i)—Theorem 2 is proved. 1 By Lemma 6, for u ∈ D(Dmax ), we always have C± (u∓ |' ) ∈ H 2 (', C4 ) and 1 it remains to prove that C± (u± |' ) ∈ H 2 (', C4 ). Using commutation relations between C± and the multiplication operator (α ·n), one obtains the following system 1 in H − 2 (', C8 ) (see [34, Equation (4.8)]): 1

 Aτ

     K(u+ |' − u− |' ) C+ (u− |' ) C+ (u+ |' ) = Bτ +F , C− (u+ |' ) C− (u− |' ) K(u+ |' − u− |' )

(16)

where Aτ , Bτ , F ∈ C 1 (', C8×8 ) and K is the anticommutator introduced in (15). 1 If ' is of class C 2 the right-hand side of (16) belongs to H 2 (', C8 ) because K is not only compact as a bounded operator in L2 (', C4 ) but also a smoothing operator 1 1 from H − 2 (', C4 ) to H 2 (', C4 ) (see [34, Proposition 2.8]). As Aτ is invertible if and only if τ #= ±2, (i)—Theorem 2 is proved. To prove (ii)—Theorem 2, one first proves that Dτ , the closure of Dτ , is Dτ∗ and the only thing left to check is that D(Dτ ) differs from D(Dτ ). Actually, any 1 1 / H 2 (', C4 ) generates a function in D(Dτ ) which f ∈ H − 2 (', C4 ) such that f ∈ does not belong to D(Dτ ) using that in this case, the matrix valued operator Aτ in (16) is not invertible (see e.g. [34, Section 4.4]).

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2.4 State of the Art on Self-Adjointness, Consequences and Open Problems In this paragraph, we give the most recent definition of the Dirac operator Dη,τ as given in [12]. This covers the previous definitions and results of Sects. 2.2 and 2.3. Definition 8 (Equation (3.1) in [12]) Let τ, η ∈ R. The Dirac operator with electrostatic interaction of strength τ and Lorentz scalar interaction of strength η denoted Dη,τ is defined as ⎧ ⎨ D(Dτ,η ) := {u = u+ ⊕ u− ∈ H 1 (*+ , C4 ) ⊕H 1 (*− , C4 ) : i(α · n)(u− |' − u+ |' ) = 12 τ 14 + ηβ (u+ |' + u− |' )}, ⎩ % $ Dτ,η u = (Du+ ) ⊕ (Du− ). Combining [12, Theorem 3.4], [34, Theorem 4.3] and [7, Theorems 1.1 & 1.2] we obtain the following result. Theorem 4 If τ 2 − η2 #= 4 the operator Dτ,η introduced in Definition 8 is selfadjoint. In the pure electrostatic case τ = ±2 and η = 0, D±2,0 is essentially self adjoint and the domain of its closure is given by D(D±2,0 )  D(D±2,0 ) := {u = u+ ⊕ u− ∈ L2 (R3 , C4 ) : (α · ∇)u± ∈ L2 (*± , C4 ) i(α · n)(u− |' − u+ |' ) = 1

 1 τ 14 + ηβ (u+ |' + u− |' )}, 2

where the transmission condition makes sense in H − 2 (', C4 ).

% $

The missing part in this program is to understand until which extent the case τ 2 − η2 = 4 share the same features as the pure electrostatic critical case τ = ±2. Let us finish this paragraph with a remark about the confinement of particles inside and outside the shell '. This condition is already stated in the seminal paper [22, Section V] and developed in [3, Section 5]. Remark 9 If η2 − τ 2 = 4 the shell generates confinement which physically means that it becomes impenetrable to the particles. This is nothing but a consequence of the fact that the traces of functions in the domain of Dη,τ are not coupled in this case and the operator can be rewritten as the direct sum of two operators with boundary conditions on ': one in *+ and another in *− (see e.g. [12, Lemma 3.1, (ii)]). In the special case τ = 0 and η = ±2, one can easily observe this fact. For example, for η = −2, D0,−2 can be rewritten as the direct sum D0,−2 = D+ ⊕ D− ,

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where the self-adjoint operators D± are defined as

D(D± ) := {u ∈ H 1 (*± , C4 ) : B± u|' = u|' }, D± u = Du,

where B± := ∓iβ(α ·n). One recognizes the direct sum of two operators introduced in the late 60s and intensively studied in the physics literature (see the initial work [13, Section IV] and the works [19–21, 27]). These operators aim to model the confinement of quarks in hadrons and they are often referred to as MIT bag operators (see [4] for a mathematical study). % $

2.5 Open Problems We conclude this section with some problems which are still open regarding the self-adjointness of the operator Dτ,η introduced in Definition 8. Open Problem 10 In the specific case τ 2 − η2 = 4 and for a smooth shell ', have the functions in the domain D(Dm,τ,η ) any Sobolev regularity? % $ Open problem 10 is partially answered in [7, Theorem 5.9] where the authors consider a pure electrostatic shell interaction (η = 0). When ' contains an open set included in a plane, a rather surprising spectral property appears: 0 belongs to the essential spectrum of Dτ,η which prevent the domain D(Dτ,η ) to be included in any Sobolev space H s (*+ , C4 ) ⊕ H s (*− , C4 ) for all s > 0. It is reminiscent of a similar phenomenon occurring in the study of metamaterials for which the geometry of the shell plays an crucial role to determine if whether or not some Sobolev regularity can be expected (see [6, 14, 18]). Open Problem 11 To our knowledge, all known results on self-adjointness of Dτ,η deal with sufficiently smooth shells ' (at least of class C 2 ). One may ask until what extent these results also hold for Lipschitz domains? In particular, the special class of corner geometries would deserve to be investigated. Indeed, it is known in the non-relativistic case that corners may generate interesting spectral features (see for instance [23] for a broken line interaction). In this direction, let us mention the recent work [35] in which the two-dimensional counterpart of D0,η is analysed for the special case of a curve with finitely many corners. % $

3 Approximation Procedure In this section we discuss the problem of the approximation of Dirac operators coupled with δ-shell interactions using regularized Hamiltonians. The main goal of

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this section is to recover Dirac operators coupled with δ-shell interactions as limit of regularized Hamiltonians as it is usually done in the non-relativistic setting (see Sect. 1.1). The one dimensional version of this problem is tackled in [37] where it is proved that the regularizing sequence of Hamiltonians converge in the norm resolvent sense to a Dirac point interaction: it answers Question (Q3) in the relativistic setting. However, when trying to answer Question (Q4), a most surprising effect appears: the coupling constant in front of the point interaction depends non-linearly of the approximating sequence. This non-linear effect, understood as a reminiscence of Klein’s Paradox, is a purely relativistic phenomenon due to the unboundedness neither from above nor below of the Dirac operator (see the original work of Klein [29] as well as [39, Section 4.5] for a detailed explication of this phenomenon). In dimension three, the problem is not entirely solved and this part of the survey focuses on the results presented in [32] where a similar non-linear effect is exhibited. Following [31], we also discuss the special case ' = S2 , for which more properties can be deduced. In Sect. 3.1 we describe the current state of the art and Sect. 3.2 explains the main tools used to obtain these results. Finally, Sect. 3.3 concludes this section with some problems which are still open about this approximation procedure.

3.1 Main Results Let us start by defining the family of approximating potentials rigorously. In this section ' is of class C 2 and for  small enough, we introduce the tubular neighborhood of ' as * := {x ∈ R3 : dist(x, ') < }, Then there exists 0 > 0 sufficiently small in order for *0 to be in one-to-one correspondence with ' × (−0 , 0 ), that is * := {x' + tn(x' ) : x' ∈ ', t ∈ (−, )}, where n(x' ) is the normal to ' in x' pointing outward *+ . Consider V ∈ L∞ (R) with support in [−0 , 0 ] and construct the family of squeezing potentials (V )0 0, we have Spdis Dη (m) = ∅. % $ Remark 16 The condition η ∈ R \ {±2} is not a restriction. Indeed, as discussed in Remark 9, the operator uncouples as the direct sum of two Dirac operators with MIT bag boundary condition. In [4], the spectral properties of this model are investigated which provides a complete picture for all η ∈ R. % $ By (v)—Theorem 8 the only interesting spectral feature that may happen in the gap (−m, m) is when mτ < 0 and by Theorems (iv)–8, without loss of generality, we can pick τ < 0 and m > 0. The main result of [26] reads as follows. Theorem 9 Let η < 0 with η #= 2. The following Weyl-type asymptotics holds:   16   τ2 #Spdis Dη (m) = |'|m2 + O m ln(m) , 2 2 π (τ + 4)

m → +∞.

Moreover, if ±μk (m) denote the eigenvalues of Dη (m) with μk (m) ≥ 0 enumerated in the non-decreasing order, then for each k ∈ N there holds μk (m) =

 ln(m)  τ 2 + 4 Ek (ϒτ ) |τ 2 − 4| m + + O , τ2 + 4 |τ 2 − 4| 2m m2

m → +∞.

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Here, Ek (ϒτ ) is the k-th eigenvalue of an m-independent Schrödinger operator ϒτ with an external Yang-Mills potential in L2 (', C2 )  ϒτ = d + i

∗    τ 2 − 4 2 4 4 τ 4 + 16 2 ω d + i ω − M 1 + K12 , 2 τ2 + 4 τ2 + 4 τ2 + 4 (τ 2 + 4)2

where K and M are the Gauss and mean curvature, respectively. The 1-form ω is given by the local expression ω := σ · (n × ∂1 n)ds1 + σ · (n × ∂2 n)ds2 .

% $

The Weyl asymptotic of Theorem 9 justifies that the larger the mass m, the more eigenvalues are created in the gap (−m, m). From a physical point of view, in the regime m → +∞, the system behaves at first order as the one for particles constrained to live on the shell ' driven by the effective Hamiltonian ϒτ . This Hamiltonian is a geometric object as it is directly seen by the expression of the Yang-Mills potential. It is reminiscent of the recent work [33], where for the MIT bag Dirac operator (obtained by setting τ = ±2, see Remark 9) the authors obtain an effective operator given by the square of the intrinsic Dirac operator on the shell ' (see e.g. [33, Theorem 1]). It leads to the following question. Open Problem 17 By analogy with [33, Theorem 1], the effective operator ϒτ looks like the square of a Dirac operator with a twisted spin connection. Can its meaning be clarified? % $ To conclude this paragraph, let us say a few words about the strategy used to prove Theorem 9. As the spectrum of Dη (m) is symmetric with respect to the origin 2  (see (ii)—Theorem 8), one can focus on the spectrum of the square Dη (m) . This 2  is done by using a variational characterization of theeigenvalues of Dη (m) via  the min-max principle. Namely, for u ∈ D(qm ) := D Dη (m) , there holds: qm (u) := Dη (m)u, Dη (m)uL2 (R3 ,C4 )   = |∇u|2 dx + m2 |u|2 dx R3 \'

+

2m τ

R3

 '

|u+ − u− |2 ds +

 '

 M|u+ |2 ds −

'

M|u− |2 ds,

where we have used the identification L2 (R3 , C4 )  u = u+ ⊕ u− ∈ L2 (*+ , C4 ) ⊕ L2 (*− , C4 ). Remark that away from the shell ', this is the quadratic form of a shifted Laplacian and that the δ-shell interaction manifests via the boundary terms.

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Then, the strategy is rather standard, though technically involved because functions in the form domain D(qm ) satisfy a specific transmission condition. First, for δ > 0 sufficiently small one constructs a tubular neighborhood of the shell ' as *δ := {x' − tn(x' ) : x' ∈ ', t ∈ (−δ, δ)}. Second, using Dirichlet and Neumann bracketing techniques, the unitary map U : L2 (*δ ) → L2 (' × (−δ, δ)),        (U u) x' − tn(x' ) := det G(x' , t) u x' − tn(x' ) := v(x' , t), where G is the metric on ' × (−δ, δ) induced by the change of variable in tubular coordinates and remarking that the volume form on ' × (−δ, δ) satisfies 

    det G(x' , t) dx' dt = 1 − 2tM(x' ) + t 2 K(x' ) dsdt,

one can bound from above and below the quadratic form q˜m (v) := qm (U −1 v) by ± defined on a form domain included in U D(q ). It reads quadratic forms qm m − + (v) ≤ q˜m (v) ≤ qm (v). qm

Hence, for the sequence of min-max levels associated to each quadratic form one gets (see [26, Lemma 4.10]): − + Ek (qm ) ≤ Ek (qm ) ≤ Ek (qm ),

k ∈ N.

+ and a The proof then relies on the obtention of a suitable upper bound for qm − suitable lower bound for qm . Let us focus on the simplest case, the upper bound. For some constant c > 0, we have:    + qm (v) := (1 + cδ)∇x' vTx '⊗C4 + (K − M 2 + cδ)|v|2 dsdt '

'×(−δ,δ)

+

  '

δ −δ

|∂t v|2 dt +

 2m |v(·, 0+ ) − v(·, 0− )|2 ds, τ

(27)

where v ∈ H 1 (' × (−δ, δ), C4 ), satisfies a transmission at the shell  condition  ', inherited from the one for functions in the domain D Dτ (m) and a Dirichlet boundary conditions at {(x' , ±δ) : x' ∈ '}. The lower bound is more involved because one has to control commutators of the surface gradient ∇x' with the projections on the eigenspace of the lowest eigenvalue of a transverse operator in the variable t and its orthogonal, see e.g. [26, Section 4.5] for details.

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Acknowledgements While writing this survey, T. O.-B. was supported by the ANR “Défi des autres savoirs (DS10) 2017” programm, reference ANR-17-CE29-0004, project molQED. F. P. is supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (grant agreement MDFT No 725528 of Mathieu Lewin).

References 1. Albeverio, S., Gesztesy, F., Hoegh-Krohn, R., Holden, H.: Solvable Models in Quantum Mechanics. Springer Science & Business Media, New York (2012) 2. Arrizabalaga, N., Mas, A., Vega, L.: Shell interactions for Dirac operators. J. Math. Pures App. 102, 617–639 (2014) 3. Arrizabalaga, N., Mas, A., Vega, L.: Shell interactions for Dirac operators: on the point spectrum and the confinement. SIAM J. Math. Anal. 47(2), 1044–1069 (2015) 4. Arrizabalaga, N., Le Treust, L., Raymond, N.: On the MIT bag model in the non-relativistic limit. Comm. Math. Phys. 354(2), 641–669 (2017) 5. Behrndt, J., Langer, M.: Boundary value problems for elliptic partial differential operators on bounded domains. J. Funct. Anal. 243(2), 536–565 (2007) 6. Behrndt, J., Krejˇciˇrík, D.: An indefinite Laplacian on a rectangle. J. Anal. Math. 134, 501–522 (2018) 7. Behrndt, J., Holzmann, M.: On Dirac operators with electrostatic δ-shell interactions of critical strength. J. Spectr. Theory 10, 147–184 (2020) 8. Behrndt, J., Exner, P., Lotoreichik, V.: Essential spectrum of Schrödinger operators with δinteractions on the union of compact Lipschitz hypersurface. Proc. Appl. Math. Mech., 523– 524 (2013) 9. Behrndt, J., Exner, P., Lotoreichik, V.: Schrödinger operators with δ and δ  interactions on Lipschitz surfaces and chromatic numbers of associated partitions. Rev. Math. Phys. 26, 1450015 (43pp) (2014) 10. Behrndt, J., Exner, P., Holzmann, M., Lotoreichik, V.: Approximation of Schrödinger operators with δ-interactions supported on hypersurfaces. Mathematische Nachrichten 290(8–9), 1215– 1248 (2017) 11. Behrndt, J., Exner, P., Holzmann, M., Lotoreichik, V.: On the spectral properties of Dirac operators with electrostatic δ-shell interactions. J. Math. Pures App. 111, 47–78 (2018) 12. Behrndt, J., Exner, P., Holzmann, M., Lotoreichik, V.: On Dirac operators in R3 with electrostatic and Lorentz scalar δ-shell interactions. Quantum Stud. Math. Found. 6, 295–314 (2019) 13. Bogolioubov, P.: Sur un modèle à quarks quasi-indépendants. Ann. I.H.P. Sec. A 8, 163–189 (1968) 14. Bonnet-Ben Dhia, A.-S., Dauge, M., Ramdani, K.: Analyse spectrale et singularités d’un problème de transmission non-coercive. C.R. Acad. Sci. Paris 328, 717–720 (1999) ˇ 15. Brasche, J.F., Exner, P., Kuperin, Yu.A., Seba, P.: Schrödinger operators with singular interactions. J. Math. Anal. Appl. 184(1), 112–139 (1994) 16. Brummelhuis, R., Duclos, P.: Effective Hamiltonians for atoms in very strong magnetic fields. J. Math. Phys. 47(3), 032103 (2006) 17. Brüning, J., Geyler, V., Pankrashkin, K.: Spectra of self-adjoint extensions and applications to solvable Schrödinger operators. Rev. Math. Phys. 20, 1–70 (2008) 18. Cacciapuoti, C., Pankrashkin, K. Posilicano, A.: Self-adjoint indefinite Laplacians. JAMA 139, 155–177 (2019). https://doi.org/10.1007/s11854-019-0057-z 19. Chodos, A.: Field-theoretic Lagrangian with baglike solutions. Phys. Rev. D (3) 12(8), 2397– 2406 (1975)

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20. Chodos, A., Jaffe, R.L., Johnson, K., Thorn, C.B.: Baryon structure in the bag theory. Phys. Rev. D 10, 2599–2604 (1974) 21. Chodos, A., Jaffe, R.L., Johnson, K., Thorn, C.B., Weisskopf, V.F.: New extended model of hadrons. Phys. Rev. D (3) 9(12), 3471–3495 (1974) ˇ 22. Dittrich, J., Exner, P., Seba, P.: Dirac operators with a spherically symmetric δ-shell interaction. J. Math. Phys. 30, 2875 (1989) 23. Duchêne, V., Raymond, N.: Spectral asymptotics of a broken δ-interaction. J. Phys. A. 47, 2014 (15) 24. Exner, P., Kovaˇrík, H.: Quantum Waveguides. Springer International, Heidelberg (2015) 25. Figotin, A., Kuchment, P.: Band-gap structure of spectra of periodic dielectric and acoustic media. II: Two-dimensional photonic crystals. SIAM J. Appl. Math. 56, 1561–1620 (1997) 26. Holzmann, M., Ourmières-Bonafos, T., Pankrashkin, K.: Dirac operators with Lorentz scalar interactions. Rev. Math. Phys. 30, 1850013 (2018) 27. Johnson, K.: The MIT bag model. Acta Phys. Pol. B(6), 865–892 (1975) 28. Kato, T.: Perturbation Theory for Linear Operators. Reprint of the Corr. Print. of the 2nd ed. 1980. Springer, Berlin (1995) 29. Klein, O.: Die Reflexion von Elektronen an einem Potentialsprung nach der relativistischen Dynamik von Dirac. Zeitschrift für Physik 53(3–4), 157–165 (1929) 30. McLean, W.: Strongly Elliptic Systems and Boundary Integral Equations. Cambridge University Press, Cambridge (2000) 31. Mas, A., Pizzichillo, F.: The relativistic spherical δ-shell interaction in R3 : spectrum and approximation. J. Math. Phys. 58 (2017). https://doi.org/10.1063/1.5000381 32. Mas, A., Pizzichillo, F.: Klein’s paradox and the relativistic δ-shell interaction in R3 . Analysis & PDE 11(3), 705–744 (2018) 33. Moroianu, A., Ourmières-Bonafos, T., Pankrashkin, K.: Dirac operators on hypersurfaces as large mass limits. Commun. Math. Phys. 374, 1963–2013 (2020) 34. Ourmières-Bonafos, T., Vega, L.: A strategy for self-adjointness of Dirac operators: applications to the MIT bag model and δ-shell interactions. Publications Matemàtiques 62(2) (2018). arXiv:1612.07058 35. Pizzichillo, F., Van Den Bosch, H.: Self-adjointness of two dimensional Dirac operators on corner domains (2019). arXiv:1902.05010 36. Reed, M., Simon, B.: Methods of Mathematical Physics. Vol. II. Functional Analysis. Academic Press, New York (1975) 37. Šeba, P.: Klein’s paradox and the relativistic point interaction. Lett. Math. Phys. 18, 705–744 (1989) 38. Thaller, B.: The Dirac Equation. Springer, Heidelberg (1992) 39. Thaller, B.: Advanced Visual Quantum Mechanics. Springer Science & Business Media, New York (2005)

Ultraviolet Properties of a Polaron Model with Point Interactions and a Number Cutoff Jonas Lampart

Abstract We discuss a model in which a nonrelativistic particle can absorb and emit bosonic particles on contact. The bosons have a constant dispersion relation, as in the related Fröhlich polaron model. We determine explicitly the domain of the Hamiltonian for finitely many bosons in terms of singular boundary conditions. The singularities occurring in this model are essentially the same as in the model with quadratic boson dispersion, with simplifications in the formulas highlighting their key features. Keywords Boundary conditions · Point interaction · Singular perturbation · Renormalisation · Polaron

1 Introduction Polaron models, in which nonrelativistic particles can absorb and emit bosons, are commonly used in physics [7]. For example, they can serve as an effective model for the interactions of electrons with phonons in a solid. Such models have also been studied in depth in mathematical physics. Apart from their direct applications in physics, they also provide a framework in which characteristic features of quantum field theories can be rigorously understood. For example, ultraviolet [5, 11–14] and infrared singularities [6, 9], spectral and scattering theory [1, 18, 19] and effective or renormalised parameters, such as the polaron mass [8, 17] are all topics of current research (see these works for a more complete picture of the literature). Since the bosons are quasi-particles, representing elementary excitations out of some equilibrium, a large class of dispersion relations and interactions can be relevant—depending on the properties of the equilibrium. In particular, the case in

J. Lampart () CNRS & Laboratoire Interdisciplinaire Carnot de Bourgogne, UMR 6303 CNRS & Université Bourgogne Franche-Comté, Dijon, France e-mail: [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. Michelangeli (ed.), Mathematical Challenges of Zero-Range Physics, Springer INdAM Series 42, https://doi.org/10.1007/978-3-030-60453-0_6

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which the bosons have a constant dispersion relation is of interest. This means that the free bosons would not propagate at all, but have non-trivial dynamics only in the interacting model. The Hamiltonian for such a model, in the case of one particle in three dimensions, is formally given by H = −Δx + N + a(vx ) + a ∗ (vx ),

(6.1)

where N = dΓ (1) is the boson-number operator, v is a distribution, x denotes the variable in the configuration space of the particle, and vx (y) = v(y − x). A proper definition of the Hamiltonian should yield an unbounded self-adjoint operator (H, D(H )) on the Hilbert space H = L2 (R3 ) ⊗ Γsym (L2 (R3 )) =

∞ 0

  L2 R3 , L2sym ((R3 )n ) ,

n=0

and clarify its relation to the formal expression (6.1). In the well-known Fröhlich model, the interaction is given by v(k) ˆ = |k|−1 . In this case, the expression for H is not well defined as an operator, since v ∈ / L2 (R 3 ), but it is defined as a quadratic form (see e.g. [10]). The domain of the operator associated to this quadratic form can be described using generalised boundary conditions, as shown in [14]. In this note we will discuss the construction of a self-adjoint operator H in a more singular variant of this model, where the interaction is a point interaction, i.e. v = δ. For this model, the formal expression (6.1) for H makes sense neither as an operator nor a quadratic form, and no rigorous definition of a self-adjoint Hamiltonian seems to be known to date. A similar model, in which the bosons have a nonrelativistic dispersion ω(k) = k 2 + 1, was recently defined in [12] using similar boundary conditions as in [14]. This construction can also be applied to the BogoliubovFröhlich Hamiltonian modelling the interaction of an impurity with excitations of a Bose-Einstein condensate [13]. Here we will use the same techniques to define the Hamiltonian for bosons with a constant dispersion. Part of the purpose of this discussion is to illustrate the singularities displayed by elements of the domain of H . The constant dispersion relation of the bosons simplifies many calculations and allows for a somewhat more explicit description of the singularities as compared to [12, 13]. However, it gives less control over the growth of certain quantities with increasing boson-number, so we will restrict ourselves to the model with at most Nmax bosons. In fact, we will focus mainly on the case with at most Nmax = 2 bosons, which is the simplest case that displays the same singularity structure as the full problem, and then indicate how the results are obtained for arbitrary Nmax < ∞. As in the recent works [12, 14, 15] our approach is to define the Hamiltonian of our model using special boundary conditions, called interior-boundary conditions, that relate sectors of the Hilbert space with different numbers of bosons.

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2 Extension of the Free Operator and Boundary Conditions In order to gain the ability to impose boundary conditions, we consider an extension of the free kinetic energy operator L = −Δx . The domain of this operator, as an operator on H , is explicitly given by D(L) =

∞ 0

H 2 (R3 , L2sym ((R3 )n )).

n=0

Elements of this domain are continuous functions of the first variable, x, and may thus be evaluated at x = yn , for appropriate n. We take L0 to be the restriction of this operator to the kernel of the annihilation operator a(δx ), that is, using the symmetry in the y-variables, D(L0 ) = {ψ ∈ D(L) : ψ (n) (x, y1 , . . . , yn )|x=yn = 0}. The extension of L we are interested in is the adjoint L∗0 , which will allow for boundary conditions on the sets {x = yj }. The domain of L∗0 can be parametrised (see Proposition 1) using the map Gμ ψ = −(L + μ2 )−1 a ∗ (δx )ψ. Explicitly, we have  Gμ ψ

(n−1)



1  (x, Y ) = − √ (−Δx + μ2 )−1 δ(yj − x)ψ (n−1) (x, Yˆj ) n n

j =1

1 = √ n

n 

fμ (x − yj )ψ (n−1) (yj , Yˆj ),

j =1

where fμ (x) = −(−Δx + μ2 )−1 δ(x) = −

e−μ|x| , 4π|x|

and Yˆj ∈ R3(n−1) denotes the vector formed by the y1 , . . . , yn without yj . From this explicit formula one easily deduces the important mapping properties of G. Lemma 1 For any positive n ∈ N, μ > 0 and 0 ≤ s < 1/2 the operator Gμ is bounded from H (n−1) to H s (R3 , L2sym ((R3 )n )).

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Proof The statement follows immediately from the fact that (−Δx + μ2 )−1+s δ ∈ L2 (R3 ) for 0 ≤ s < 1/2. Note,√ however, that the estimate for the norm of Gμ : H (n−1) → H (n) will grow like n/(2μ). We then have a characterisation of D(L∗0 ), by standard arguments from the theory of abstract boundary value problems (see e.g. [2]). We only sketch the proof since, for the construction of the operator H , we do not really need this characterisation and it is sufficient to work on a subset suitably parametrised by Gμ . Proposition 1 For any positive n ∈ N and μ > 0 we have D(L∗0 ) ∩ H (n) = (D(L) ∩ H (n) ) ⊕ Gμ H (n−1) , i.e., for every ψ ∈ D(L∗0 ) there is a unique ϕμ(n−1) ∈ H Gμ ϕμ(n−1) is an element of H 2 (R3 , L2 (R3n )).

(n−1)

so that ψ (n) −

Proof (sketch) First observe that this characterisation is independent of μ, since fμ (x) − fν (x) ∈ H 2 (R3 ). Since L is a positive self-adjoint extension of L0 , we have D(L∗0 ) = D(L0 ) ⊕ ker(L∗0 + i) ⊕ ker(L∗0 − i) = D(L) ⊕ ker(L∗0 + μ2 ) for μ > 0. Now let ϕ (n) ∈ D(L0 ) ∩ H (n) and ψ (n−1) ∈ H (n−1) , then Gμ ψ (n−1) , (L0 + μ2 )ϕ (n) H (n) = ψ (n−1) , G∗μ (L + μ2 )ϕ (n) H (n) 1  (n−1) = −√ ψ (yj , Yˆj ), ϕ (n) |x=yj H (n−1) n n

j =1

= 0. Thus Gμ ψ (n−1) is in the kernel of L∗0 + μ2 . The point is now to show that we have equality, ker(L∗0 +μ2 ) = Gμ H (n−1) . To achieve this, first observe that, by the same logic as above, for any ξ ∈ ker(L∗0 + μ2 ) ∩ H (n) , (L + μ2 )ξ annihilates D(L0 ) in the pairing of D(L−1 ) × D(L). The domain D(L0 ) is exactly the kernel of a(δx ) in D(L), so its annihilator is the closure of the range of a ∗ (δx ) : H (n−1) → D(L−1 ) (cf. [3, Chap. 2.7]), i.e. ξ = (L2 + μ2 )−1 η for some η ∈ ran a ∗ (δx ). To complete the argument, one shows that the range of a ∗ (δx ) is closed in D(L−1 ) by proving that Gμ ψ ≥ Cψ for some C > 0. We will not go into this technical point here. An argument, for a slightly different situation, that can be adapted to our case is given in [4, Lem. B2].

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2.1 Boundary Values On D(L∗0 ) we can define a boundary value operator that extracts the singular part of a function by   √ Bψ (n+1) (x, Y ) = −4π n + 1 lim |x − yn+1 |ψ (n+1) (yn+1 , Y, x). yn+1 →x

(6.2)

Clearly, we have Bψ = 0 for ψ ∈ D(L) and BGψ = ψ. This shows that B is a well-defined operator from D(L∗0 ) ∩ H (n+1) to H (n) , by the characterisation of the domain in Proposition 1. The other relevant boundary value in our problem is the annihilation operator, i.e. the evaluation at x = yn+1 . Since functions in the range of Gμ diverge at x = yn+1 , this is naturally defined on D(L), but not D(L∗0 ). We thus need to find an appropriate extension A of this operator to D(L∗0 ). As in [12, 15] this extension to singular functions is obtained by considering the expansion near the set {x = yn+1 }, fμ (x − yn+1 ) = −

μ 1 + + O(|x − yn+1 |), 4π|x − yn+1 | 4π

and taking only the value of the constant part. That is, the extended annihilation operator A acts on Gμ ψ (n−1) as  μ (n) ψ (x, Y ) + (x, Y ) = fμ (x − yj )ψ (n) (yj , Yˆj , x). 4π n

AGμ ψ

(n)

(6.3)

j =1

The action of A is equivalently given by the formula     √ 1 (n+1) (n+1) (n+1) (Bψ Aψ (x + r, Y, x) + )(x, Y ) . (x, Y ) = n + 1 lim ψ r→0 4πr (6.4) One easily sees that this formula yields the action of a(δx ) for ψ ∈ D(L) and (6.3) on the range of Gμ . The extension A of a(δx ) given by (6.4) is clearly local, in the sense that the value of Aψ (n+1) (x, Y ) depends only on the values of ψ (n+1) in a small neighbourhood of the point (x, Y, x) ∈ R3(n+2) , on the “boundary” {x = yn+1 }.

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2.2 The Hamiltonian for Nmax = 1 We will now give a short exposition of the construction of the operator H1 with Nmax = 1. This is considerably easier than the other cases, and very similar models were already discussed by Lévy-Leblond [16] and Thomas [20]. Let H1 be the operator given by

(H1 ψ)(n)

⎧ 0 ⎪ ⎪ ⎨ = L∗0 ψ (1) ⎪ ⎪ ⎩ − Δψ (0) + Aψ (1)

n>1 n=1 n=0

with the domain D(H1 ) = {ψ ∈ H : ψ (0) ∈ H 2 (R3 ), ψ (1) ∈ D(L∗0 ), Bψ (1) = ψ (0) }. Note that the boundary condition Bψ (1) = ψ (0) can equivalently be written as ψ (1) − Gμ ψ (0) ∈ D(L), by Proposition 1. Proposition 2 The operator (H1 , D(H1 )) is self-adjoint and non-negative. (0) (0) Proof Let G(0) = Gμ ψ (0) , μ : H → H be the operator given by Gμ ψ (0) (n) (0) (0) Gμ ψ = 0, n > 0. Since Gμ ψ is in the kernel of L∗0 + μ2 , we have (0) 2 (1) L∗0 ψ (1) = (L + μ2 )(ψ (1) − G(0) μ ψ )−μ ψ .

(6.5)

∗ 2 Using that (G(0) μ ) (L + μ ) = −a(δx ) = −A on D(L), we also obtain ∗ 2 (1) (0) (1) (0) (1) −(G(0) − G(0) − G(0) − μ ) (L + μ )(ψ μ ψ ) = A(ψ μ ψ ) = Aψ

μ (0) ψ . 4π

Inserting these identities for L∗0 ψ (1) and Aψ (1) in the definition of H1 , we obtain the formula ∗ 2 (0) 2 H1 ψ = (1 − G(0) μ ) (L + μ )(1 − Gμ )ψ − μ ψ +

μ (0) ψ . 4π

(0) (0) 2 Because (G(0) μ ) = 0, the operator 1 − Gμ is invertible (with inverse 1 + Gμ ). (0) ∗ 2 Consequently, (1 − G(0) μ ) (L + μ )(1 − Gμ ), and thus also H1 , are self-adjoint. It 2 is clearly bounded from below by −μ , for arbitrary μ > 0, so it is non-negative.

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139

2.3 The Hamiltonians for Nmax > 1 Having understood the Hamiltonian for Nmax = 1, there is an obvious generalisation to Nmax > 1. However, this turns out not to give a well-defined operator, due to additional singularities on the sets where the positions of more than one boson and the particle coincide. To see this, let us consider the case Nmax = 2. The obvious guess would be to set H ψ (n) = L∗0 ψ (n) + Aψ (n+1) for n = 0, 1 and H ψ (2) = L∗0 ψ (2) , as above, on the domain where Bψ (n+1) = ψ (n) for n = 0, 1. If ψ (0) #= 0, this boundary condition implies that ψ (1) has the singularity 1/|x − y|. More precisely, it can be written as ψ (1) = φ (1) + Gμ ψ (0) = φ (1)(x, y) + fμ (x − y)ψ (0) (y) with a regular function φ (1) ∈ H 2 (R3 , L2 (R3 )). The same condition also relates the functions ψ (2) and ψ (1) , and we have that ψ (2) = φ (2) + Gμ ψ (1) = φ (2) + Gμ φ (1) + G2μ ψ (0) with φ (2) ∈ H 2 (R3 , L2 (R6 )). Now, spelling out the last term,   G2μ ψ (0) (x, y1 , y2 )  1  = √ fμ (x − y2 )fμ (y1 − y2 )ψ (0) (y1 ) + fμ (x − y1 )fμ (y1 − y2 )ψ (0) (y2 ) , 2 1 near the set {x = y1 = y2 } we see that ψ (2) has a singularity like |x−y1 ||x−y 2| (0) whenever ψ #= 0. This singularity is square-integrable, of course, so it does not pose a problem for the satisfiability of he boundary conditions. However, if we apply the extended annihilation operator A using (6.3) to such a term, we obtain

μ Gμ ψ (0) (x, y) + fμ (x − y)Gμ ψ (0) (y, x) 4π μ fμ (x − y)ψ (0) (y) + fμ2 (x − y)ψ (0) (x). = 4π

AG2μ ψ (0) (x, y) =

The last term here has the singularity 1/|x − y|2 which is not square-integrable. Hence, our tentative operator does not map its domain into the Hilbert space H , and our guess cannot be correct. Note that this ansatz does not even define a quadratic form, since fμ3 is not integrable. Another way to look at this is that the operator A, together with the boundary condition, introduces (up to a permutation of arguments) an interaction fμ (x − y) between the bosons and the particle. Such a potential does not map the extended

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domain D(L∗0 ) to H , so L∗0 + A is not an operator from D(L∗0 ) to H , either. However, multiplication by fμ (x − y) does map the free domain D(L) to H and is infinitesimally bounded by L, so we will be able to address the problem by grouping these interactions together with the free operator in a certain way. Let Tμ = AGμ be the operator given by (6.3). It is infinitesimally L-bounded, so the operator Kμ := L + Tμ is self-adjoint on D(L), and for μ sufficiently large (depending on n), Kμ is invertible on H (n) . The construction now proceeds in a similar way to the case Nmax = 1, but replacing the free operator L by Kμ . We first restrict this operator to the functions in its domain for which ψ (n) (x, Y )|x=yj = 0 for any j = 1, . . . , n, defining (Kμ )0 := Kμ |D(L0 ) . We then pass to the extension (Kμ )∗0 , and parametrise elements of its domain using the following modification of Gμ for n < Nmax ;μ ψ (n) = −(Kμ + μ2 )−1 a ∗ (δx )ψ (n) , G where μ is chosen large enough for Kμ + μ2 to be invertible on H (n+1) for all n < Nmax . ;μ to Gμ , and many important Since K is a perturbation of L we can relate G properties carry over to the modified operator. By the resolvent formula we have ;μ ψ (n) = −(Kμ + μ2 )−1 a ∗ (δx )ψ (n) G ;μ ψ (n) . = Gμ ψ (n) − (−Δx + μ2 )−1 Tμ G

(6.6)

;μ for From this and Lemma 1 we immediately obtain the boundedness of G sufficiently large μ. Lemma 2 For any positive n ∈ N, μ > 0 large enough and 0 ≤ s < 1/2 the ;μ is bounded from H (n−1) to H s (R3 , L2sym ((R3 )n )). operator G We also have a characterisation of D((Kμ )∗0 ), which follows from the same arguments as for L∗0 and the equivalence of norms on D(K) = D(L) and their duals. Proposition 3 For any positive n ∈ N and μ > 0 large enough we have D((Kμ )∗0 ) ∩ H

(n)

;μ H (n−1) , = (D(L) ∩ H (n) ) ⊕ G

Ultraviolet Properties of a Polaron Model with Point Interactions and a Number Cutoff

i.e., for every ψ ∈ D((Kμ )∗0 ) there is a unique ϕμ ;μ ϕμ(n−1) is an element of H 2 (R3 , L2 (R3n )). G

(n−1)

141

∈ H (n−1) so that ψ (n) −

;μ have different singularities than those in the However, functions in the range of G ∗ range of Gμ , and the domains of L0 and K0∗ are different. We will now analyse these singularities in more detail and define the analogues of the boundary value operators ;μ . A and B on the range of G The operator (−Δx +μ2 )−1 Tμ in (6.6) is regularising, so the principal singularity is still given by the first term fμ (x − yn+1 )ψ (n) (y). In fact, one easily sees that it maps H (n) into H 1 (R3 , L2 (R3(n+1))) and that consequently B(−Δx + μ2 )−1 Tμ is well defined and equal to zero. Hence the boundary operator B, given by the ;μ and B G ;μ ψ (n) = ψ (n) , as for Gμ . expression (6.2), is defined on the range of G The same is not true, however, for the extension of the annihilation operator A. ;μ ψ, Eq. (6.6), is not sufficiently regular to be evaluated at The second term in G x = yn+1 , where it has a logarithmic singularity, as we will now see. The analogue ;μ ψ near of A is then defined by neglecting the divergent terms in the expansion of G x = y, as was done for A. More precisely, in the case n = 0, the singular behaviour ;μ ψ (0) (x, y) at |x − y| = r = 0 is given by of G ;μ ψ (0) (x, y) = Gμ ψ (0) (x, y) − (−Δx + μ2 )−1 G = g(r)ψ (0) +

 μ ;μ ψ (0) (x, y) + fμ (x − y) G 4π

μ ψ + Sψ (0) + o(1), 4π

where g(r) = −

1 log r + , 4πr 16π 2

μ the constant 4π corresponds of course to Tμ on H (0) and S is a bounded operator from H  (R3 ) to H (0) for any  > 0. The logarithmic divergence in g(r) originates from the term

(−Δx + μ2 )−1 fμ (x − y)2 ψ (0) (x) ;μ ψ (0) after using the resolvent formula as above. We will give a that appears in G proof of this asymptotic behaviour later, but we may already observe, by scaling, that this function should behave like a homogeneous function of degree zero for small values of |x − y|. ; of a(δx ) to D((Kμ )∗ ) in an analogous way to A, We now define the extension A 0 Eq. (6.4),     √ ; (n+1) (x, Y ) = n + 1 lim ψ (n+1) (x + r, Y, x) − g(r)(Bψ (n+1) )(x, Y ) . Aψ r→0

(6.7)

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As with A, this is a local boundary value operator that restricts to a(δx ) on D(L). Its important properties as an operator on the range of Gμ are as follows. ;μ to Proposition 4 The expression (6.7) defines a map from the range of G ;G ;μ − Tμ =: Sμ defines a symmetric operator on the H −1 (R3 , L2 (R3n )) and A domain D(S) = H  (R3 , L2sym ((R3 )n )) for any  > 0. We will postpone the proof of this proposition, and first explain how this allows us to define the operator HNmax and prove its self-adjointness. We set

 (n) HNmax ψ

⎧ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ L∗0 ψ (n) ⎪ ⎪ ⎨ = (Kμ )∗0 ψ (n) + Aψ (n+1) − Tμ ψ (n) ⎪ ⎪ ⎪ ⎪ (K )∗ ψ (n) + Aψ ; (n+1) − Tμ ψ (n) ⎪ μ 0 ⎪ ⎪ ⎪ ⎪ ⎩ ; (1) Lψ (0) + Aψ

n > Nmax n = Nmax n = Nmax − 1 0 < n < Nmax − 1 n = 0.

on the domain ⎧ ⎪ ⎪ ⎪ ⎨

! (n) ! ψ ∈ D(L∗ ) and Bψ (n) = ψ (n−1) 0 ! ! ! (n) D(HNmax ) = ψ ∈ H ! ψ ∈ D((Kμ )∗0 ) and Bψ (n) = ψ (n−1) ⎪ ! ⎪ ⎪ ! (0) ⎩ ! ψ ∈ D(L)

for n = Nmax for 0 < n < Nmax

⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭

.

The condition Bψ (n) = ψ (n−1) on D((Kμ )∗0 ) is of course equivalent to ψ (n) − ;μ ψ (n−1) ∈ D(L), as in the case Nmax = 1. In this definition μ must be taken G large enough for Lemma 2 and Proposition 3 to hold but is otherwise arbitrary. On easily checks that HNmax and its domain are independent of μ. Theorem 1 For any positive Nmax , the operator Hmax is self-adjoint and bounded from below. Proof To keep the notation simple, we will focus on the case Nmax = 2. The general case is a straightforward generalisation. The operator for Nmax = 2 reads

(H2 ψ)(n) =

⎧ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ L∗0 ψ (n) ⎪ ⎪ (Kμ )∗0 ψ (n) + Aψ (n+1) − Tμ ψ (n) ⎪ ⎪ ⎪ ⎪ ⎩ ; (1) Lψ (0) + Aψ

n>2 n=2 n=1 n = 0,

;μ ψ (0) and ψ (0) be and the conditions on D(H2 ) are that ψ (2) − Gμ ψ (1) , ψ (1) − G ;μ is contained in the kernel of (Kμ )∗ + μ2 on the in D(L) = D(K). The range of G 0

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143

sector with than n = 1 bosons, and in the kernel of L∗0 + μ for n = 2 bosons. We thus have for ψ ∈ D(H2 ) ;μ ψ (0) ) − μ2 ψ (0) , (Kμ )∗0 ψ (1) = (L + Tμ + μ2 )(ψ (1) − G and the analogue with only L, as in Eq. (6.5), on the sector with n = 2 bosons. We also have ; (1) = A(ψ ; (1) − G ;μ ψ (0) ) + A ;G ;μ ψ (0) = a(δx )(ψ (1) − G ;μ ψ (0) ) + (Tμ + Sμ )ψ (0) Aψ Aψ (2) = a(δx )(ψ (2) − Gμ ψ (1) ) + Tμ ψ (1)

When we insert these identities into the definition of H2 , the terms ±Tμ ψ (1) will cancel each other. We thus have ∗ (1) 2 (0) 2 ;(1) ;(1) H2 = (1 − G μ ) (L + Tμ + μ )(1 − Gμ ) + Sμ − μ

(6.8)

;(1) on the sectors with at most Nmax = 2 bosons. Here, G μ is given by

(n) ;(1) G μ ψ

⎧ Gμ ψ (1) n=1 ⎪ ⎪ ⎨ ;μ ψ (0) = G n=0 ⎪ ⎪ ⎩ 0 otherwise,

the operator Tμ(1) acts as Tμ on the sectors with at most one boson, Sμ(0) acts as Sμ on H (0), and zero on all other sectors. ;(1) ;(1) The operator G μ is nilpotent and thus 1 − G μ is invertible. Consequently, the first term in Eq. (6.8) is a self-adjoint operator and bounded from below. The operator S (0) is relatively bounded w.r.t. this operator by Proposition 4. In fact, by Eq. (6.12) below, S (0) is essentially a Fourier multiplier of logarithmic growth. Since ∗ (0) ;(1) ;(1) it acts non-trivially only on H (0) , we have Sμ(0) = (1 − G μ ) Sμ (1 − G μ ) and the relative bound is obvious. This completes the proof for Nmax = 2. For Nmax > 2 one needs to use in the final step the fact that, by Proposition 4 ;μ is a bounded operator. and Lemma 1, Sμ G We finally come to the proof of Proposition 4. Proof (of Proposition 4) From Eq. (6.6) we can immediately conclude that the sum of the first term in (6.6) and the first term in g(x − yn+1 ) has a limit, and this acts in the same way as AGμ = Tμ , mapping H (n) to H −1 (R3 , L2 (R3(n−1) )). We then need to analyse the second term in (6.6), i.e. the negative of ;μ ψ (0) . (−Δx + μ2 )−1 Tμ G

(6.9)

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We give the details of this analysis only in the case n = 0 and comment on the adjustments for the general case in the end. The analysis for n = 0 is sufficient for the construction of the model with Nmax = 2. By the regularising properties of (−Δx + μ2 )−1 , the difference of (6.9) with (−Δx + μ2 )−1 fμ (x − y)Gμ ψ (0) (y, x) = (−Δx + μ2 )−1 fμ2 (x − y)ψ (0) (x) (6.10) is an element of H 2 (R3 , L2 (R3 )), and can thus be evaluated at x = y, yielding a bounded operator on H (0) that will be absorbed into Sμ . We now calculate the asymptotics of (6.10) near x = y using the Fourier representation. One can explicitly calculate the Fourier transform of fμ2 , 2 f, μ (q) =

1 1 arctan(|q|/2μ), 3/2 4π(2π) |q|

and thus the Fourier transform of (6.10) is 1 1 arctan(|q|/2μ)ψˆ (0)(p + q). 3/2 2 4π(2π) (p + μ)|q|

(6.11)

Taking the inverse transform then leads to 1 (6.10) = 4π(2π)9/2 1 = 4π(2π)9/2

 dpdq

eipx+iqy arctan(|q|/2μ)ψˆ (0)(p + q) (p2 + μ2 )|q|

dkdq

eikx+iq(y−x) arctan(|q|/2μ)ψˆ (0)(k). ((q − k)2 + μ2 )|q|



Setting x − y = 0 we would (formally) have the action of a Fourier multiplier on ψ (0) , but this makes no sense since the q-integral does not converge. We must thus show that the difference of this expression and the logarithmic term in g(x − y) has a limit, and this will act as an appropriate Fourier multiplier on ψ (0) . As the difficulty stems from the insufficient decay of the integrand at q → ∞, we  −1 may replace arctan(|q|/2μ) by its limit π/2. We also replace (q − k)2 + μ2   2 −1 by q + k 2 + μ2 . The errors arising from both replacements give absolutely convergent integrals that evaluate to bounded Fourier multipliers. It then remains to determine the singular behaviour of   ∞ 1 1 sin(tr) 1 eiq(x−y) dt , = dq r t 2 + k 2 + μ2 8(2π)3 (q 2 + k 2 + μ2 )|q| 16π 2 0 as r → 0. Changing variables to s = tr, one reduces this to the calculation of 

∞

ds 0

s2

sin(s) = − log r − + r 2 (k 2 + μ2 )

1 2

log(k 2 + μ2 ) + O(1),

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where the remainder is convergent as r → 0 and uniformly bounded in k. Consequently,  Ssing (k) := =

lim

|x−y|→0

1 − 4π(2π)3



log |x − y| eiq(x−y) − dq 2 2 2 (q + k + μ )|q| 16π 2

log(k 2 + μ2 ) + O(1) 32π 2



(6.12)

defines a Fourier multiplier that gives rise to a symmetric operator on D(S), as claimed. This completes the argument for n = 0. For n ≥ 1, Tμ contains additional terms that, in the analogue of (6.10) lead to terms of the form (with Y ∈ R3(n+1)) (−Δx + μ2 )−1 fμ (x − yj )fμ (yj − yi )ψ (n) (yi , Yˆi,j , x), (−Δx + μ2 )−1 fμ2 (x − yi )ψ (n) (x, Yˆi ),

(6.13)

with j ≤ n + 1, i ≤ n and j #= i. These terms are less singular than (6.10) at x = yn+1 , since at least some of the singularities in fμ concern different directions. In fact, they can be evaluated at x = yn+1 and this evaluation defines a bounded operator on H (n) . We will give a short proof of this statement for the term (6.13), the argument for the other one is very similar. A detailed exposition of similar arguments may be found in [12, App. A]. Using the Fourier transform (6.11) the evaluation of (6.13) at x = yn+1 has the Fourier representation (now with Q ∈ R3n ) 1 2(2π)4

 dk

arctan(|qi |/2μ) ψˆ (n) (p − k + qi , Qˆ i , k). ((p − k)2 + μ2 )|qi |

This integral operator can be bounded by an argument similar to the well-known Schur test. We have for ϕ ∈ H (n) ! !  ! ! arctan(|qi |/2μ) ˆ (n) (p − k + qi , Qˆ i , k)! (6.14) ˆ Q) ψ dQdpdk !!ϕ(p, ! 2 2 ((p − k) + μ )|qi |   π |ϕ(p, ˆ Qˆ i , # − p)|2 |# − p|s−1 ≤ d Qˆ i dpdkd# 4 ((p − k)2 + μ2 ) |# − k|s ˆ − k, Qˆ i , k)|2 |# − k|s  |ψ(# . ((p − k)2 + μ2 ) |# − p|s+1

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Now for 1 < s < 2,  

((p

− k)2

dk ≤ C|# − p|1−s + μ2 )|# − k|s

((p

− k)2

dp ≤ C|# − k|−s , + μ2 )|# − p|s+1

for some constant C. This implies that (6.14) is bounded by C(ϕ2H (n) +ψ2H (n) ), which proves boundedness of the corresponding operator by standard arguments. The symmetry of Sμ is easy to check from the Fourier representation by changing variables and summing over the indices i, j .

References 1. Ammari, Z.: Asymptotic completeness for a renormalized nonrelativistic Hamiltonian in quantum field theory: the Nelson model. Math. Phys. Anal. Geom. 3(3), 217–285 (2000) 2. Behrndt, J., Micheler, T.: Elliptic differential operators on Lipschitz domains and abstract boundary value problems. J. Funct. Anal. 267(10), 3657–3709 (2014) 3. Brezis, H.: Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer, New York (2010) 4. Correggi, M., Dell’Antonio, G., Finco, D., Michelangeli, A., Teta, A.: A class of Hamiltonians for a three-particle fermionic system at unitarity. Math. Phys. Anal. Geom. 18(1), 32 (2015) 5. Deckert, D.A., Pizzo, A.: Ultraviolet properties of the spinless, one-particle Yukawa model. Commun. Math. Phys. 327(3), 887–920 (2014) 6. Derezi´nski, J., Gérard, C.: Scattering theory of infrared divergent Pauli-Fierz Hamiltonians. Ann Henri Poincaré 5(3), 523–577 (2004) 7. Devreese, J.T.: Polarons. In: Trigg, G.L., Immergut, E.H. (eds.) Encycl. Appl. Phys., vol. 14, pp 383–409 (1996) 8. Dybalski, W., Spohn, H.: Effective mass of the polaron—revisited. Preprint (2019). arXiv:190803432 9. Fröhlich, J.: Existence of dressed one electron states in a class of persistent models. Fortschr. Phys. 22(3), 159–198 (1974) 10. Griesemer, M., Wünsch, A.: Self-adjointness and domain of the Fröhlich Hamiltonian. J. Math. Phys. 57(2), 021,902 (2016) 11. Griesemer, M., Wünsch, A.: On the domain of the Nelson Hamiltonian. J. Math. Phys. 59(4), 042,111 (2018) 12. Lampart, J.: A nonrelativistic quantum field theory with point interactions in three dimensions. Ann. H Poincaré 20, 3509–3541 (2019a) 13. Lampart, J.: The renormalised Bogoliubov-Fröhlich Hamiltonian. Preprint (2019b). arXiv:190902430 14. Lampart, J., Schmidt, J.: On Nelson-type Hamiltonians and abstract boundary conditions. Commun. Math. Phys. 367(2), 629–663 (2019) 15. Lampart, J., Schmidt, J., Teufel, S., Tumulka, R.: Particle creation at a point source by means of interior-boundary conditions. Math. Phys. Anal. Geom. 21(2) (2018) 16. Lévy-Leblond, J.M.: Galilean quantum field theories and a ghostless Lee model. Commun. Math. Phys. 4(3), 157–176 (1967)

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17. Lieb, E.H., Yamazaki, K.: Ground-state energy and effective mass of the polaron. Phys. Rev. 111, 728–733 (1958). https://doi.org/10.1103/PhysRev.111.728. https://link.aps.org/doi/ 10.1103/PhysRev.111.728 18. Miyao, T.: Monotonicity of the polaron energy ii: General theory of operator monotonicity. J. Stat. Phys. 153(1), 70–92 (2013) 19. Møller, J.S.: The polaron revisited. Rev. Math. Phys. 18(05), 485–517 (2006) 20. Thomas, L.E.: Multiparticle Schrödinger Hamiltonians with point interactions. Phys. Rev. D 30, 1233–1237 (1984). https://doi.org/10.1103/PhysRevD.30.1233. https://link.aps.org/doi/10. 1103/PhysRevD.30.1233

Zero Modes and Low-Energy Resolvent Expansion for Three Dimensional Schrödinger Operators with Point Interactions Raffaele Scandone

Abstract We investigate the low-energy behavior of the resolvent of Schrödinger operators with finitely many point interactions in three dimensions. We also discuss the occurrence and the multiplicity of zero energy obstructions. Keywords Point interactions · Resolvent expansion · Schrödinger operators · Threshold obstructions Mathematical Subject Classification 2020: 35B25, 35B34, 35J10, 47A75, 81Q10.

1 Introduction and Main Results A central topic in quantum mechanics is the study of quantum systems subject to very short-range interactions, supported around a submanifold of the ambient space. A relevant situation occurs when the singular interaction is supported on a set of points in the Euclidean space Rd . This leads to consider, formally, operators of the form  −Δ+ μy δy (·), (1) y∈Y

where Y is a discrete subset of Rd , and μy , y ∈ Y , are real coupling constants. Heuristically, (1) can be interpreted as the Hamiltonian for a non-relativistic quantum particle interacting with “point obstacles” of strengths μy , located at y ∈ Y.

R. Scandone () GSSI - Gran Sasso Science Institute, L’Aquila, Italy e-mail: [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. Michelangeli (ed.), Mathematical Challenges of Zero-Range Physics, Springer INdAM Series 42, https://doi.org/10.1007/978-3-030-60453-0_7

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From a mathematical point of view, Schrödinger operators with point (delta-like) interactions have been intensively studied, since the first rigorous realization by Berezin and Faddeev [5], and subsequent characterizations by many other authors [3, 8, 16, 17, 23, 32] (see the surveys [2, 12], the monograph of Albeverio, Gesztesy, Høegh-Krohn, and Holden [4], and references therein for a thorough discussion). In this work we focus on the case of finitely many point interactions in three dimensions. Our aim is to provide a detailed spectral analysis at the bottom of the continuous spectrum, i.e. at zero energy. A similar analysis has been done in [6] for the two dimensional case, with application to the Lp -boundedness of the wave operators. We start by recalling some well-known facts on the construction and the main properties of Schrödinger operators with point interactions. We fix a natural number N  1 and the set Y = {y1 , . . . , yN } ⊆ R3 of distinct centers of the singular interactions. Consider TY := (−Δ)  C0∞ (R3 \{Y })

(2)

as an operator closure with respect to the Hilbert space L2 (R3 ). It is a closed, densely defined, non-negative, symmetric operator on L2 (R3 ), with deficiency index N. Hence, it admits an N 2 -real parameter family of self-adjoint extensions. Among these, there is an N-parameter family of local extension, denoted by ! {−Δα,Y ! α ≡ (α1 , . . . , αN ) ∈ (R ∪ {∞})N },

(3)

whose domain of self-adjointness is qualified by certain local boundary conditions at the singularity centers. The self-adjoint operators −Δα,Y provide rigorous realizations of the formal Hamiltonian (1), the coupling parameters αj , j = 1, . . . , N, being now proportional to the inverse scattering length of the interaction at the center yj . In particular, if for some j ∈ {1, . . . , N} one has αj = ∞, then no actual interaction is present at the point yj , and in practice things are as if one discards the point yj . When all αj = ∞, one recovers the Friedrichs extension of TY , namely the self-adjoint realization of −Δ on L2 (R3 ). Owing to the discussion above, we may henceforth assume, without loss of generality, that α runs over RN . We review the basic properties of −Δα,Y , from [4, Section II.1.1] and [26] (see also [9, 12, 13, 18]). We introduce first some notations. For z ∈ C and x, y, y  ∈ R3 , set y

Gz (x) :=

eiz|x−y| 4π|x − y|

,

yy 

Gz

⎧ iz|y−y  | ⎪ ⎨ e := 4π|y − y  | ⎪ ⎩ 0

if y  #= y if y  = y ,

(4)

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and Γα,Y (z) :=

  iz  y y δj,k − Gz j k αj − . j,k=1,...,N 4π

(5)

The function z → Γα,Y (z) has values in the space of N × N symmetric, complex valued matrices and is clearly entire, whence z → Γα,Y (z)−1 is meromorphic in C. It is known that Γα,Y (z)−1 has at most N poles in the open upper half-plane C+ , which are all located along the positive imaginary semi-axis. We denote by E + the set of such poles. Moreover, we denote by E 0 the set of poles of Γα,Y (z)−1 on the real line. Observe that E 0 is finite and symmetric with respect to z = 0. Actually, either E 0 = ∅ or E 0 = {0}. This follows by a generalization of the Rellich Uniqueness Theorem [29, Theorem 2.4], valid for a large class of compactly supported perturbations of the Laplacian, introduced by Sjöstrand and Zworski in [30]. For an introduction to the classical theory of the Rellich Uniqueness Theorem, we refer to the monograph of Lax and Phillips [22]. More recently, the absence −1 of non-zero real poles for Γα,Y has been proved through different techniques by Galtbayar-Yajima [14], and by the author in collaboration with Michelangeli [24]. The following facts are known. Proposition 1 (i) The domain of −Δα,Y has the following representation, for any z ∈ C+ \E + : D(−Δα,Y ) =

N % &  y g = Fz + (Γα,Y (z)−1 )j k Fz (yk )Gz j , Fz ∈ H 2 (R3 ) . j,k=1

(6) Equivalently, for any z ∈ C+ \E + ,

D(−Δα,Y ) =

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

g = Fz +

N  j =1

y

cj G z j

⎫ ! ! Fz ∈ H 2 (R3 ) ⎪ ⎪ ! ⎪ N ⎪ ! (c , . . . , c ) ∈ C ⎪ 1 N ⎬ !⎛ ⎞ ⎞ ⎛ ! Fz (y1 ) c 1 . ! !⎜ . ⎟ .. ⎟⎪ ⎪ !⎝ .. ⎠ = Γα,Y (z) ⎜ ⎪ ⎠ ⎝ ⎪ . ⎪ ! ⎭ ! F (y ) c z

N

N

(7) At fixed z, the decompositions above are unique. (ii) With respect to the decompositions (6)–(7), one has (−Δα,Y − z2 ) g = (−Δ − z2 ) Fz .

(8)

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(iii) For z ∈ C+ \E + , we have the resolvent identity 2 −1

(−Δα,Y − z )

2 −1

− (−Δ − z )

=

N 

y

(Γα,Y (z)−1 )j k |Gz j Gz k | . y

(9)

j,k=1

(iv) The spectrum σ (−Δα,Y ) of −Δα,Y consists of at most N non-positive eigenvalues and the absolutely continuous part σac (−Δα,Y ) = [0, ∞), the singular continuous spectrum is absent. Parts (i) and (ii) of Proposition 1 above originate from [17] and are discussed in [4, Theorem II.1.1.3], in particular (7) is highlighted in [12]. Part (iii) was first proved in [16, 17] (see also [4, equation (II.1.1.33)]). Part (iv) is discussed in [4, Theorem II.1.1.4], where it is stated that σp (−Δα,Y ) ⊂ (−∞, 0). An errata at the end of the monograph (see also [11, 15]) specifies that a zero eigenvalue embedded in the continuous spectrum can actually occur: in fact for every N ≥ 2 one can find a configuration Y of the N centers, and coupling parameters α1 , . . . αN such that 0 ∈ σp (−Δα,Y )—see the discussion in Sect. 4. Next, let us discuss in detail the spectral properties of −Δα,Y , whose resolvent is characterized by (9) as an explicit rank-N perturbation of the free resolvent. For negative eigenvalues, the situation is well-understood [4, Theorem II.1.1.4]. Proposition 2 There is a one to one correspondence between the poles iλ ∈ E + of Γα,Y (z)−1 and the negative eigenvalues −λ2 of −Δα,Y , counting the multiplicity. The eigenfunctions associated to the eigenvalue −λ2 < 0 have the form ψ =

N 

y

cj Giλj ,

j =1

where (c1 , . . . , cN ) ∈ Ker Γα,Y (iλ) \ {0}. Our main purpose is to analyze the spectral behavior of −Δα,Y at z = 0, and more generally when z approaches the real line. The starting point is a classical version of the Limiting Absorption Principle for the free Laplacian. Given σ > 0, we consider the Banach space Bσ := B(L2 (R3 , x2+σ dx); L2 (R3 , x−2−σ dx)), (10) / where x := 1 + |x|2, and B(X; Y ) denotes the space of linear bounded operators from X to Y . We have the following result [1, 19, 21]. Proposition 3 (Limiting Absorption Principle for −Δ) Let σ > 0. For any z ∈ C+ , we have (−Δ − z2 )−1 ∈ Bσ . Moreover, the map C+  z → (−Δ − z2 )−1 ∈ Bσ can be continuously extended to the real line.

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Owing to the resolvent formula (9), and observing that for any z ∈ C+ ∪ R the y y projectors |Gz j Gz k | belong to Bσ , it is easy to deduce that also −Δα,Y satisfies a Limiting Absorption Principle. Proposition 4 (Limiting Absorption Principle for −Δα,Y ) Let σ > 0. For every z ∈ C+ \ E + , we have (−Δα,Y − z2 )−1 ∈ Bσ . Moreover, the map C+ \ E +  z → (−Δα,Y − z2 )−1 ∈ Bσ can be continuously extended to R \ E 0 . Our main result is a resolvent expansion in a neighborhood of z = 0, which in view of the previous discussion is the only possible singular point on the real line for the map z → (−Δα,Y − z2 )−1 ∈ Bσ . Theorem 1 In a (real) neighborhood of z = 0, we have the expansion (−Δα,Y − z2 )−1 =

R−2 R−1 + R0 (z), + 2 z z

(11)

where R−2 , R−1 ∈ Bσ and z → R0 (z) is a continuous Bσ -valued map. Moreover, R−2 #= 0 if and only if zero is an eigenvalue for −Δα,Y . Remark 1 For Schrödinger operators of the form −Δ + V , the Limiting Absorption Principle and the analogous of Theorem 1 can be proved under suitable short-range assumptions on the scalar potential V (see e.g. the classical papers [1, 19]). In this case, moreover, it is well-known that R−1 #= 0 if and only if there exists a generalized eigenfunction at z = 0 (a zero-energy resonance for −Δ + V ), namely a function ψ ∈ L2 (R3 , x−1−σ dx) \ L2 (R3 ), for any σ > 0, which satisfies (−Δ + V )ψ = 0 as a distributional identity on R3 . As it will be clear from the proof of Theorem 1, a similar characterization holds true also for −Δα,Y (see Remark 2).

2 Asymptotic for Γα,Y (z)−1 as z → 0 We fix N ≥ 1, α ∈ RN and Y ⊆ R3 , and we set Γ (z) := Γα,Y (z). We shall use the notation O(zk ), k ∈ Z, to denote a meromorphic M N (C)-valued function whose Laurent expansion in a neighborhood of z = 0 contains only terms of degree ≥ k. In particular, O(1) denotes an analytic map in a neighborhood of z = 0. We also write Θ(zk ) to denote a function of the form Azk , with A ∈ M N (C)\{0}. In a neighborhood of z = 0, we can expand Γ (z) = Γ0 + zΓ1 + z2 Γ2 + O(z3 ). Explicitly, we have y y

(Γ0 )j k = αj δj k − G0 j k ,

(Γ1 )j k = (4πi)−1 ,

(Γ2 )j k = (8π)−1 |yj − yk |.

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In particular, Γ0 , Γ2 are real symmetric matrices, while Γ1 is skew-Hermitian, i.e. Γ1∗ = −Γ1 . Our aim is to characterize the small z behavior of Γ (z)−1 . Preliminary, we recall the following useful result due to Jensen and Nenciu [20]. Lemma 1 (Jensen-Nenciu) Let A be a closed operator in a Hilbert space H and P a projection, such that A + P has a bounded inverse. Then A has a bounded inverse if and only if B = P − P (A + P )−1 P has a bounded inverse in P H , and in this case A−1 = (A + P )−1 + (A + P )−1 P (B  P H )−1 P (A + P )−1 . We can state now the main result of this section. Proposition 5 In a neighborhood of z = 0 we have the Laurent expansion Γ (z)−1 =

A−2 A−1 + O(1), + 2 z z

(12)

where A−2 , A−1 ∈ M N (C). Moreover, # 0 if and only if Ker Γ0 ∩ Ker Γ1 #= {0}, (i) A−2 = (ii) A−1 = # 0 if and only if Ker Γ0 #⊆ Ker Γ1 . Proof If Γ0 = Γ (0) is non-singular, then Γ (z)−1 is analytic in a sufficiently small neighborhood of z = 0. Assume now that Γ0 is singular. Let us distinguish two cases: Case 1: Ker Γ0 ∩ Ker Γ1 = {0}. Let us set Γ≤1 (z) := Γ0 + zΓ1 , and observe that for z small enough, z #= 0, the matrix Γ≤1(z) is non-singular. Suppose indeed that Γ≤1 (z)v = 0 for some v ∈ CN . If Γ0 v #= 0, then for small z we also have Γ≤1 (z)v #= 0, a contradiction. Hence Γ0 v = 0, which for z #= 0 implies Γ1 v = 0, and using the hypothesis Ker Γ0 ∩ Ker Γ1 = {0} we deduce that v = 0. Observe also that for z small enough, z #= 0, the matrix Γ (z) in non-singular, with Γ (z)−1 = Γ≤1 (z)−1 + O(1). In order to invert Γ≤1 (z), we use the Jensen-Nenciu Lemma. Let P : CN → CN be the orthogonal projection onto Ker Γ0 . Since Γ0∗ = Γ0 , we have that Γ0 + P is non-singular, whence the same is Γ≤1 (z) + P for small z, with (Γ≤1 (z) + P )−1 = O(1). More precisely, (Γ≤1 (z) + P )−1 = [I + z(Γ0 + P )−1 Γ1 ]−1 [Γ0 + P ]−1 = [I − z(Γ0 + P )−1 Γ1 ][Γ0 + P ]−1 + O(z2 ).

(13)

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By Lemma 1 we get Γ≤1 (z)−1 = (Γ≤1 (z) + P )−1  −1  + (Γ≤1 (z) + P )−1 P P − P (Γ≤1 (z) + P )−1 P  P CN P (Γ≤1 (z) + P )−1 .

(14) Observe that (Γ0 + P )−1 P = P , and since Γ0∗ = Γ0 we also have P (Γ0 + P )−1 = P . Using these relations and (13), we compute P − P (Γ≤1 (z) + P )−1 P = zP Γ1 P + O(z2 ). Substituting into (14) we obtain −1 (z) = (Γ≤1 (z) + P )−1 Γ≤1

+ (Γ≤1 (z) + P )−1 P

  −1 z P Γ 1 P  P CN + O(1) P (Γ≤1 (z) + P )−1

= z−1 P (P Γ1 P  P CN )−1 P + O(1) = Θ(z−1 ) + O(1). (15) Case 2: Ker Γ0 ∩ Ker Γ1 #= {0}. We start by proving that Ker Γ1 ∩ Ker Γ2 = {0}. Since Γ2 is real symmetric, and Γ1 is purely imaginary and skew-symmetric, it is sufficient to show that the quadratic form associated to Γ2 is strictly negative on 7 8 ( Ker Γ1 ∩ RN ) \ {0} = v ∈ RN \ {0} | v1 + . . . + vN = 0 . To this aim, we prove preliminary that for any v ∈ RN , with v1 + . . . + vN = 0, Γ2 v, v := (8π)−1



|yj − yk |vj vk ≤ 0.

(16)

1≤j,k≤N

The key point is to use the so called averaging trick. By rotational and scaling invariance, we can see that there exists a positive constant c such that, for any y ∈ R3 ,  |w, y|dw = c|y|. S2

It follows that (8π)

−1

 1≤j,k≤N

|yj − yk |vj vk = (8πc)

−1



 S 2 1≤j,k≤N

|w, yj − yk |vj vk dw, (17)

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and then it is sufficient to prove that, for a fixed w ∈ S 2 , 

|y˜j − y˜k |vj vk ≤ 0,

1≤j,k≤N

where we set y˜j := w, yj  for j = 1, . . . , N. We have 



|y˜j − y˜k |vj vk = 2

1≤j,k≤N

max {y˜j − y˜k , 0}vj vk

1≤j,k≤N



=2



t ∈R 1≤j,k≤N

(18) [y˜k < t < y˜j ]vj vk ,

where we use the Iverson bracket notation [P ], which equals 1 if the statement P is true and 0 if it is false. So it is enough to prove that, for almost every t ∈ R, 

vj vk ≤ 0.

y˜k y˜N . Owing to (18)–(19), we deduce that  vj vk = 0, (21) y˜ k 0 and s ∈ (−1/2, 3/2). Then Tλ ξ H s−1 ≤ Cξ H s ,

∀ξ ∈ H s .

(14)

In particular, Tλ maps continuously H s (R3 ) into H s−1(R3 ) for such s’s. The same holds true for s = {−1/2, 3/2} if one restricts its attention to angular sectors # ≥ 1. Together with the obvious symmetry with respect to the standard L2 scalar product, we obtain that Tλ is a densely defined symmetric operator on L2 (R3 ) with domain H s for every s ≥ 1. • The dependence on m of Tλ is not immediately clear. However, in [5] it is proven that for any given ξ for which the quadratic form ξ, Tλ ξ  is well defined, the map m → ξ, Tλ ξ  is increasing. Owing to (12), we restrict our attention to the ξ ’s in H 1/2 such that Tλ ξ ∈ H 1/2 . For such ξ ’s, the expression Aλ,α ξ = 2Wλ−1 (Tλ + α)ξ is meaningful in that Wλ maps H 1/2 into H −1/2, and one has uξ , Aλ uξ H −1/2 = ξ, (Tλ + α)ξ H −1/2 ,H 1/2 .

(15)

Wλ

This quadratic form plays a major role in the study of semi-boundedness: in particular, in [9] it is shown that its positivity (depending on α and λ) lead to the

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α bound Hα ≥ 0 for α ≥ 0, and Hα ≥ − 4π 4 (1−Λ(m)) 2 if α < 0. The function Λ(m) is

an explicit monotone decreasing function, and m∗ := Λ−1 (1) is precisely the value for which the positivity of Tλ +α cannot be achieved for any λ, α. We are now ready to state the reduction lemma from [5].   2 α α2 Theorem 2 Given α < 0 and λ ∈ 4π 4 , 4π 4 (1−Λ(m))2 , the following are equivalent: • −λ is an eigenvalue for Hα with corresponding eigenfunction g. • g = uλξ where ξ ∈ H 1/2 is an eigenfunction for Tλ with eigenvalue −α. • g = uλξ , where . ξ (p) = . ξ˜ ( √p ) and ξ˜ ∈ H 1/2 is an eigenfunction for T1 with eigenvalue

|α| √ . λ

λ

The restrictions for α and λ are consistent with the results about the essential spectrum we will present later and the lower bound on Hα . Indeed, combining the two it iseasy to show that the discrete spectrum may appear only in the energy α2 α2 window 4π 4 , 4π 4 (1−Λ(m))2 , and only in the repulsive case α < 0. The reduction lemma is useful because it relates the discrete spectrum of Hα and that of T1 . An easy application of min-max shows that the absence of discrete spectrum for Hα is equivalent to the positivity of T1 − 2π 2 .

4 Essential Spectrum We now address the problem of determine the essential spectrum of Hα . We recall that is plays a major role in that the long time behavior of the evolution eit Hα , and in particular the associate scattering theory, are related to the structure of the essential spectrum of Hα . The picture looks different in the case α ≥ 0 and α < 0: the former correspond to a repulsive interaction, the latter to an attractive one. Once more, we stress that Hα is not a small perturbation of the free Laplacian in three dimension, therefore we do not expect to have necessary the same essential spectrum of the free Laplacian. Indeed, here is the result from [5]. Theorem 3 Let m > m∗ . One has • σess (Hα ) = [0, +∞) if α ≥ 0. α2 • σess (Hα ) = [− 4π 4 , +∞) if α < 0. For a full proof, we refer to [5], while here we only give a sketch. For the first part, one notices that the quadratic form associated to Hα is positive for α positive. This implies, in particular, that σ (Hα ) ⊂ [0, +∞). Moreover, one can construct a singular sequence at each λ ≥ 0 by choosing a singular sequence for the free Laplacian (whose essential spectrum is [0, +∞)) that also belongs to D(Hα ). This

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is easy to do since H02 (R3 × R3 ) ⊂ D(Hα ), and singular sequences for the free Laplacian can be chosen to have value 0 at 0. For the second part, the situation is different. The same proof shows that σess (Hα ) ⊃ [0, +∞), while some works needs to be done in order to find a α2 singular sequence at λ ∈ [− 4π 4 , 0). For a given λ, the candidate is a g = uξn 1/2

where ξn ∈ H# (R3 ), and its radial part is an approximated delta at r0 . Here r0  is defined to be the unique solution to 2π 2 νr02 + λ = α. A careful use of the mapping properties of Tλ , Wλ show that this is indeed a singular sequence at λ.

5 Discrete Spectrum: Finiteness and Existence of Eigenvalues So far, we have ruled out the possibility of Thomas effect for the 2+1 system in the regime m > m∗ , owing to the semi-boundedness of the operator Hα . In this section, we aim to prove the absence of Efimov effect as well in the same regime of mass. We will actually prove a stronger statement, namely the finiteness of the discrete spectrum for m > m∗ . We point out that this result is not expected to hold for masses below the threshold (cfr [28]). The argument given in the previous section already shows that no discrete spectrum can appear in the regime α ≥ 0. Concerning the case α < 0, here is the main result from [5]. Theorem 4 Let m > m∗ , α < 0. Then σdisc (Hα ) consists of finitely many eigenvalues (each of which has finite multiplicity). While Yoshitomi in [31] proved the result for masses large enough, we pushed this up to the (expected) optimal value m∗ . Again, we refer to [5] for a complete proof. Here, we only sketch the main idea: first of all, we have already seen how every eigenvalue −λ of Hα below the bottom of the essential spectrum corresponds to an eigenvalue −α for Tλ . Then, by means of a Birman-Schwinger type of argument, one can reduce the problem to that of the behavior of the off-diagonal part of Tλ + α relative to the diagonal part. Concretely, one has the identity ξ, Tλ + αξ H 1/2 ,H −1/2 = ξ, (1 − Zλ,n )ξ H λ,n , where Hλ,n is a suitable Hilbert space, and Zλ,n is an Hilbert-Schmidt (hence compact) operator on Hλ,n . Using the finiteness of the spectrum of a compact operator outside a compact containing the origin, one can obtain the result. The core of the proof we just outlined relies on the finiteness of the discrete spectrum away from 0 for a compact operator. In particular, it is not clear whether eigenvalues may exist. Equivalently, using Theorem 2, the question becomes T1 whether the operator 2π 2 − 1 may fail to be positive for same value of m.

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This question (and more) are also answered in [5]: the first (trivial) observation is that this cannot happen for even angular sectors, exploiting. The second (non-trivial) observation is that the same holds for angular odd sectors with # ≥ 3. Therefore, the question can be formulated for the angular sector # = 1, and here the mass starts playing a role. Rather than presenting the proof, we want to give an heuristic of why the problem is non trivial, and how an eigenfunction should look like. If ξ(p) = f (|p|)pY1 (Ω), where Ω is a variable on the sphere, Y1 is the first √ harmonic function in R3 and f ∈ L2 (R+ , r 2 + 1), then the quadratic form T1 ξ, ( 2π 2 − 1)ξ  becomes (after integrating in the angular variables) ξ, (

T1 − 1)ξ  = 2π 2



+∞

/ ( νr 2 + 1 − 1)|f (r)|2 dr +

0

1 2μπ

+∞  ∞

 0

drdr  f (r)f (r  )G(z),

0

(16) where z=

r 2 + r 2 + 1 , μrr 

G(z) = 2 − z ln

z + 1 . z−1

Notice that z ∈ (1 + m, +∞). The left extreme correspond to r = r  → +∞, insofar as z → μ2 = 1 + m, while the right extreme corresponds to r → 0. The function G(z) is easily seen to be negative, strictly increasing, approaching −∞ when z → 1− and approaching 0 as z → +∞. In order for the quadratic form (16) to be negative, we need to trade between the mass that f puts close to 0, for which the diagonal part is positive and small but the off-diagonal part is also small, and the mass that f puts at ∞, where the diagonal term becomes huge while the off-diagonal term is significantly larger. In particular, it is clear that m being smaller and smaller corresponds to z taking values closer and closer to the critical value z = 1, where the off-diagonal term becomes arbitrary big (in absolute value). By carefully checking this trade-off, in [5] we explicit exhibit ξ ’s on which the quadratic form (16) have negative expectation (i.e., presence of eigenvalues) for small values of m > m∗ , and we proved the positivity of the quadratic form (i.e., absence of eigenvalues) when the mass is large enough. Summarizing, in [5] we proved the existence of masses m∗ < M∗ ≤ M ∗ , with the following properties: 1. For m∗ < m < M∗ , there exists a bound states with energy below the bottom of the essential spectrum for the Hamiltonian Hα . 2. For m > M ∗ , no bound states appear below the bottom of the essential spectrum. 3. One has M∗ = M ∗ , and the analytic estimates (8.587)−1 ≤ M ∗ ≤ (2.617)−1. The first two points, as well as the lower and upper bound, originate from the proof of the above results. In the next section, we will deal with the equivalence of M ∗ and M∗ and the accuracy of our bounds.

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6 Monotonicity and Number of Eigenvalues: Rigorous Results and Open Questions Since, for fixed ξ , the quadratic form m → ξ, (Tλ + α)ξ  is monotone increasing, one may expect, in the spirit of the Hellmann-Feynman-type theorems, the monotonicity of the ground state energy (and other bound states energy) itself in the mass parameter. Moreover, experiments (see, e.g., [10]) naturally lead to the same conjecture, which we eventually proved in [5]. An immediate consequence of this is the equivalence of M∗ and M ∗ , since the monotonicity of eigenvalues implies that their existence below the bottom of the essential spectrum occurs in a whole interval. Another interesting fact emerging from our analysis is that the upper bound (which originates from the variational argument) seems to be almost sharp from the point of view of the experiments (cfr [28]). It would be interesting to increase the lower bound and narrow the window for the threshold mass M ∗ , which is a current work in progress. Here is another natural question for both physicists and mathematicians. Despite its mathematical elegance and simplicity, the Birman-Schwinger-like argument used in the proof of the finiteness of the discrete spectrum is not immediately suited to draw further conclusions about the number of bound states, like for instance estimating their number, which would follow if one was able to extract an analogous estimate out from the auxiliary compact operator Zλ,n . The monotonicity argument ensures that for m close to M ∗ only the ground state appears, but of much more interest is the region m close to m∗ : is the number of eigenvalues uniformly bounded in m, or does it explode to infinity? In the experiment (e.g., [10]), no more than two bound states are observed, regardless for the value of α. In this sense, m∗ seems to represent a real phase transition, since the number of bound states goes from infinity to 2 crossing this value.

7 N+1/N+M Fermionic System: Known Results and Open Questions In this section we briefly survey known results and open questions about general fermionic models of N+M fermions. These are topical models, of great physical relevance, and for which the 2+1 is a prototype, consisting of two different species of identical fermions with an inter-species zero-range two-body interaction. In particular, the N+1 model describes a polaron particle embedded in a fermionic gas of different species.Once more, because of Pauli exclusion principle, interactions can only take place among different particles. The operator-theoretic construction of the model should follow the same lines as for the 2+1 Hamiltonian: however, to the author knowledge, no such construction within the framework of KVB theory has been done, though in [8, 11] these systems are studied via a quadratic form approach. Consequently, a complete classification

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of local (TMS in the two-body channels) self-adjoint extensions is lacking, as well as the presence/absence of other boundary conditions to determine a well-posed Hamiltonian. Even putting aside the classification problem and focusing only on relevant selfadjoint realisations (e.g., the one obtained in [8, 11]), some natural questions arise: • Are those system stables? And for which values of the mass ratio m between the mass of the probe particle and the mass of each of the N fermions? • In which regimes of m do interesting spectral phenomena such as the Thomas or the Efimov effect occur? As for the first question, it was first proved in [8] that above a threshold m = m(N) a self-adjoint Hamiltonian can be constructed by quadratic form methods which is also bounded from below, and subsequently in [26] such a threshold was ameliorated from a growing-to-infinity m(N) to a uniform-in-N one, thus ruling out the occurrence of the Thomas effect for sufficiently large mass ratios. However such bounds are still far from the values expected from theoretical-physical heuristic arguments and from experiments. For the few-body problem, sharper results are avaiable via a numerical approach [14, 15]. So far no rigorous derivation of the sharp mass threshold for stability has been established. Regarding the Efimov effect, the picture is even more challenging. To our knowledge, no rigorous result on the presence of Efimov effect is appeared in the mathematical literature (even in the 2 + 1 system). An extremely inspiring demonstration of such an effect is proposed in the series of works [20–23], however with a mistake in the identification of the correct space of charges, as was first noticed in [16, 17], which makes the argument not immediately adjustable into the correct setting. An amount of heuristic arguments are being made within the counterpart physical literature, and we refer to the very recent and comprehensive review [28]. In particular: 1. For the 3 + 1 system, there exists a mass m∗3 > m∗ such that in the region m ∈ (m∗ , m∗3 ), four-body Efimov effect is expected to occur. Notice that the condition m > m∗ comes from the very definition of four-body Efimov effect (no three-body subsystem should be linked). As in the 3-body case, eigenvalues only seems to occur in the angular sector # = 1. 2. Various partial conjectures and numerical calculations are proposed for generic N, not supported yet by rigorous mathematics. 3. Almost nothing is known about higher N. We trust to tackle these and further questions in future works!

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References 1. Albeverio, S., Gesztesy, F., Høegh-Krohn, R., Holden, H.: Solvable Models in Quantum Mechanics. Texts and Monographs in Physics. Springer, New York (1988) 2. Basti, G., Teta, A.: Efimov effect for a three-particle system with two identical fermions. Ann. Henri Poincaré. 18(12), 3975–4003 (2017) 3. Basti, G., Cacciapuoti, C., Finco, D., Teta, A.: The three-body problem in dimension one: From short-range to contact interactions. J. Math. Phys. 59, 072104 (2018) 4. Bazak, B., Petrov, D.S.: Five-body efimov effect and Universal Pentamer in Fermionic mixtures. Phys. Rev. Lett. 118, 083002 (2017) 5. Becker, S., Michelangeli, A., Ottolini, A.: Spectral analysis of the 2+ 1 fermionic trimer with contact interactions. Math. Phys. Anal. Geometry 21(4), 35 (2018) 6. Bethe, H., Peierls, R.: Quantum theory of the diplon. Proc. R. Soc. Lond. A Math. Phys. Sci. 148, 146–156 (1935) 7. Castin, Y., Werner, F.: The unitary gas and its symmetry properties. The BCS-BEC Crossover and the Unitary Fermi Gas. In: Zwerger, W. (ed.) Lecture Notes in Physics, vol. 836, pp. 127– 191. Springer, Berlin, Heidelberg (2012) 8. Correggi, M., Dell’Antonio, G., Finco, D., Michelangeli, A., Teta, A.: Stability for a system of N fermions plus a different particle with zero-range interactions. Rev. Math. Phys. 24, 1250017, 32 (2012) 9. Correggi, M., Dell’Antonio, G., Finco, D., Michelangeli, A., Teta, A.: A class of Hamiltonians for a three-particle fermionic system at unitarity. Math. Phys. Anal. Geom. 18, 1–36 (2015) 10. Endo, S., Naidon, P., Ueda, M.: Universal physics of 2+1 particles with non-zero angular momentum. Few-Body Syst. 51, 207–217 (2011) 11. Finco, D., Teta, A.: Quadratic forms for the fermionic unitary gas model. Rep. Math. Phys. 69, 131–159 (2012) 12. Gallone, M., Michelangeli, A., Ottolini, A.: Kre˘ın-Višik-Birman self-adjoint extension theory revisited. SISSA preprint 25/2017/MATE (2017) 13. Griesemer, M., Hofacker, M., Linden, U.: From Short-Range to contact interactions in the 1d Bose Gas. https://arxiv.org/pdf/1908.05705.pdf 14. Michelangeli, A., Schmidbauer, C.: Binding properties of the (2+1)-fermion system with zerorange interspecies interaction. Phys. Rev. A 87, 053601 (2013) 15. Michelangeli, A., Pfeiffer, P.: Stability of the (2+2)-fermionic system with zero-range interaction. J. Phys. A Math. Theor. 49, 105301 (2016) 16. Michelangeli, A., Ottolini, A.: On point interactions realised as Ter-Martirosyan-Skornyakov Hamiltonians. Rep. Math. Phys. 79, 215–260 (2017) 17. Michelangeli, A., Ottolini, A.: Multiplicity of self-adjoint realisations of the (2+ 1)-fermionic model of Ter-Martirosyan–Skornyakov type. Rep. Math. Phys. 81(1), 1–38 (2018) 18. Minlos, R.A.: On the point interaction of three particles. In: Applications of selfadjoint extensions in quantum physics (Dubna, 1987), vol. 324 of Lecture Notes in Phys., pp. 138– 145. Springer, Berlin (1989) 19. Minlos, R.A.: On pointlike interaction between N fermions and another particle. In: Dell’Antonio, A., Figari, R., Teta, A. (eds.) Proceedings of the Workshop on Singular Schrödinger Operators, Trieste 29 September - 1 October 1994, ILAS/FM-16 (1995) 20. Minlos, R.A.: On point-like interaction between n fermions and another particle. Mosc. Math. J. 11, 113–127, 182 (2011) 21. Minlos, R.A.: On point-like interaction between three particles: two fermions and another particle. ISRN Math. Phys. 2012, 230245 (2012) 22. Minlos, R.A.: A system of three pointwise interacting quantum particles. Uspekhi Mat. Nauk 69, 145–172 (2014) 23. Minlos, R.A.: On point-like interaction of three particles: two fermions and another particle. II. Mosc. Math. J. 14, 617–637, 642–643 (2014)

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24. Minlos, R.A., Faddeev, L.D.: On the point interaction for a three-particle system in quantum mechanics. Sov. Phys. Dokl. 6, 1072–1074 (1962) 25. Minlos, R.A., Shermatov, M.K.: Point interaction of three particles. Vestnik Moskov. Univ. Ser. I Mat. Mekh., 7–14, 97 (1989) 26. Moser, T., Seringer, R.: Stability of a fermionic N+1 particle system with point interactions. Commun. Math. Phys. 356, 329 (2017) 27. Moser, T., Seringer, R.: Stability of the 2+2 fermionic system with point interactions. https:// arxiv.org/pdf/1801.07925 28. Pascal, N., Shimpei, E.: Efimov Physics: a review. Rep. Prog. Phys. 80 (2017) 056001 29. Skornyakov, G.V., Ter-Martirosyan, K.A.: Three body problem for short range forces. I. Scattering of low energy neutrons by deuterons. Sov. Phys. JETP 4, 648–661 (1956) 30. Teta, A.: Quadratic forms for singular perturbations of the Laplacian. Publ. Res. Inst. Math. Sci. 26, 803–817 (1990) 31. Yoshitomi, K.: Finiteness of the discrete spectrum in a three-body system with point interaction. Math. Slovaca 67, 1031–1042 (2017)

Born-Oppenheimer Type Approximation for a Simple Renormalizable System Haci Akbas and O. Teoman Turgut

Abstract We discuss a simple singular system in two dimension, two heavy particles interacting with a light particle via an attractive contact interaction. Although intuitively clear the actual application of the Born-Oppenheimer type approximation to this problem is quite subtle. The light particle ground state energy is not bounded from below when we consider it as a function of heavy particles’ separation. Nevertheless, with due care, we calculate the leading term of the Born-Oppenheimer type approximation and indicate how to get the higher order corrections. Keywords Renormalization · Singular interaction · Born-Oppenheimer approximation

1 Introduction Born-Oppenheimer approximation [1] is the basic tool of molecular physics [2– 4]. When all the particles interact via Coulomb forces, as is the typical case of molecular physics, the Born-Oppenheimer vibrational energy levels go with (m/M)1/2, where m/M refers to the light mass to heavy mass ratio. One can consider rotational energy levels as well as anharmonic vibrations as higher order corrections, since they turn out to be of order (m/M). In the usual approach the relevant expansion parameter is thought to be (m/M)1/4, yet only even powers seem to show up in the expansions. There is a large literature on the stationary level calculations in the Born-Oppenheimer approximation, we will not be able to cover most of it, we only mention some of the more rigorous works, since in our present work, we also aim to get an approximation scheme in which we can control the errors that we make in a self-consistent way. The time-dependent Born-

H. Akbas () · O. T. Turgut Department of Physics, Bo˘gaziçi University, Istanbul, Turkey e-mail: [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. Michelangeli (ed.), Mathematical Challenges of Zero-Range Physics, Springer INdAM Series 42, https://doi.org/10.1007/978-3-030-60453-0_9

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Oppenheimer approximation is a very interesting and closely related subject, the reader can consult the review articles [5–7] for more information. However, our system is highly singular and requires renormalization as is well known in the literature[8–15]. Therefore, one does not expect the existing results on well-behaved potentials to hold in this regime. An interesting toy model in which two heavy and one light particle all interact via harmonic oscillator potentials is presented by R. Seiler in [16], where the assumptions of the Born-Oppenheimer approach is carefully tested. Following this, some rigorous aspects of Born-Oppeheimer approximation is presented in [17]. It is not at all clear that the eigenvalues of the light degrees of freedom, that one computes, under the influence of potentials when the heavy centers are clamped, actually define well-behaved nonintersecting surfaces when one considers the heavy degrees of freedoms as parameters. This difficult problem is solved by Hunziker in [18], where even for Coulomb type potentials energy eigenfunctions are shown to be essentially analytic functions of the heavy coordinates. These problems further investigated in a series of papers by Hagedorn [19–21]. In the usual potentials, higher order corrections to the Born-Oppenheimer approximation are rigorously investigated by Hagedorn in a series of papers [22, 23]. Further investigations along similar lines are presented in [24]. They typically correspond to higher order corrections to the effective potentials generated by the light degrees of freedom. The reader can find a large collection of references and various mathematically precise statements on Born-Oppenheimer approximation in a recent review by Jecko [7]. A different attempt to include higher order corrections to Born-Oppenheimer approximation is given by Weingert and Littlejohn [25] as an example of their diagonalization technique in the deformation quantization approach. They discover derivative terms in the higher order corrections. As our problem concerns a highly singular system, we get kinetic energy corrections, very similar to the ones found in [25], even at the leading order. In a similar spirit to [16], a slightly simpler model for its pedagocial value was proposed by G. Gangopadhyay and B. Dutta-Roy in [26] where the authors consider a light particle coupled to a heavy particle via a delta function potential and the whole system is confined to a box in one dimension. This system has a distributional potential yet it is not truly singular, since an analytical treatment is still possible one can test the Born-Oppenheimer approximation. We propose to consider a toy model in a highly nontrivial situation, which requires a coupling constant renormalization. Recently, we considered a simpler version of this problem in one dimension, in which there is no renormalization, consequently we could analyse this case in much more detail [27]. The existing literature on Born-Oppenheimer approximation is tailored to deal with interaction potentials which are sufficiently regular. It is not obvious that these methods would suffice to deal with somewhat more singular cases, especially the problems which require renormalization. In view of the fact that most quantum field theories require renormalization this is an especially important problem which has direct physical implications. The insight that one may gain even from some simple models is likely to shed light on the methods to be developed for more realistic

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systems. The model we have in mind looks very simple, we consider two heavy particles in two dimensions interacting via a contact term with a third light particle. To make the calculations tractable we use a nonrelativistic approach (nevertheless, see the remark below), a more interesting version would be to treat the light particle as a relativistic one. The relativistic model may be relevant for understanding the following problem, just as in the case of nucleons interacting with each other via exchange of mesons, the interaction of two heavy quarks may be energetically more favorable by exchange of glueballs, which are to be modeled as light relativistic particles. In any case, we see that the basic idea of separating heavy (slow) and light (fast) modes does indeed work, yet in a rather nontrivial manner. One does not find a simple expansion scheme, where corrections manifest themselves easily. It has to be worked out carefully, and even the leading term requires a delicate set of computations. We see that the leading term is actually also large, this is an effect due to the nonlinear nature of renormalization, the light degree of freedom also has very large energy. What is then the relevant expansion parameter? It turns out that the separation between the heavy particles is very small, as compared to single center binding length scale. This length scale is determined by the binding energy of the light particle to a single heavy center, this is our input parameter for renormalization, coupling constant is traded off with this binding energy. The ratio of these two length scales is our expansion parameter in a very subtle way as we see below. One may compute some of the lower order terms as well, a few contain the logarithm of the large parameter M/m, there are also next lower order terms one can in principle compute. A many body approach could be more natural to develop a systematic expansion but this seems to be much more challenging at the moment. Having completed the present work, we learned through the work of D. S. Rosa et al. [28] (and references therein) that three particle problem within the Born-Oppenheimer approximation was originally introduced by Fonseca, Reddish and Stanley as an analytic model to study the Effimov effect, they use regular separable potentials between the heavy and light particles [29]. Our delta-function potential model is essentially worked in the Jacobi type coordinates within the BornOppenheimer approximation in [30], in the main part they assume some reasonable potential between the heavy centers. Moreover these ideas are further elaborated in [28] mainly for large separation of the heavy centers, with such atomic systems in mind. In the limit of large separation, in which the large z expansion becomes important, the correction terms that we compute are actually small and they can be neglected. These ideas are further developed in arbitrary dimensions in the recent work [31] using the Born-Oppenheimer approximation, albeit in this work corrections due to small distance are not taken into account. When there is no repulsive potential between the heavy centers, we see that it is crucial to develop an improved expansion.

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2 Two Heavy Particles Interacting with a Light One We consider two heavy particles interacting with a single light particle via a contact interaction in two dimensions. We write down the Schrödinger equation for this simple model; 

−

 h¯ 2 2 h¯ 2  2 ∇i − ∇x − λδ(x − x1 ) − λδ(x − x2 )] 1(x; x1 , x2 ) = E1(x; x1 , x2 ). 2M 2m i

(1) Here x1 , x2 refer to the heavy particles coordinates and x refers to the light one (for simplicity ∇xi is written as ∇i ). The choices of the masses also reflect this difference. Let us pretend that the Born-Oppenheimer approximation can be applied to this system, thus we use a decomposition of the wave function into fast and slow degrees of freedom: 1(x; x1, x2 ) = φ(x|x1 , x2 )ψ(x1 , x2 ).

(2)

We assume that this decomposition respects the translational invariance of the system, moreover we assume for the time being that the delta-functions are actually regularized since we know that there is a divergence hidden in this problem. We substitute the proposed solution into the Schrödinger equation, 

−

   h¯ 2  2 h¯ 2  2 ∇i ψ(x1 , x2 ) φ(x|x1 , x2 ) + − ∇i φ(x|x1 , x2 ) ψ(x1 , x2 ) 2M 2M i

−

h¯ 2  ∂φ ∂ψ 2 + 2M ∂xi ∂xi i

i



−

  h¯ 2 2 ∇x − λδ(x − x1 ) − λδ(x − x2 ) φ(x|x1 , x2 ) ψ(x1 , x2 ) 2m = Eφ(x|x1 , x2 )ψ(x1 , x2 ).

(3) In general, we should assume an expansion, in terms of a complete set of solutions to the system relative to the light particle. Here the separation between the light binding energy for the ground state configuration and the other levels is very large, we expect negligible overlap with other solutions so we ignore them. Ordinarily we would assume that we could find the solution to the equation below, −

h¯ 2 2 ∇ φ(x|x1 , x2 ) − λ[δ(x − x1 ) + δ(x − x2 )]φ(x|x1 , x2 ) = E(x1 , x2 )φ(x|x1 , x2 ). 2m x (4)

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That would mean that the heavy particles would act like fixed centers and the light particle would move in this background. Then we would find, 

−

 h¯ 2  h¯ 2  2 ∇i ψ(x1 , x2 ) + E(x1 , x2 )ψ(x1 , x2 ) φ(x|x1 , x2 ) − ∇i φ · ∇i ψ 2M M i

i

  h¯ 2  2 ∇i φ(x|x1 , x2 ) ψ(x1 , x2 ) = Eφ(x|x1 , x2 )ψ(x1 , x2 ), + − 2M

(5)

i

and if we could neglect the last two terms on the lefthand side, we would end up with the Born-Oppenheimer result, 

−

 h¯ 2  2 ∇i + E(x1 , x2 ) ψ(x1 , x2 ) = Eψ(x1 , x2 ). 2M

(6)

i

Nevertheless we will see that this would be wrong, and we would end up with a divergent result. The expression 

−

 h¯ 2  2 ∇i φ(x|x1, x2 ) 2M

(7)

i

contains a term that we can convert into 

−

 h¯ 2 ∇x 2 φ(x|x1 , x2 ) 2M

(8)

hence should be incorporated into the term (4), resulting into −

 h2 h¯ 2  2 ¯ + ∇ φ(x|x1 , x2 ) − λ[δ(x − x1 ) + δ(x − x2 )]φ(x|x1 , x2 ) = E(x1 , x2 )φ(x|x1 , x2 ), 2m 2M x

(9)

to be renormalized all together in this two-dimensional case. Note that because of the divergence, the usual rule of ignoring m/M terms at this order does not work here. Moreover, the effective potential generated from the derivative terms contain a z12 term, where z is the relative coordinate for the two heavy particles. This term cannot be added as a perturbation since it changes the character of the wave function at the origin, hence should be used in the leading order Born-Oppenheimer type approximation. To set up the formalism, we will introduce an ansatz for the solution of the light degrees of freedom, assuming for the time being that the delta functions are properly regularized—one possibility is to use the heat kernel itself, this will preserve the

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translational invariance of the whole system, for the time being we will proceed formally:   φ(x|x1, x2 ) = A(x1, x2 ) η+ (x|x1, x2 ) + η− (x|x1, x2 ) .

(10)

Later, we use as coordinates, the center of mass of the heavy particles and the relative position, X=

x1 + x2 2

and z = x1 − x2 ,

we will see that these are the natural coordinates for our system. Let us search for the solution in the following form, 

∞

φ(x|x1 , x2 ) = N 0

ν2 dt Kt (x, x1 )e− h¯ t + h¯



∞ 0

 ν2 dt Kt (x, x2 )e− h¯ t , h¯

where −

∂Kt (x, y) h¯ 2 2 = 0, ∇ Kt (x, y) + h¯ 2m∗ x ∂t

(11)

is the usual heat equation in two-dimensions corresponding to a mass, 1 1 1 + , = m∗ m M

(12)

the solution of which is the well-known Gaussian when we demand that it goes to a delta-function as “time” variable t goes to zero. Here, with the mentioned divergence in mind, we choose a corrected mass m∗ for the heat kernel. Consequently, to get a solution (assuming the cut-off being removed), we need to satisfy the equation, 1 1 − λ h¯



∞

ν2

dtKt (x1 , x1 )e− h¯ t =

0

1 h¯



∞

ν2

dtKt (x1 , x2 )e− h¯ t .

0

As a result to cure the divergence we need to choose the coupling constant as 1 1 = h¯ λ



∞

2

dtKt (x1 , x1 )e− h¯ t ,

0

in which an arbitrary bound state energy  2 appears for the system. This is the binding energy between a heavy particle and a light particle interacting with a contact interaction. Instead of the coupling constant this is what we need to choose, or we think of this as the measured quantity. The energy of the system when the two heavy particles interacting with a light particle is to be determined from this

Born-Oppenheimer Type Approximation for a Simple Renormalizable System

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input. Here the dimensionless coupling constant is to be traded off with this binding energy, for this particular choice there is no need to introduce a finite part to the coupling constant. Thus we find a well-defined expression, 2m∗ 4π h¯ 2



∞ 0

ν2 dt −  2 t 2m∗ (e h¯ − e− h¯ t ) = t 4π h¯ 2



∞ 0

dt − ν 2 t − 2m∗ z2 4ht ¯ e h¯ . t

Consequently we find the equation to be satisfied for the binding energy of the total system, in this approximation, 

ν2 ln 2 



 √ 2m∗ ν|z| , = 2K0 h¯

where K0 denotes the well-known modified Bessel function. Let us keep this expression in mind and calculate the resulting normalized wave functions, this requires evaluating N2 h¯ 2

1=



∞

0

ν2

dt1 dt2 [Kt1 +t2 (x1 , x1 ) + Kt1 +t2 (x2 , x2 )]e− h¯ (t1 +t2 ) 

∞

+2 0

 ν2 dt1 dt2 Kt1 +t2 (x1 , x2 )e− h¯ (t1 +t2 ) ,

where we use the reproducing property of the Gaussian expression for the heat kernels. The integrals can be done easily by transforming to the variables, t = t1 + t2 , s = t1 − t2 1=

N2 h¯ 2



∞

ν2

dt te− h¯ t [Kt (x1 , x1 ) + Kt (x2 , x2 )] + 2

0



∞

 ν2 dttKt (x1 , x2 )e− h¯ t ,

0

or equivalently, 1=2

N2 h¯ 2



2m∗ 4π h¯

 

∞

ν2

dte− h¯ t +

0



∞

ν2

dte− h¯ t −

2m∗ z2 4ht ¯

 .

0

This gives us, 1 √ N = √ h¯ ν 2

=

4π h¯ # 2m∗ 

1

1+

√  √ 2m∗ 2m∗ ν|z|K ν|z| 1 h¯ h¯

,

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where K1 (.) refers to the modified Bessel function. Consequently, we have the normalized wave function, φ(x|x1 , x2 ) = #

√ 2m∗ h¯ ν √  √ 2m∗ 2m∗ h¯ ν|z|K1 h¯ ν|z| √1 2π

1+

 K0

×

√ √   2m∗ 2m∗ ν|x − x1 | + K0 ν|x − x2 | h¯ h¯

where K0 (.) again is the zeroth order modified Bessel function. Using the more 2 natural variables, X = x1 +x and z = x1 − x2 , we can rewrite 2 φ(x|X, z) = #

√1 2π

1+

√

2m∗ h¯ ν √

√ 2m∗ h¯ ν|z|K1

2m∗ h¯ ν|z|

 ×

 √   √ z z 2m∗ 2m∗ K0 ν|x − X + | + K0 ν|x − X − | 2 2 h¯ h¯   = A(z) η+ (ν(z), x − X + z/2) + η− (ν(z), x − X − z/2) . Here we use, 1 A(z) = √ 2π

√

√ η± (ν, x − X ± z/2) = K0

2m∗ ν# h¯

1 1+

√

2m∗ h¯ ν|z|K1

2m∗ !! ν !x − X ± h¯

√

2m∗ h¯ ν|z|



 z !! ! . 2

Let us emphasize again that here ν = ν(|z|), so it also depends on the distance |z| between the two heavy centers and this will be crucial in our computations. As we will see, heavy particle limit implies that |z| can be considered as small relative to the light particle length scale to be made precise below (in a subtle way). We now take a step back and write equation (3) in the new coordinates with the proposed wave functions in mind, 

h¯ 2  h¯ 2 2 ∇X − ∇z2 ψ(X, z)φ(x|X, z) 4M M  h 2   ¯ + − ∇x2 − λδ(x − x1 ) − λδ(x − x2 ) φ(x|X, z) ψ(X, z) 2m = Eφ(x|X, z)ψ(X, z).

−

(13)

Born-Oppenheimer Type Approximation for a Simple Renormalizable System

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We can write this as, 

−

   h¯ 2  h¯ 2  h¯ 2 2 h¯ 2 2 ∇X − ∇z2 ψ(X, z) φ(x|X, z) + ∇X − ∇z2 φ(x|X, z) ψ(X, z) − 4M M 4M M

h¯ 2 h¯ 2 ∇X ψ(X, z) · ∇X φ(x|X, z) − ∇z ψ(X, z) · ∇z φ(x|X, z) 4M M  h2   ¯ + − ∇x2 − λδ(x − x1 ) − λδ(x − x2 ) φ(x|X, z) ψ(X, z) = Eφ(x|X, z)ψ(X, z). 2m −

Let us consider the second term and using the decomposition of the wave function we see that the first Laplacian becomes (−

h¯ 2 2 h¯ 2 2 ∇X )φ(x|X, z) = A(z)(− ∇ )[η+ (ν(z), x − X + z/2) + η− (ν(z), x − X − z/2)] 4M 4M X

= A(z)(− = (−

h¯ 2 2 ∇ )[η+ (ν(z), x − X + z/2) + η− (ν(z), x − X − z/2)] 4M x

h¯ 2 2 ∇ )φ(x|X, z). 4M x

(14)

This term is actually singular and should be renormalized with the light particle solution. The other term, ∇z2 φ, requires more care, let us note that the functions η± depend on z in two ways, one is through the difference x − X ± z/2 the other is through the term ν(z). It is the first dependence that we should pay more attention to, let us divide the z derivatives acting on these functions as follows, ∇z = ∇z |ν + (∇z ν)

∂ . ∂ν

Therefore, we have ∇z2 = ∇z2 |ν + (∇z2 ν)

∂2 ∂ ∂ + (∇z ν)2 2 + 2(∇z ν) ∇z |ν . ∂ν ∂ν ∂ν

Again, the important term here is the first one, acting on the η± part of the wave function, ∇z2 |ν (η+ + η− ) =

1 2 (∇ η+ + ∇x2 η− ), 4 x

(15)

because, ∇z η± = ± 12 ∇x η± . Hence, we have this part, named as the singular part, to be separated from the heavy system as claimed, 

−

 h¯ 2 2 h¯ 2 2 ∇z φ(x|X, z) ∇ φ(x|X, z). =− sing M 4M x

(16)

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As a result, adding the terms (14) and (16) into the light particle solution, we have 

−

   h¯ 2  h¯ 2 2 h¯ 2  ∇X − ∇z2 ψ(X, z) φ(x|X, z) + φ(x|X, z) ψ(X, z) − ∇z2 reg 4M M M h¯ 2 h¯ 2 ∇X ψ(X, z) · ∇X φ(x|X, z) − ∇z ψ(X, z) · ∇z φ(x|X, z) 4M M    2 2 h¯ h¯ + − ∇x2 − ∇x2 − λδ(x − x1 ) − λδ(x − x2 ) φ(x|X, z) ψ(X, z) 2M 2m −

= Eφ(x|X, z)ψ(X, z).

Here, we named the remaining piece as the regular part. To simplify the computations we may assume ψ(X, z) = eiQX 1(z), that would essentially remove all the center of mass terms of the heavy particles from this expression, and we take the expectation value with the light degrees of freedom (note that we are projecting to the ground state of the light system for fixed centers, this is so much below from the other levels that we expect errors to be very small): 

    h¯ 2 h¯ 2  dxφ(x|X, z) − ∇z2 1(z) φ(x|X, z) + − ∇z2 φ(x|X, z) 1(z) reg M M  h¯ 2 h¯ 2 − 1(z)Q · ∇X φ(x|X, z) − ∇z 1(z) · ∇z φ(x|X, z) 4M M      2 h¯ h¯ 2 2 + dx φ(x|X, z) − ∇x2 − ∇x − λδ(x − x1 ) − λδ(x − x2 ) φ(x|X, z) 1(z) 2M 2m ( )* + −ν 2 (|z|)



= E−

after renormalization

h 2 Q2  ¯ 4M

1(z) = δE1(z).

Thus we should work out all the individual terms here, we will see that the effective potential does not only come from the expansion of the binding energy −ν 2 (|z|), the cross terms will also matter, especially the repulsive |z|12 generated by these terms will make the wave function to vanish mildly at the origin. As a result,  h¯ 2 2 h¯ 2 dxφ(x − X|z)∇z φ(x − X|z) · ∇z 1(z) ∇z 1(z) − ν 2 (|z|)1(z) − M M     h2  h¯ 2 ¯ dxφ(x − X|z) − ∇z2 dxφ(x − X|z)Q · ∇X φ(x − X|z) 1(z) + φ(x − X|z) − reg M 4M

−

= δE1(z).

Born-Oppenheimer Type Approximation for a Simple Renormalizable System

187

Note that we deliberately wrote φ(x|X, z) = φ(x − X|z) to emphasize this translational invariance of x, as a result ∇X φ(x − X|z) = −∇x φ(x − X|z), furthermore, we can redefine x → x − X, hence the apparent dependence on X disappears, we end up with, −

 h¯ 2 2 h¯ 2 ∇z 1(z) − ν 2 (|z|)1(z) − d 2 xφ(x|z)∇z φ(x|z) · ∇z 1(z) M M     h2  h¯ 2 ¯ Q · d 2 xφ(x|z)∇x φ(x|z) 1(z) φ(x|z) + + d 2 xφ(x|z) − ∇z2 reg M 4M )* + ( being an exact differential=0

= δE1(z).

(17)

In the subsequent pages we will complete this delicate calculation and obtain the effective potential. Let us reiterate that the seemingly negligible character of these terms are misleading, the resulting potential being rather singular and repulsive, this influences the behaviour of the wave function significantly around the origin. As we will see, δE is not small, it contains the energy of the light as well as the heavy system as it is written, it is the length scale associated to the heavy particle pair which is small compared to the light particle’s spread of the wave function when the light particle is bound to an isolated heavy particle. The reader who is interested in the resulting potential can skip the calculations presented in the next section; they are essentially of a technical nature.

3 Effective Potential for the Heavy Degrees of Freedom In this section, we provide some of the calculational details for the effective potential acting between the two heavy particles. Let us start with the regular expression 

 h2  ¯ dxφ(x|z) − ∇z2 φ(x|z), reg M

(18)

we recall that   φ(x|z) = A(z) η+ (x + z/2) + η− (x − z/2) .

(19)

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Let us explicitly write this expectation value as h¯ 2 − M =−



h¯ 2 M

   dxA(z)(η+ + η− ) ∇z2 A(z)(η+ + η− ) 



reg

  + (η+ + η− )∇z2 A(z) d 2 xA(z)(η+ + η− ) A(z) ∇z2 (η+ + η− ) reg )* + ( ( )* + (2) (1)



+ 2∇z A(z) · ∇z (η+ + η− ) )* + ( (3)

so after the integration the numbered terms produce the following results:    h¯ 2 2 A d 2 x(η+ + η− ) ∇z2 (η+ + η− ) reg M   h¯ 2 1 2 ∇ A (2) = − M A z  h¯ 2 A d 2 x(η+ + η− )∇z A · ∇z (η+ + η− ) (3) = − M/2 (1) = −

(20)

Let us recall that acting on η± the regular part of the z derivatives are given by   ∂ ∂ ∂2 + (∇z ν)2 2 + 2(∇z ν) · ∇z |ν η± , [∇z2 ]reg η± = (∇z2 ν) ∂ν ∂ν ∂ν 2

∂ 1 ∂ here in cylindrical coordinates z, θ , ∇z2 = ∂z 2 + z ∂z , note that not to complicate the notation we use the same letter z for the radial coordinate. This will not lead to any confusion, since pure z derivatives only appear in ∇z , and usually converted into radial derivatives for the particular solution. Note that this expression can be equivalently written

[∇z2 ]reg η±

  ∂ν 2 ∂ 2 ∂ ∂ν ∂ 2 +( ) zˆ · ∇z |ν η± . = (∇z ν) +2 ∂ν ∂z ∂ν 2 ∂z ∂ν

This is the form that we will be using. For some of the calculations to follow, we remind the equation satisfied by ν below, 

ν2 ln 2 



 √ 2m νz . = 2K0 h¯

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189

As√ a crucial step in our Born-Oppenheimer type approximation, we assume ∗ that 2m h¯ z  1, the validity of which is to be justified later. We note that z is small compared to the distance scale defined by the wave function of the √ light particle, 2m∗ /h¯ , however we √ have a stronger result, thanks to the form of the eigenvalue expression, indeed 2mνz/h¯ is small. If we assume ν to be large, we need a logarithmic increase on the other side as well, which turns out to be a consistent assumption. Then we expand the equation giving the bound state energy ν as a function of z, using the well-known short distance behaviour of K0 (x) ≈ − ln(xeγ /2) + . . ., 

ν2 ln 2 



√  √  2m∗ 2 2 2m∗ 2m∗ γ νze − 2 ln νzeγ −1 = −2 ln ν z + ..., 2h¯ 2h¯ 4h¯ 2

Then we collect the similar terms together,   1 2m∗ 2 2 1+ ν z + . . . 8 h¯ 2 √   √  2m∗ 2m∗ 2m∗ 2 2 γ ze − 2 ln zeγ −1 = −2 ln ν z + ..., 2h¯ 2h¯ 4h¯ 2 

2 ln

ν2 2

note that in the second logarithm term the factor eγ −1 can be replaced with eγ since this constant factor being multiplied with z2 is of lower order, as long as we keep up to the second order expansion, so to simplify the expression we change this term to eγ . Hence we can rewrite all of it as, ⎡ ⎤ √   2 1 + 2m2∗ ν 2 z2 + . . . 2m∗ ν 4 h γ ¯ ⎣ ⎦ ln ze . ln 2 = − 2h¯  1 + 2m2∗ ν 2 z2 + . . . 8h¯

By expanding again the denominator, 

ν2 ln 2 



 √   2m∗ 2m∗ 2 2 γ ze ν z . . . ln =− 1+ . 2h¯ 8h¯ 2

Let us reorganize this expression, to this purpose we define  ξ=

ν2 2

√

2m∗ z, h¯

then,  γ e 1 ln ξ = − ξ e−γ x ln x + . . . , 2 4

(21)

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H. Akbas and O. T. Turgut √

γ ∗ here we introduced x = 22m h¯ ze . This equation can be solved order by order by iteration, since x ln(x)  1, we see that ξ ≈ 2e−γ , to a first approximation, which defines our zeroth order expression. We shall content ourselves with computing the correct first order result, namely

h¯ h¯ 2 − e−2γ x ln x 2 ν ≈√ e 2 =√ γ ze ze 2m∗ 2m∗ γ 2

√ √    2m∗ e−γ 2m∗ z ln z + . . . , 1− h¯ h¯ 4

where we dropped the precise constant in the logarithmic term. Notice that we h¯ have the emergence of the characteristic length scale ζ0 = √2m , which describes ∗ the spread of the light particle-heavy particle bound state wave function in our calculations. Then we see immediately that indeed √ 2m ν(z)z  1, h¯ is true, hence we have the desired consistency for all our expansions below. Knowing the exact relation for ν, one can immediately find the resulting derivative of ν with respect to the variable z: ∂ν = ∂z

√  √ 2m∗ 2 2m∗ ν K νz 1 h¯ h¯ √  . − √ 2m∗ 2m∗ 1 + h¯ νzK1 νz h¯

(22)

We remind for the convenience of the reader the small argument expansion of Bessel function K1 (x), K1 (x) ≈

1 x 1 1 + x ln( ) + . . . 2 (x/2) 2 2

as x → 0+ .

(23)

The following next order term is of order x which we neglect, however this leads to an ambiguity in the logarithmic part, any multiplicative constant can be admitted in the argument since it only affects the result at the neglected next order. At small distance, using the above approximation, we find √  √ 1ν 1  2m∗ 2 3 ∂ν 2m∗ ≈− − νz , ν z ln ∂z 2z 8 2h¯ h¯

(24)

which should be further simplified by using the expansion of ν(z) to first order. For the time being we only need the zeroth order expressions, so we will keep it as it is given. We also need the expression for the second derivative, which we can find exactly and present in Appendix A, we write here the leading order expansion, using

Born-Oppenheimer Type Approximation for a Simple Renormalizable System

191

the radial variable z, as follows ∂ 2ν 3 ν ≈ + ... ∂z2 4 z2

(25)

Note that here we should expand ν to first order again to find a consistent expansion. We remark that for simplicity we kept m∗ as it is, it can be replaced with m at various places, but we may do this at the end. Let us go back and work out each term separately, as we numbered them, the first piece becomes the following;  2     1 ∂ν h¯ 2 2 ∂ν ∂ 2ν  ∂ν2 (η+ + η− ) A + 2 ∂ν + d 2 x(η+ + η− ) 2∇z ν · ∇z |ν ∂ν + M z ∂z ∂z ∂z   h¯ 2 = − A2 d 2 x ∇z ν · η+ ∇x ∂ν η+ − ∇z ν · η− ∇x ∂ν η− + ∇z ν · η− ∇x ∂ν η+ − ∇z ν · η+ ∇x ∂ν η− ( )* + M

(1) = −

(1b )



  1 ∂ν ∂ν ∂ 2ν  + (η+ + η− ) ∂ν2 (η+ + η− ) , + 2 ∂ν (η+ + η− ) + (η+ + η− ) z ∂z ∂z ∂z )* + ( ( )* + (1c )

2

(26)

(1d )

note that we used ∇z |ν η± = ± 12 ∇x η± in the first few terms. Shifting the integration variable by ±z/2, one can check that  d 2 x [∇z ν · η+ ∇x ∂ν η+ − ∇z ν · η− ∇x ∂ν η− ] = 0.

(27)

Let us consider the other cross term, by an explicit computation of the derivative terms we find out that,   ∂ν ∂ν d 2 x η− zˆ · ∇x ∂ν η+ − η+ zˆ · ∇x ∂ν η− ∂z ∂z  h¯ 2 2m∗ ∂ν = − A2 2 νz d 2 xη+ η− M ∂z h¯ √  √ h¯ 2 2m∗ ∂ν π h¯ 2m∗ 2m∗ νz . zK1 = − A2 2 νz h¯ M ∂z 2m∗ ν h¯

(1b ) = −

h¯ 2 2 A M



(28)

Up to this point we have an exact calculation. Let us now expand each term to leading order under the small z/ζ0 assumption as before, (1b ) ≈ −

1 m∗ 2 ζ 0 2m∗ 2 ∂ν z π h¯ 2 1 m∗ 2 1 m∗ h¯  A ν ≈ =  , (29) ≈ √ M ∂z ν 2m∗ 2M 2 M 2m∗ z 2M z

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H. Akbas and O. T. Turgut

which shows that it is a small perturbation to the leading terms. We now calculate the next term,  h¯ 2 2  1 ∂ν ∂ 2ν  d 2 x(η+ + η− )∂ν (η+ + η− ) (1c ) = − A + 2 M z ∂z ∂z  h¯ 2 2  1 ∂ν h¯ 2 2  1 ∂ν ∂ 2ν  ∂2ν  1 d 2 x∂ν (η+ + η− )2 = − =− A + 2 A + 2 ∂ν 2 2M z ∂z ∂z 2M z ∂z ∂z A   2 2 1 ∂ν h¯ ∂ ν =− (30) + 2 (A2 ∂ν A−2 ). 2M z ∂z ∂z Again this is an exact computation. Let us now find the leading contribution of this expression, we write below the expansion of A2 ∂ν A−2 to leading order, A2 ∂ν A−2 ≈ −

2 +... ν

(31)

The details of the calculation are given in the Appendix A, using our leading order ∂2ν expansions for ∂ν ∂z and ∂z2 given by (24) and (25) respectively, as well as the expansion above, we find h¯ 2 1 +... M 4z2

(1c ) ≈

(32)

In a similar way, the last part (1d ) is computed, −

h¯ 2 2 A M



∂ν ∂z

2  d 2 x(η+ + η− )∂ν2 (η+ + η− )

h¯ 2 2 A M





2 

1 2 ∂ (η+ + η− )2 − (∂ν (η+ + η− ))2 2 ν      h¯ 2 2 ∂ν 2 1 2 1 2 2 =− A ∂ − d x(∂ν (η+ + η− )) . M ∂z 2 ν A2

=−

∂ν ∂z



d 2x

Up to this point there is no approximation, we calculate the last part separately:  d 2 x(∂ν (η+ + η− ))2 =

=

 ν2 8ν 2 π 2 h¯ 2 ∞ dt1 dt2 t1 t2 e− h¯ (t1 +t2 ) 2 2 m h¯ 0   Kt1 +t2 (0) + Kt1 +t2 (z/2, −z/2) ν 2 π 2 h¯ 2

 t

ds(t 2 − s 2 )

 ∞

h¯ 2 m2∗ −t 0   Kt (0) + Kt (z/2, −z/2)

dte−

ν2 h¯ t

Born-Oppenheimer Type Approximation for a Simple Renormalizable System

=

4ν 2 π 2 h¯ 2

 ∞

3h¯ 2 m2∗

dt t 3 e−

ν2 h¯ t

193



 Kt (0) + Kt (z/2, −z/2)

0    ∞ 2 2 2m∗ z2 4ν π h¯ 2 e− νh¯ t 1 + e− 4ht ¯ = dt t 3h¯ 2 2m∗ 0

=

√  √  2m∗ ν 4ν 2 π h¯ 2h¯ 3 h¯ 3  2m∗ ν 3 + K z z . 3 h¯ h¯ 4ν 6 3h¯ 2 2m∗ ν 6

(33)

Finally, h¯ 2 (1d ) = − A2 M



∂ν ∂z

2 

√    √ 1 2 1 4ν 2 π h¯ 2h¯ 3 h¯ 3  2m∗ ν 3 2m∗ ν z K3 z . ∂ − 2 + 6 h¯ h¯ 2 ν A2 4ν 3h¯ 2m∗ ν 6

(34) As a result of a careful calculation, the details of which are given in Appendix A, we find the leading order of A2 ∂ν2 A−2 term A2 ∂ν2 A−2 ≈

6 +... ν2

(35)

After an expansion of all these terms, again using the small argument behavior of Bessel functions, to the leading order, we find that   3 h¯ 2 5 h¯ 2 1 − 2 + 2 ≈− (1d ) ≈ +... M 4z 3z M 12z2

(36)

As a result the leading term of the first part labeled as (1) above, is found as   h¯ 2 1 5 (1) ≈ − +... M 4z2 12z2

(37)

The next term here is labeled as (2), h¯ 2 1 2 h¯ 2 1 ∇z A = − − MA MA



 ∂2 1 ∂ A. + ∂z2 z ∂z

(38)

Therefore, we need these derivatives, we collect the calculations in Appendix A, and we just state the result of these expansions. The first derivative term, to the leading order, is   1 1 1 ∂z A ≈ − 2 + . . . . (39) A z 2z

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H. Akbas and O. T. Turgut

The next derivative becomes, again to the leading order, 1 2 3 ∂z A ≈ 2 + . . . A 4z

(40)

As a consequence, the total derivative of normalization can be expanded at the leading order as −

h¯ 2 1 h¯ 2 1 2 ∇z A ≈ − + .... MA M 4z2

(41)

There is one more term, which corresponds to the expression number (3):   ∂ν ∂ν (η+ + η− ) d 2 x(η+ + η− ) ∂z |ν (η+ + η− ) + ∂z   h¯ 2 = − 2A∂z A d 2 x η+ ∂z |ν η+ + η− ∂z |ν η− + η+ ∂z |ν η− + η− ∂z |ν η+ M  ∂ν + (η+ + η− )∂ν (η+ + η− ) . ∂z

(3) = −

h¯ 2 2A∂z A M



To simplify our calculations, we make the following observation:  2 2 + η− ) = 0. d 2 x(η+

∂z |ν

(42)

As a result, equation (3) becomes (3) = −

h¯ 2 2A∂z A M



  1 ∂ν ∂ν (η+ + η− )2 . d 2 x ∂z |ν η+ η− + 2 ∂z

(43)

We divide the above term into two parts, the first of which can be easily found as (3a ) = − =− ≈

h¯ 2 ∂ν A∂z A ∂ν M ∂z

 d 2 x(η+ + η− )2 = −

1 h¯ 2 ∂ν A∂z A ∂ν 2 M ∂z A

h¯ 2 1 ∂ν 1 ∂z A A2 ∂ν 2 MA ∂z A

h¯ 2 1 + ..., M 2z2

(44)

Born-Oppenheimer Type Approximation for a Simple Renormalizable System

195

where in the last line we present the short distance expansion of this part. In a similar way, we work out the second part, h¯ 2 (3b ) = − 2A∂z A∂z |ν M =−

 d 2 xη+ η−

π h¯ h¯ 2 2A∂z A ∂z |ν M 2m∗

 √ √ 2m∗ 2m∗ νz zK1 h¯ ν

(45)

This expression is worked out in Appendix A, its expansion under the small z/ζ0 approximation leads to (3b ) ≈ O

m

∗ 2

M

ν



+ ...,

(46)

hence it is of lower order. Consequently the whole sum for (3) becomes in the leading approximation (3) ≈

h¯ 2 1 +... M 2z2

(47)

We will now consider the cross term which contains first order derivative of the heavy particle wave functions in the averaged out Schrödinger equation given in equation (17): (∗) := −

h¯ 2 M

 d 2 xφ(x|z)∇z φ(x|z) · ∇z 1(z).

(48)

We evaluate each term here, (∗) = − =−

h¯ 2 M

=−

h¯ 2 M

=−

h¯ 2 M

  ∂ ∂ d 2 xA(η+ + η− ) (η+ + η− ) A + A (η+ + η− ) ∂z ∂z    2 h¯ ∂1 2 ∂ν ∂1 2 d 2 x(η+ + η− ) ∂z |ν + ∂z A − 2A ∂ν (η+ + η− ) ∂z A M ∂z ∂z    2 h¯ ∂1 2 1 1 ∂ν 2 2 2 + η− ) + ∂z |ν η+ η− + ∂z A∂z 1 − 2A ∂z |ν (η+ ∂ν (η+ + η− )2 d2x A M ∂z 2 2 ∂z √ √     ! 2 2m∗ 1 h¯ 2 π h¯ ∂ ! 2m∗ 1 ∂ν ∂z A∂z 1 − ∂z 12A2 |z|K1 ∂ν 2 . νz + ! A M 2m∗ ∂z ν ν 2 ∂z A h¯

h¯ 2 ∂1 2 M ∂z



We now evaluate the derivative and use the Bessel function identity xKn−1 (x) − xKn+1 (x) = −2nKn (x)

(49)

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H. Akbas and O. T. Turgut

for the resulting K2 (·), to arrive at   √ √ h¯ 2 2 2m∗ 2m∗ ¯ 2 πh νz ∂z A + A K1 − ∂z 1 h¯ M A 2m∗ ν √ √ √ √  2m∗ 2m∗ ν  2m∗ ν −1 2m∗ − z z K1 νz ν h¯ h¯ h¯ √ √  √  2m∗ 2m∗ ν 2m∗ ∂ν 2 1 z A ∂ν 2 − K0 νz + (50) ν ∂z A h¯ h¯ After canceling K1 -terms, we expand this term to leading order, using the result for A2 ∂ν A−2 that we found before, we find, m∗ 2 ν z∂z 1 ln (∗) ≈ C1 M

√  2m∗ νz + . . . , h¯

(51)

where C1 is a constant that we can explicitly compute. This is quite a remarkable result, we see that there is no 1z term multiplying ∂z 1 term. To get a well-defined operator we need to symmetrize this term, and the absence of 1z implies that such a symmetrization will not lead to another z12 correction to the potential term.

4 Energy Levels of the System In the previous section we worked out all the leading terms for our system. We finally write down the leading order (effective) Schrödinger equation for the heavy degrees of freedom,  h¯ 2 2 h¯ 2 2 ∇ d 2 xφ(x|z)∇z φ(x|z) · ∇z 1(z) − 1(z) − ν (|z|)1(z) − M z M    h2  ¯ φ(x|z) 1(z) + d 2 xφ(x|z) − ∇z2 reg M ≈−

 h   h2 1  2 h¯ 2 2 ¯ ¯ ∇z 1(z) − √ + . . . 1(z) + + . . . 1(z) + . . . = δE1(z). γ 2 M M 12z 2m∗ ze

This is a very interesting result, since the leading energy is not only given by the potential term that we have in the usual Born-Oppenheimer approximation but gets a nonperturbative contribution from the effective potential generated by the kinetic energy operator acting on the wave function of the light degrees of freedom. The energy that one finds is by no means small, it grows with the mass ratio of the heavy one to the one of the light particle, namely M/m. Nevertheless, it has still an order by order expansion, the real expansion is being done with respect to

Born-Oppenheimer Type Approximation for a Simple Renormalizable System

197

the smallness of the average separation of the heavy particles relative to the spread of the light particle’s true wave function for the given self-consistent configuration (which is again determined by z, albeit at a lower order). The solution to the above equation is well-known in terms of the generalized Laguerre polynomials, being familiar from the hydrogenic atoms. In the ground state wave function we have no dependence on the angular coordinate therefore no angular momentum contribution. We write the resulting equation as   h¯ 2 ∂ 2 β2 1 ∂ α − − 2 1(z) − 1(z) = δE1(z). + M ∂z2 z ∂z z z

(52)

By defining the following parameters and the new coordinate r δE = − r=

E0 K 2 (n)

z z0 K(n)

E0 =

where z0 =

where

h¯ 2 Mα

α2 M 4h¯ 2 and

also

β 2 = 1/12

2h¯  α= √ , 2m∗ eγ

(53)

we have a transformed equation: 

 β2 1 1 ∂ K ∂2 − 2 + − + Rn = 0. ∂r 2 r ∂r r r 4

(54)

Considering the r → ∞, one can see that we can write the solution in the form Rn = C(n)e− 2 ψ(r), r

(55)

and putting this back into the equation, ψ(r) must satisfy the equation ∂ 2ψ + ∂r 2



   ∂ψ 1 1 β2 1 −1 + K− ψ − 2 ψ = 0. r ∂r r 2 r

(56)

We investigate the small r behavior of the equation, and look for a solution by plugging r δ into the most singular part of the equation (for r → 0+ ): 1 δ(δ − 1)r δ−2 + δr δ−1 − (β 2 )r δ−2 = 0. r We can easily see that δ2 = β 2

(57)

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H. Akbas and O. T. Turgut

and we admit the regular solution, δ = β, so that the wave function ψ(r) should be of the form ψ(r) = r β g(r)

(58)

for a regular function g(r). Let us plug this back into equation (56), to get r

  1 ∂g(r) ∂ 2 g(r) + K − − β g(r) = 0. + (2β + 1 − r) ∂r 2 ∂r 2

(59)

A general solution which is regular at r = 0 of this equation is 2β (r). K− 12 −β

g(r) = L

(60)

where Lαγ refers to the generalized Laguerre functions. In the absence of restrictions to the parameters the Laguerre functions grow exponentially at infinity; the normalizability of the wave function imposes the restriction on the allowed values as K(n) −

1 − β = n, 2

for any integer n ≥ 0. Thus the resultant wave function becomes r

Rn = C(n)r β e− 2 L2β n (r).

(61)

Consequently the energy becomes En = −

α2 M 4h2 ¯

 n+

1 1 2

+β

2 = −

2M m∗

e2γ

2  2 . 1 n+ 2 +β

When n = 0 this gives the true ground state Egs = −

2M e−2γ  2 , m (1 + 2β)2

where we replaced m∗ with m to have the correct leading order expression. Moreover, we have the ground state wave function r

R0 = Cr β e− 2

and here

r=

2z . z0 (1 + 2β)

Born-Oppenheimer Type Approximation for a Simple Renormalizable System

199

The properly normalized heavy particles wave function becomes = 1(z) =

2 π&(2β + 2)z02 (1 + 2β)2



2z z0 (1 + 2β)

β e

z 0 (1+2β)

−z

.

(62)

These are finally our main results. It is essential to note that the energy goes with the M/m ratio. This is important because, as we let the heavy particle mass becomes infinite, the two heavy centers coalesce into a single center, renormalization that we perform does not allow this configuration, thus this limit leads to a divergence. A crucial point to check for the consistency is the expectation value of the separation z within our approximation, that is with the solution we have found for the heavy particle separation. Not surprisingly we find  = 0 eγ

∞  2π

zdzdθ |1(z)|2z

0

  h¯ &(2β + 3) 2m∗ (1 + 2β) √ 4 &(2β + 2) M 2m∗    γ m∗ e &(2β + 3) (1 + 2β) ζ0 , = 2 &(2β + 2) M =

(63)

where ζ0 refers to the characteristic length scale of the light particle binding energy introduced in renormalization (it is the binding to a single heavy center). More properly we should compute the expectation value of the light particle binding energy. Which means we should find the leading order expectation value of 2 h¯ . − ν2 = − √ 2m∗ eγ z

(64)

As a result, the expectation value of the binding energy is: G

H M e−2γ  2 M e−2γ  2 − ν 2 = −4 ≈ −4 = 2Egs . 2 m∗ (1 + 2β) m (1 + 2β)2

(65)

This shows that the binding is very large due to the light particle, the heavy centers are confined into a small region and thus they have very large kinetic energy due to extreme localization of the position, the sum comes out to be about half the value of this binding energy, as a result the system is strongly bound. If we think of this in terms of the time scales, our intuition is correct, the light particle is responding much more quickly than the heavy ones, even though they have comparable energies, thanks to the large mass difference. It remains as a challenge to corroborate this approach and gain a deeper understanding of the dynamics by means of a many body approach. We are not able to do it at the moment.

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In the last Appendix we compute the next order corrections to the energy coming from the higher order expansions. These are not computed completely but some terms are found to illustrate the consistency of our approximations.

5 Conclusions In a singular system which has two heavy and one light particle, we can apply a Born-Oppenheimer type approximation under the assumption that the spread of the wave function of the light particle is much larger than the spread of the heavy particles relative coordinate wave function, here the relative coordinate is characterized by the variable z. Contrary to the usual Born-Oppenheimer expansions, we find that the contributions of the heavy degrees of freedom to the total energy of the system is very large, of order M/m, same order as the light one, moreover correction terms are also large. Yet there is still an order by order expansion of the total energy of the system in terms of the expectation values of increasing powers of z and logarithms of z in a proper sense. It would be very important to understand this from a many body perspective as we have done in our previous work for a one dimensional version of this problem. Finally, we remark that from a physical point of view the model is not completely consistent, we assume all particles are nonrelativistic, so light-heavy binding satisfies,  2  mc2 . As we show, the resulting total ground state energy comes out as (M/m) 2  Mc2 , yet we cannot conclude that the binding energy of the light particle remains much smaller than mc2 . Of course such inherent consistency problems appear from the beginning, renormalized problem has infinite kinetic energy, thus hinting that a realistic treatment of kinetic energy is better suited in these problems. Acknowledgements O. T. Turgut would like to thank F. Erman, J. Hoppe, A. Moustafazadeh, M. Znojil for discussions. Part of this work is completed while the second author is visiting Mathematics Department of KTH, Stockholm, he is grateful to J. Hoppe for this extremely kind invitation to work there. Our interest in this problem is rekindled thanks to the encouragements we received from A. Michelangeli while the second author was in the conference organized in INdAM, Sapienza, Rome. We would like to thank him for his interest in our work and for the kind invitation to contribute to this unique collection.

Appendix A: Small Distance Expansions Here, we provide the detailed computations of all the derivative terms and their expansions. Short distance expansions are used to find the leading order solution within this modified Born-Oppenheimer approximation. For all the approximations

Born-Oppenheimer Type Approximation for a Simple Renormalizable System

201

below we make use of the Bessel function expansion for n ≥ 2, Kn (x) ≈

1 1 (n − 1)! +... 2 (x/2)n

(66)

for small x and also the expansion of K0 (x) that we mentioned in the text. Moreover in some cases K1 (x)’s next order term in the expansion may have importance and it is given by K1 (x) ≈

1 x 1 1 + x ln( ) + . . . 2 (x/2) 2 2

as x → 0+ ,

(67)

In our calculations we need the second derivative of the square-root of the binding energy ν(z), √  √ 2 2m∗ ∂ν 2m∗ ν ∂z K1 ∂ 2ν h¯ h¯ νz √  = − √ 2m∗ ∂z2 ∗ 1 + 2m h¯ νzK1 h¯ νz  √   √   2m∗ 2m∗ 2m∗ 2 ∂ν √ h¯ z ν + ν K νz + K νz 0 1 2 h¯ h¯ ∂z 2m∗ νz h  √  + ¯ √ 2m∗ 2m∗ 1 + h¯ νzK1 h¯ νz +

   √ 2 2m∗ z ∂ν + ν K νz 1 h¯ ∂z  √ 2 √ 2m∗ ∗ 1 + 2m νzK νz 1 h¯ h¯

2m∗ 2 ν h¯ 2

  √  2m∗ ν 3 |z| z ∂ν + ν K νz 1 h¯ ∂z −  √ 2 √ 2m∗ 2m∗ 1 + h¯ νzK1 h¯ νz    √ √ 2m∗ 2m∗ h¯ νz + √ νz K1 K0 h¯ h¯ 2m∗ νz √  √ 2 2m∗ ∂ν 2m∗ ν K νz 1 h¯ h¯ ∂z √  = − √ 2m∗ 2m∗ 1 + h¯ νzK1 νz h¯   √    √ 2m∗ 2m∗ 2m∗ 2 ν z ∂ν + ν K0 νz + √2mh¯ νz K1 νz 2 h h ∂z ¯ ¯ h ∗  √  + ¯ √ 2m∗ 2m∗ 1 + h¯ νzK1 h¯ νz 3

(2m∗ ) 2 h¯ 3

√    2m∗  √ ν 3 z z ∂ν + ν K νz 1 h¯ ∂z 2m∗ νz K 0  √ 2 √ h¯ 2m∗ ∗ 1 + 2m h¯ νzK1 h¯ νz 3

−

(2m∗ ) 2 h¯ 3

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√  2m∗ ∂ν 2m∗ ν K νz 1 ∂z h¯ h¯ 1 ∂ν √  − − √ 2m∗ 2m∗ z ∂z 1 + h¯ νzK1 h¯ νz  √  ⎡ 2m∗ 2m∗ 2 ν z ∂ν + ν K0 νz 2 h ∂z ¯ h √  ⎣1 − + ¯ √ 2m∗ 2m∗ 1 + h¯ νzK1 νz h¯ √

=

√  ⎤ √ 2m∗ 2m∗ h¯ νzK1 h¯ νz  √  ⎦ √ 2m∗ ∗ 1 + 2m νzK νz 1 h¯ h¯

√  √  1 2m∗ 3 2m∗ 1 2m∗ ν νz =− ν z ln ν − − 2 h¯ 2z 8 h¯ 2 2h¯ √ √   h¯ 1 2m∗ 2m∗ νz ln νz + √ 2h¯ 2m∗ νz 4 h¯  √ ν 2m∗ 1 2m∗ 3 + 2+ νz ν ln 2z 8 h¯ 2 2h¯

    √ √ 1 2m∗ 3 2 1 2m∗ 2 2 1 2m∗ 2 ν 2m∗ 2m∗ − νz 1 − νz + ν ν z ln ν z ln 2 h¯ 2 2 8 h¯ 2 2h¯ 4 h¯ 2 2h¯  √    √ 1 1 2m∗ 2 2 2m∗ 2m∗ × − ln νz − γ − νz ν z ln 2h¯ 2 8 h¯ 2 2h¯  √ 2m∗ 3 2m∗ 3 2m∗ 3 ν = νz + .... + a ν + a ν ln 1 2 2 2 4 z2 2h¯ h¯ h¯

As a result we see that 1 ∂ 2ν 3 = 2 +O ν ∂z2 4z



   √ 2m∗ 2 2m∗ 2 ν|z| , + O ν ν ln 2h¯ h¯ 2 h¯ 2

2m∗

(68)

where a1 , a2 are constants that we can find explicitly, moreover we did not further simplify the ν terms to see the pattern more explicitly, otherwise they should also be expanded. In our computations we also need various derivatives of the normalization constant A. We will now start with the derivative of the inverse with respect to ν only keeping z constant, √ √     √ √ 2m∗ 2m∗ 2m∗ h¯ 2 1 2m∗ 2 νzK1 νz + 2 zK1 νz − 3 1+ 2m∗ h¯ h¯ h¯ h¯ ν ν     √ √ 1 2m∗ 2 2m∗ 2m∗ h¯ − z K0 νz + √ K1 νz ν h¯ 2 h¯ h¯ 2m∗ νz √     √ √ 2m∗ 2m∗ 2m∗ 1 2m∗ 2 2 h¯ 2 z K0 νzK1 νz − νz . − 3 1+ = 2π 2m∗ ν h¯ 2 h¯ h¯ h¯ ν

∂ν A−2 = 2π

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Let us now write the expression we need, and then expand for small z; A2 ∂ν A−2

 √ 2m∗ K0 ν|z| 2m 2 h ¯ ∗  √ = − − νz2 2  √ 2m∗ ν ∗ h¯ 1 + 2m h¯ νzK1 h¯ νz

    √  √ 2 1 2 2m∗ 1 2m∗ 2 2 2m∗ 2m∗ − νz νz − γ 1 − νz + . . . ν z ln − ln ν 2 2h¯ 4 h¯ 2 2h¯ h¯ 2  √ 2 2m∗ 2m∗ 2m∗ ≈ − + b1 2 νz2 + b2 2 νz2 ln νz + . . . , (69) ν 2h¯ h¯ h¯

≈−

where b1 , b2 again are some constants we can determine. Furthermore, we did not expand the ν terms: in principle they should also be expanded, however this form is better since in our computations some combinations of ν’s will then cancel to give us a simpler result. Let us now consider the following second ν derivative of A−2 that we need in our calculations: √ √   √ √   2m∗ 2m∗ 2m∗ h¯ 2 6 2 2m∗ 1 + νzK νz − zK νz 1 1 2m∗ ν 4 ν 3 h¯ h¯ h¯ h¯  √ √   2 2m∗ 2m∗ 2m∗ h¯ + 2 2 z2 K0 νz + √ νz K1 ν h¯ h¯ h¯ 2m∗ νz √ √   3 1 (2m∗ ) 2 3 1 2m∗ 2 2m∗ 2m∗ νz + νz . z K1 + 2 2 z K0 ν h¯ 3 h¯ h¯ ν h¯

∂ν2 A−2 = 2π

We now write the combination we need and then expand as usual keeping ν terms as they are, so as to simplify later calculations:

A2 ∂ν2 A−2

  √ √ 3 2 2m∗ 2m∗ νz3 (2m∗3) K1 K0 νz νz h h 6 2m ¯ ¯ ∗ h ¯  +   √ √ = 2 + 3z2 2  √ √ 2m∗ 2m∗ 2m∗ ν ∗ h¯ νzK νz νzK νz 1 + 2m 1 + 1 1 h¯ h¯ h¯ h¯

   √  √ 6 1 2m∗ 2 2 3 2 2m∗ 2m∗ 2m∗ ≈ 2 + z νz − γ 1 − νz ν z ln − ln ν 2 2h¯ 4 h¯ 2 2h¯ h¯ 2 √ √    √ 3  νz3 (2m∗ ) 2 2m∗ ν 2 z2 2m∗ νz 2m∗ 2m∗ h¯ + ln νz 1 − νz √ ln + 2h¯ 2h¯ 2h¯ 2m∗ νz 4h¯ 3 4h¯ 2  √ 6 2m∗ 2m∗ 2m∗ ≈ 2 + c1 z2 2 + c2 z2 2 ln νz + . . . , (70) 2h¯ ν h¯ h¯

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where we have well-defined constants c1 , c2 , and ν is kept as it is without further expansion. We now write down the combinations we encounter in our computations, one of them is this expression:  ∂ν 2 1 2 2 −2 A ∂ν A ∂z 2 √ √  2     ν 6 2m∗ νz 2m∗ νz h¯ 2 2m∗ ν 4 2m∗ z2 2m∗ z2 ≈− + ln + + ln + . . . + . . . 2M 4z2 2h¯ ν2 2h¯ 8h¯ 2 h¯ 2 h¯ 2   √   2 m∗ 2 2m∗ m∗ 2 h¯ 3 ν +O ν ln νz + ... +O (71) ≈− M 4z2 M M 2h¯ −

h¯ 2 M



We write the final expanded version that we use. Note that here instead of the constants in front we indicate the order of magnitude of the term in the expansion with big-O symbol. We also need the combination below, hence we write it explicitly and use the expansions we have obtained:  1 ∂ν ∂ 2ν + 2 A2 ∂ν A−2 z ∂z ∂z     √ √ ν 3ν h¯ 2 2m∗ ν 3 2m∗ ν 3 2m∗ ν 3 2m∗ νz 2m∗ νz − 2 − + + . . . ≈− ln + + ln 2M 2z 2h¯ 4z2 2h¯ 8h¯ 2 h¯ 2 h¯ 2    √ 2m∗ νz 2 2m∗ 2m∗ + ... × − + 2 νz2 + 2 νz2 ln ν 2h¯ h¯ h¯  √  m  h¯ 2 1 m∗ 2 2m∗ ∗ 2 ≈ ν +O ν ln νz +... +O (72) M 4z2 M M 2h¯ −

h¯ 2 2M



Note that the logarithmic terms by themselves are ambiguous, one should keep the next order term to find the exact expansion, however we will not be able to compute all the corrections even at the first order due to complicated nature of these expressions, this is why we are essentially emphasizing the order of each term rather than the precise numerical factors. Our second purpose is to find derivatives of A, with respect to z, to begin with, we look at the first derivative of this normalization constant; 1 ∂z A = √ 2π

1 − √ 2 2π

1 + √ 4 2π

√

√ 2m∗ ∂ν # h¯ ∂z

√

1 1+

√

2m∗ h¯ νzK1

√

2m∗ h¯ νz



 √  ⎤ 2m∗ 2m∗ ∂ν z + ν K νz 1 ⎥ ⎢ h¯ 2m∗ ⎢ h¯ ∂z ⎥ ν⎣ 3 ⎦   √ √ h¯ 2 2m∗ ∗ 1 + 2m νzK νz 1 h¯ h¯ ⎡√

⎡

2m∗ ⎢ ν⎢ ⎣ h¯ 1+





2m∗ νz z ∂ν ∂z + ν h¯ 2 √  3 √ 2 2m∗ 2m∗ h¯ νzK1 h¯ νz

⎤

 √ √   ⎥ 2m∗ 2m∗ ⎥ K0 νz + K νz . 2 ⎦ h¯ h¯

(73)

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205

We consider the following term ⎡ √ 1 2m∗ ⎢ √ ν⎢ ⎣ 4 2π h¯



1+

⎡

√ 1 2m∗ ⎢ = √ ν⎢ ⎣ 2 2π h¯

⎤



2m∗ νz z ∂ν ∂z + ν h¯ 2 √  3 √ 2 2m∗ 2m∗ h¯ νzK1 h¯ νz



1+

   √ √ ⎥ 2m∗ 2m∗ ⎥ K0 νz + K νz 2 ⎦ h¯ h¯



2m∗ νz z ∂ν ∂z + ν h¯ 2 √  3 √ 2 2m∗ 2m∗ νzK νz 1 h¯ h¯

⎤

 √ √   ⎥ 2m∗ νz 2m∗ νz hK ¯ 1 ⎥ K0 + √ ⎦ h¯ h¯ 2m∗ νz

(74)

.

Here we recognized a term corresponding to ∂ν ∂z . As a result of these computations, the first derivative of normalization constant is found to be √ 1 2m∗ ∂ν 1 # ∂z A = √ √  √ 2π h¯ ∂z 2m∗ ∗ 1 + 2m h¯ νzK1 h¯ νz ⎡ √ 1 2m∗ ⎢ + √ ν⎢ ⎣ 2 2π h¯ 1+



⎤



2m∗ ν|z| z ∂ν ∂z + ν h¯ 2  3 √ √ 2 2m∗ 2m∗ νzK νz 1 h¯ h¯

⎥ ⎥ K0 ⎦

 √ 2m∗ νz . (75) h¯

The term we are interested in is given below, by using the exact expression for A again, we get 1 A



⎡



1 1 ∂ν 1 ⎣ ∂z A = + z zν ∂z 2z 1+





⎤

2m∗  √ νz z ∂ν ∂z + ν 2m∗ h¯ 2 ⎦   νz . K0 √ √ 2m∗ 2m∗ h¯ h¯ νzK1 h¯ νz

Let us consider its small z expansion, we therefore find 1 A



1 ∂A z ∂z



⎡ 1 ⎣ 1 ∂ν  + = νz ∂z 2z 1+





⎤

2m∗  √ νz z ∂ν ∂z + ν 2m∗ h¯ 2 ⎦ √  K0 νz √ 2m∗ 2m∗ h¯ νzK νz 1 h¯ h¯

 √  1 1 2m∗ 3 2m∗ ν ≈ νz ν z ln − − νz 2z 8 h¯ 2 2h¯     √ √ 1 2m∗ 1 2m∗ 3 2 1 2m∗ 2 2 ν 2m∗ 2m∗ + − νz 1− νz ν ν z ln ν z ln 4 h¯ 2 2 8 h¯ 2 2h¯ 4 h¯ 2 2h¯

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   √ 2m∗ νz − γ × − ln 2h¯     √ 1 2m∗ 2 2m∗ 2m∗ 2 ≈ − 2 + b1 νz + b2 ν ln ν + ..., 2h¯ 2z h¯ 2 h¯ 2 where the coefficients b1 , b2 can be calculated from above, since we will not compute all the higher order corrections their precise values are not important. We may therefore write the required expansion as h¯ 2 1 − MA



1 ∂A z ∂z



h¯ 2 1 +O ≈ M 2z2



m∗ 2 ν ln M

√

2m∗ νz 2h¯

 +O

m

∗ 2

M

ν



+ . . . (76)

In the last part, we look at the more complicated derivative expressions of A: ∂z A2 =

1 2m∗ ∂ν  2ν 2π h¯ 2 ∂z 1 + 3

−

1 (2m∗ ) 2 2π h¯ 3

1

√  √ 2m∗ 2m∗ h¯ νzK1 h¯ νz  √   2m∗ z ∂ν + ν K1 νz h ∂z ¯ ν2  √ 2 √ 2m∗ ∗ 1 + 2m h¯ νzK1 h¯ νz

1 (2m∗ )2 3 + ν z 2π h¯ 4 1+

   √ √   z ∂ν ∂z + ν 2m∗ 2m∗ h¯ √ K K νz + νz 0 1 √ 2 √ h¯ h¯ 2m∗ νz 2m∗ 2m∗ h¯ νzK1 h¯ νz

1 2m∗ ∂ν 1 √ 2ν  √  2m∗ 2π h¯ 2 ∂z 1 + 2m∗ νzK 1 h¯ h¯ νz   ∂ν + ν √  2 z 2m∗ 1 (2m∗ ) 3 ∂z ν z K + νz √  √ 2 0 2π h¯ 4 h¯ 2m∗ ∗ 1 + 2m h¯ νzK1 h¯ νz

=

 √ √    2m∗ 2m∗ ν 1 2m∗ 3 1 2m∗ 2 2 1 2m∗ ν − ν z ln ν z ln − νz 1 − νz 2π h¯ 2 2z 8 h¯ 2 2h¯ 4 h¯ 2 2h¯ √   √    2 2m∗ 2m∗ 1 2m∗ 3 2 1 2m∗ 2 2 1 (2m∗ ) 3 ν ν z ν z ln ν z ln − νz 1− νz + 8π h¯ 4 2 8 h¯ 2 2h¯ 2 h¯ 2 2h¯    √ 2m∗ νz − γ × − ln 2h¯    √  2 2 (2m∗ )2 4 (2m∗ ) 4 1 2m∗ ν 2m∗ =− ν z +O ν z ln +O νz + ... 2π h¯ 2 2z 2h¯ h¯ 4 h¯ 4 =

(77)

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207

In our computations this term appears in the following combination, which comes from η+ η− term and it is denoted by (3b ) in the main text: √ √  h¯ 2 2m∗ 2m∗ π h¯ zK1 − 2A∂z A ∂z |ν νz = M 2m∗ ν h¯ √ √  h¯ 2 2m∗ 2m∗ π h¯ K1 =− 2A∂z A νz M 2m∗ ν h¯  √    √ 2m∗ 2m∗ h¯ 2m∗ K1 z K0 νz + √ νz − h¯ h¯ h¯ 2m∗ νz   √  h¯ 2 2 π h¯ 2m∗ 2m∗ zK0 νz = ∂z A A2 h¯ M A 2m∗ h¯ √      m  m∗ 2m∗ 4 2 2m∗ νz m∗ 2 ∗ 2 ≈O ν +O ν ln +O ν z M M 2h¯ M h¯ 2  √  m∗ 2m∗ 4 2 2m∗ νz +O ν z ln . M h¯ 2 2h¯ The most complicated combinations come from the second derivative of the normalization constant A with respect to z. This straightforward, yet long computation can be simplified into nicer blocks by using some of the relations we have found. Let us recall the first derivative we have, 1 ∂z A = √ 2π

√ 2m∗ ∂ν # h¯ ∂z

1 1+

⎡ √ 2m∗ ⎢ 1 ν⎢ + √ ⎣ 2 2π h¯ 1+

√

2m∗ h¯ νzK1

√

2m∗ h¯ νz



 ⎤



2m∗ νz z ∂ν ∂z + ν h¯ 2 √  3 √ 2 2m∗ 2m∗ νzK νz 1 h¯ h¯

⎥ ⎥ K0 ⎦

√  2m∗ νz . h¯

We compute the second derivative, after tedious calculations and some simplifications, we arrive at 1 ∂z2 A = √ 2π

√

2m∗ ∂ 2 ν # h¯ ∂z2

1 1+

√ 2m∗ ∂ν 1 + √ 2 2π h¯ ∂z  1+

√

2m∗ h¯ νzK1



√

2m∗ h¯ νz





2m∗ νz z ∂ν ∂z + ν h¯ 2 √  3 √ 2 2m∗ 2m∗ νzK νz 1 h¯ h¯

K0

 √ 2m∗ νz h¯

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√ 2m∗ ∂ν 1 + √ 2 2π h¯ ∂z  1+





2m∗ νz z ∂ν ∂z + ν h¯ 2 √  3 √ 2 2m∗ 2m∗ νzK νz 1 h¯ h¯

   2 2ν ∂ν ∂ν ∂ 2 2 2 √ √   2z ∂z + νz ∂z2 + 5νz ∂z + ν 1 2m∗ 2m∗ 2m∗ + √ νK0 νz  3  √ √ h¯ h¯ 2 2 2π h¯ 2 2m∗ ∗ νzK νz 1 + 2m 1 h¯ h¯ 3 + √ 4 2π

√  √ 2 2m∗ 2m∗ 2m∗ ν K0 νz h¯ h¯ h¯ 2 

 2 2m∗ 2 2 ν z z ∂ν +ν 2 ∂z h¯ √  5 √ 2 2m∗ ∗ 1 + 2m νzK νz 1 h¯ h¯

.

What we need is the following combination:   ⎤ ∂ν √  + ν z z ∂z 1 1 ∂ν 2m∗ ⎣ 1 2 2m∗ ⎦   √ ∂ A= νz + K0 √ 2m∗ A z ν ∂z2 2 ∂z h¯ 2 ∗ h¯ 1 + 2m h¯ νzK1 h¯ νz   ⎡ ⎤ z z ∂ν ∂z + ν 1 ∂ν 2m∗ ⎣  ⎦  √ + √ 2m∗ 2m∗ 2 ∂z h¯ 2 1 + h¯ νzK1 h¯ νz ⎤ ⎡  2 ∂ν 2 ∂ν 2 ∂2ν + ν2  √ + 5νz + νz 2z 2m∗ ⎢ 2m∗ ∂z ∂z 1 ∂z2 ⎥  ⎦ √ νz + K0 √ ⎣  2 2m 2m 2 ∗ h¯ h¯ 1 + h¯ ∗ νzK1 h¯ νz ⎡

∂ 2ν

⎡

+

3 4

⎤

2  2m∗ 2 2  √ 2 ν z z ∂ν +ν 2 ∂z 2m∗ ⎢ 2m∗ h¯ ⎥ K νz . ⎣ ⎦ 0    √ √ 2 h¯ h¯ 2 2m∗ ∗ νzK νz 1 + 2m 1 h¯ h¯

(78)

In our approach, we need the small-z expansion of A1 ∇z2 A which, in cylindrical coordinates, can be expressed with the first and second z derivatives of A. As one may appreciate the expression above is fairly complicated, its exact expansion requires care and patience, since we are not actually computing the second order correction, we will only point out the types of terms we encounter in this expansion,

Born-Oppenheimer Type Approximation for a Simple Renormalizable System

209

with some undetermined coefficients in front. As a consequence, the Laplacian part of normalization is found to be h¯ 2 1 2 h¯ 2 1 m∗ 2 − + c1 ∇z A ≈ − ν ln MA M 4z2 M

 √ 2m∗ νz h¯

2  √ m∗ 2 m∗ 2m∗ 4 2 2m∗ νz ν z + . . . , (79) ν + c3 ln + c2 M M h¯ 2 h¯

where c1 , c2 , c3 are constant that can be found explicitly, moreover m∗ should be replaced with m to this order of accuracy. These are the results that we use in the main text. In the subsequent Appendix to illustrate the consistency of our approximations as well as to propose a scheme to compute higher order corrections we evaluate the expectation values of some of the next order terms within first order perturbation theory. This of course is an asymptotic expansion and hopefully presents a reliable description of the dynamics.

Appendix B: Perturbative Corrections In order to verify the consistency of our approximations we should first show that the expectation value of z is much smaller than the spread of the light particles h¯ wave functions, which is characterized by ζ0 = √2m . This is related to z0 that  ∗

∗ we introduced previously, one can see easily that z0 = 2m M ζ0 . For this, we need to normalize our wave function, written in terms of the variable z:

1(z) = C

z 2β z β − e z0 (1+2β) . β (z0 (1 + 2β))

(80)

The normalization constant C is then found as 22β C 2π (z0 (1 + 2β))2β



=

∞

2

dzz

2z 2β+1 − z0 (1+2β)

e

=1

and

C=

0

2 . π&(2β + 2)z02 (1 + 2β)2

(81) Next we look at the expectation value of z (within our approximation) to check the consistency of our assumption =

22β+2 (z0 (1 + 2β))2β+2 &(2β + 2)



∞ 0

dzz2β+2 e

m  h¯ eγ &(2β + 3) ∗ (1 + 2β) √ = . 2 &(2β + 2) 2m∗  M

−z

2z 0 (1+2β)

=

1 &(2β + 3) z0 (1 + 2β) 2 &(2β + 2)

(82)

Let us emphasize that this is a key result for the consistency of our approximations.

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It is also instructive to calculate next order corrections to the binding energy of the heavy-light system as a function of the distance between the heavy centers: h¯ 2 2 + ln −ν ≈ − √ 2m∗ eγ z 2e2γ 2

√  2m∗ γ ze + . . . . 2h¯

The first term is the original potential part that we used in the effective description of the heavy system, so let us look at the expectation value of the second term as a perturbation on our solution: G

2 ln 2e2γ

√ H 2m∗ 22β+2 2 zeγ = 2γ 2h¯ 2e (z0 (1 + 2β))2β+2 &(2β + 2) √   ∞ 2m∗ − 2z zeγ dzz2β+1 e z0 (1+2β) ln 2h¯ 0      M (1 + 2β)e2γ 2 + ln = 2γ ψ(2β + 2) − ln 2m∗ 8 2e   2 M  (83) ≈ − 2γ ln . 2e m∗

We can easily see that this expectation value is of lower order compared to the leading order energy of the heavy particle. We emphasize that the constant terms are ambiguous as long as we keep to this order, they can be fixed if the next order terms (z without the logarithms) are also taken into account. An interesting estimate will be to calculate the expectation values of the correction terms resulting from the heavy particle relative coordinate kinetic energy term (Laplacian with respect to z) operating on the normalization constant A, the expansion of which is given in the previous Appendix in equation (79). Since we have not found the precise constant in front, we compute the expectation values of each basic part separately without the constants in front: √  2 2m∗ 2m∗ 2 2m∗ h¯ 2m∗  2 γ (a) = √ ln ν ≈ − ze + ... M M M 2e2γ 2h¯ 2m∗ eγ z √   √ 2m∗ 2 2m∗ 1 2m∗ h¯ 2m∗ 2 γ γ ≈ (b) = ν ln νze ln ze √ M 2h¯ 2 M 2h¯ 2m∗ eγ z √  2m∗ 1 2m∗ 2 2 γ  ln ze + . . . − 2γ M 2h¯ 4e √  √  2m∗ 2m∗ 2m∗  2 2 2m∗ 2m∗ 4 2 2 γ γ ν z ln ln νze ze ≈ + . . . . (84) (c) = M h¯ 2 2h¯ M e2γ 2h¯

Born-Oppenheimer Type Approximation for a Simple Renormalizable System

211

We first look at the expectation value of (a) the log-term has already been calculated above so let us focus on the expectation of the first term of (a), that we denote by a subscript as (a)1 , G < (a)1 > = = =

2 2m∗ h¯ √ M 2m∗ eγ z

H

2m∗ h¯ 2 22β+2 √ γ M 2m∗ e (z0 (1 + 2β))2β+2&(2β + 2)



∞

dzz2β e

2z 0 (1+2β)

−z

0

8 &(2β + 1) 2 2m∗ h¯ 4 = 2γ , √ γ M e (1 + 2β)2 2m∗ e &(2β + 2)z0 (1 + 2β)

which is negligible to this order. In a similar way, we find the expectation value of the second term (b) √ H 2m∗ zeγ 2h¯ √   ∞ 2z  22β+2 2m∗ 2m∗ h¯ 2β e− z0 (1+2β) ln γ √ dzz ze = M 2h¯ 2m∗ eγ (z0 (1 + 2β))2β+2 &(2β + 2) 0      &(2β + 1) M (1 + 2β)e2γ 2 2m∗ h¯ √ ψ(2β + 1) − ln = + ln M 2m∗ 8 2m∗ eγ z0 (1 + 2β) &(2β + 2)       M (1 + 2β)e2γ 2 4 = 2γ ψ(2β + 1) − ln + ln 2m∗ 8 e (1 + 2β)2 G

2m∗ h¯  < (b) >= √ ln γ M e 2m∗ z

  1 4 2 M . ≈ − 2γ ln m∗ e (1 + 2β)2

(85)

The final expectation value is that of (c) part, it must be negligible due to m/M term and no inverse powers of z appearing in it: G < (c) >=

2m∗  2 2 ln M e 2γ

√

2m∗ ze γ 2h¯

H

√   ∞ 2z 22β+2 2m∗ 2m∗  2 2β+1 − z0 (1+2β) 2 γ ze dzz e ln = M e 2γ (z0 (1 + 2β))2β+2 &(2β + 2) 0 2h¯        2 M (1 + 2β)e 2γ 2m∗  2 ψ(2β + 2) − ln + ln = + ζ(2, 2β + 2) , (86) M e 2γ 2m∗ 8

which is negligible to the order that we are interested in, as expected. To gain more insight, we can go one step further and calculate some of the expectation values resulting from the heavy particle kinetic energy partially operating on the light particle wave function and generating a mixed gradient term, after our

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simplifications, this residual term is found to be 2m∗ 2 ∂1 ν z ln M ∂z

√ √   m∗ h¯ 2 ∂1 2m∗ 2m∗ γ νzeγ ≈ ln ze √ 2h¯ M 2m∗ eγ ∂z 2h¯ √   2m∗  2 ∂1 2 2m∗ γ − ln ze + . . . z M 4e2γ ∂z 2h¯

We compute the expectation value of this expression (to properly identify the corrections within first order perturbation theory we must symmetrize this expression, ∂ thus we get a term coming from the anticommutator of ∂z and the log expression, we ignore this subtlety for now since we are not aiming for an exact computation). Let us start with the first part:

= = = ≈

 √  ∞ ∂1 2m∗ h¯  2m∗ γ 2π dzz1 ln ze √ M ∂z 2h¯ 2m∗ eγ 0 √    ∞ m∗ h¯ ∂1 2  2m∗ γ 2π dzz ln ze √ M 2m∗ eγ ∂z 2h¯ 0  √ √    ∞   ∞ ! m∗ h¯  2m∗ 2m∗ 2 γ !∞ 2 γ 2 2π 1 z ln − dz1 ln dz1 − ze ze √ 0 M 2m∗ eγ 2h¯ 2h¯ 0 0 √    ∞ m∗ h¯ 1  2m∗ − dz1 2 ln < > +2π zeγ √ M 2m∗ eγ z 2h¯ 0   2 2 M ln (87) . m∗ e2γ (1 + 2β)2

The second part of the expansion should be small, let us see this by an explicit computation (within first order perturbation theory): √

 2m∗ γ ln ze dzz ∂z 2h¯ 0  √  √   ∞ ! 2m∗  2 2m∗ 2m∗ 2 2 2 γ !∞ 2 2 γ ze ze =− 2π z 1 ln −2 dzz1 ln 0 M 8e2γ 2h¯ 2h¯ 0 √   ∞ 2m∗ 2 γ ze dzz1 ln −2 2h¯ 0 √ H √ H G G 2m∗ 2m∗ 2m∗  2 2m∗  2 γ 2 γ ze ze ln ln (88) = + , M 4e2γ 2h¯ M 4e2γ 2h¯ 2m∗  2 − 2π M 8e2γ



∞

2 ∂1

2

2

all of which are computed above and clearly of smaller order. This completes our short digression on calculating higher order corrections. As we emphasize in principle it is possible to compute all these corrections in first order perturbation theory, nevertheless it requires precise expansions of all the terms to first order (in terms of z) which we have not done. If we intend to go beyond the first order

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terms in the expansions and evaluate their contributions, the possibility of second order perturbation of the first order terms becoming equally important should be discussed. We leave these questions to the future.

References 1. Born, M., Oppenheimer, J.R.: Zur Quantentheorie der Molekeln [On the Quantum Theory of Molecules]. Annalen der Physik 389(20), 457–484 (1927) 2. Landau, L.D., Lifshitz, E.M.: Quantum Mechanics: Nonrelativistic Theory. Course of Theoretical Physics. Addison-Wesley Series in Advanced Physics, vol. 3. Addison-Wesley Publishing Co., Inc., Reading (1958) 3. Bethe, H.A., Jackiw, R.: Intermediate Quantum Mechanics. Advanced Books Classics, 3rd edn. Westview Press, Nashville (1977) 4. Weinberg, S.: Lectures on Quantum Mechanics, 2nd edn. Cambridge University Press, Cambridge (2015) 5. Panati, G., Spohn, H., Teufel, S.: The time-dependent Born-Oppenheimer approximation. ESAIM: Math. Model. Numer. Anal. 41, 297–314 (2007) 6. Hagedorn, G.A., Joye, T.: Mathematical Analysis of Born-Oppenheimer Approximations, Spectral Theory and Mathematical Physics: A Festschrift in Honor of Barry Simon’s 60th Birthday: Quantum Field Theory, Statistical Mechanics and Non-relativistic Quantum Systems, edited by F. Gesztesy et al. AMS Publications, Providence (2007) 7. Jecko, T.: On the mathematical treatment of the Born-Oppenheimer approximation. J. Math. Phys. 55, 053504 (2014) 8. Thorn, C.: Quark confinement in the infinite momentum frame. Phys. Rev. D19, 639 (1979) 9. Beg, M.A.B., Furlong, R.C.: λφ 4 theory in the nonrelativistic limit. Phys. Rev. D31, 1370 (1985) 10. Jackiw, R.: Delta-function potentials in two and three dimensional quantum mechanics. In: Ali, A., Hoodbhoy, P. (eds.) M.A.B.Beg Memorial Volume. World Scientific, Singapore (1991) 11. Fernando Perez, J., Coutinho, F.A.B.: Schrödinger equation in two dimensions for a zero range potential and a uniform magnetic field: An exactly solvable model. Am. J. Phys. 59, 52 (1991) 12. Mead, L.R., Godines, J.: An analytical example of renormalization in quantum mechanics. Am. J. Phys. 59, 935 (1991) 13. Gosdzinsky, P., Tarrach, R.: Learning quantum field theory from elementary quantum mechanics. Am. J. Phys. 59, 70 (1991) 14. Albeverio, S., Gesztesy, F., Hoegh-Krohn, R., Holden, H.: Solvable Models in Quantum Mechanics, 2nd edn. AMS Chelsea Publishing, Providence, RI (2004) 15. Mitra, I., Dasgupta, A., Dutta-Roy, B.: Regularization and renormalization in scattering from Dirac delta potentials. Am. J. Phys. Am. J. Phys. 66, 1101 (1998) 16. Seiler, R.: Does the Born-Oppenheimer approximation work? Helvetica Physica Acta 46, 230 (1973) 17. Combes, J.M., Duclos, P., Seiler, R.: The Born-Oppenheimer approximation. In: Wightman, Velo (eds.), Rigorous Atomic and Molecular Physics Proceedings, vol. 1980, pp. 185. Plenum, New York (1981) 18. Hunziker, W.: Distortion analyticity and molecular resonance curves. Ann. Inst. H. Poincare Sect. A. 45, 339 (1986) 19. Hagedorn, G.A.: Classification and normal forms for quantum eigenvalue crossings. Asterisque 210, 115–134 (1993) 20. Hagedorn, G.A.: Molecular propagation through electron energy level crossings. Memoirs Am. Math. Soc. 111(536), 1–130 (1994)

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21. Hagedorn, G.A.L.: Classification and normal forms for avoided crossings of quantum mechanical energy levels. J. Phys. A. 31, 369–383 (1998) 22. Hagedorn, G.A.: High order corrections to the time-independent Born-Oppenheimer approximation I: smooth potentials. Ann. Inst. H. Poincare Sect. A. 47, 1–16 (1987) 23. Hagedorn, G.A.: High order corrections to the time-independent Born-Oppenheimer approximation II: diatomic coulomb systems. Commun. Math. Phys. 116, 23–44 (1988) 24. Klein, M., Martinez, A., Seiler, R., Wang, X.: On the Born-Oppenheimer expansion for polyatomic molecules. Commun. Math. Phys. 143, 607–639 (1992) 25. Weigert, S., Littlejohn, R.G.: Diagonalization of multicomponent wave equations with a BornOppenheimer example. Phys. Rev. A 45, 3506 (1993) 26. Gangopadhyay, G., Dutta-Roy, B.: The Born-Oppenheimer approximation: the toy model. Am. J. Phys. 72, 389 (2004) 27. Akbas, H., Turgut, O.T.: Born-Oppenheimer approximation for a singular system. Arxiv-160202811. Submitted for publication 28. Rosa, D.S., Bellotti, F.F., Jensen, A.S., Krein, G., Yamashita, M.T.: Bound states of a light atom and two heavy dipoles in two dimensions. Phys. Rev. A 94, 062707 (2016) 29. Fonseca, A.C., Reddish, E.F., Shenley, P.E.: Effimov effect in an analytically solvable model. Nuclear Physics A320, 273–288 (1979) 30. Bellotti, F.F., Frederico, T., Yamashita, M.T., Fedorov, D.V., Jensen, A.S., Zinner, N.T.: Mass imbalanced three body systems in 2-dimensions. J. Phys. B 46, 055301 (2013) 31. Rosa, R.D., Federico, T., Krein, G., Yamashita, M.T.: Effimov effect in a D-dimensional BornOppenheimer approach. J. Phys. B At. Mol. Opt. Phys. 52, 025101 (2019)

Spectral Isoperimetric Inequality for the δ  -Interaction on a Contour Vladimir Lotoreichik

Abstract We consider the problem of geometric optimization for the lowest eigenvalue of the two-dimensional Schrödinger operator with an attractive δ  interaction of a fixed strength, the support of which is a C 2 -smooth contour. Under the constraint of a fixed length of the contour, we prove that the lowest eigenvalue is maximized by the circle. The proof relies on the min-max principle and the method of parallel coordinates. Keywords Schrödinger operator · δ  -Interaction · Lowest eigenvalue · Min-max principle · Eigenvalue optimization

1 Introduction 1.1 The State of the Art and Motivation The question of optimizing shapes in spectral theory is a rich subject with many applications and deep mathematical insights; see the monographs [1, 2] and the references therein. In this note, we consider the problem of shape optimization for the lowest eigenvalue of the two-dimensional Schrödinger operator with a δ  interaction supported on a closed contour in R2 . This problem can be regarded as a counterpart of the analysis performed in [3] for δ-interactions. In the recent years, the investigation of Schrödinger operators with δ  -interactions supported on hypersurfaces became a topic of permanent interest—see, e.g. , [4– 11]. The Hamiltonians with δ  -interactions and some of their generalizations appear, for example, in the study of photonic crystals [12, 13] and in the analysis of the Dirac operator with Lorentz scalar shell interactions [14]. The boundary condition

V. Lotoreichik () ˇ Department of Theoretical Physics, Nuclear Physics Institute, Czech Academy of Sciences, Rež, Czech Republic e-mail: [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. Michelangeli (ed.), Mathematical Challenges of Zero-Range Physics, Springer INdAM Series 42, https://doi.org/10.1007/978-3-030-60453-0_10

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corresponding to the δ  -interaction arises in the asymptotic analysis of a class of structured thin Neumann obstacles [15, 16]. Finally, the same boundary condition pops up in the computational spectral theory; see [17] and the references therein. The proofs in [3] and in related optimization problems for singular interactions on hypersurfaces [18–23] rely on the Birman-Schwinger principle, which can also be viewed as a boundary integral reformulation of the spectral problem. In this note, we do not pass to any boundary integral reformulation. Instead, we combine the min-max principle and the method of parallel coordinates on the level of the quadratic form for the Hamiltonian, in the spirit of the recent analysis for the Robin Laplacian [24–28]. Our main motivation is to show that this approach initially developed for the Robin Laplacian can also be adapted for a much wider class of optimization problems involving surface interactions. The convenience of this alternative method is particularly visible for δ  -interactions, because the operator arising in the corresponding Birman-Schwinger principle (cf. [6, Rem. 3.9]) is more involved than for δ-interactions.

1.2 Schrödinger Operator with a δ  -Interaction on a Contour In order to define the Hamiltonian, we need to introduce some notation. In what follows we consider a bounded, simply connected, C 2 -smooth domain *+ ⊂ R2 , whose boundary will be denoted by ' = ∂*+ . The complement *− := R2 \ *+ of *+ is an unbounded exterior domain with the same boundary '. For a function u ∈ L2 (R2 ) we set u± := u|*± . We also introduce the first order L2 -based Sobolev space on R2 \ ' as follows H 1 (R2 \ ') := H 1 (*+ ) ⊕ H 1 (*− ), where H 1 (*± ) are the conventional first-order L2 -based Sobolev spaces on *± . Given a real number ω > 0, we consider the spectral problem for the self-adjoint operator Hω,' corresponding via the first representation theorem to the closed, densely defined, symmetric, and semi-bounded quadratic form in L2 (R2 ), 1 12 1 12 hω,' [u] = 1∇R2 \' u1L2 (R2 ;C2 ) −ω1[u]' 1L2 (') , dom hω,' = H 1 (R2 \'),

(1)

where ∇R2 \' u := ∇u+ ⊕ ∇u− and [u]' := u+ |' − u− |' denotes the jump of the trace of u on '; cf. [4, Sec. 3.2]. The operator Hω,' is usually called the Schrödinger operator with the δ  -interaction of strength ω supported on '. It acts as the minus Laplacian on functions satisfying the transmission boundary condition of δ  -type on the interface ' ∂ν+ u+ |' = −∂ν− u− |' = ω[u]' ,

(2)

Spectral Isoperimetric Inequality for the δ  -Interaction on a Contour

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where ∂ν± u± |' denotes the trace onto ' of the normal derivative of u± with the normal vector ν± at the boundary of *± pointing outwards; see Sect. 2 for more details. Recall that the essential spectrum of Hω,' coincides with the set [0, ∞) and that its negative discrete spectrum is known to be non-empty; see Proposition 1 below. By λω1 (') we denote the spectral threshold of Hω,' , which is an isolated negative eigenvalue.

1.3 The Main Result The aim of this note is to demonstrate that λω1 (') is maximized by the circle C ⊂ R2 , among all contours of a fixed length. A precise formulation of this statement is the content of the following theorem. Theorem 1 For any ω > 0, one has max λω1 (') = λω1 (C),

|'|=L

where C ⊂ R2 is a circle of a given length L > 0 and the maximum is taken over all C 2 -contours of length L. The proof of Theorem 1 relies on the min-max principle and the method of parallel coordinates. The latter method has been proposed in [29] by L. E. Payne and H. F. Weinberger in order to obtain inequalities being reverse to the celebrated Faber-Krahn inequality [30, 31] with some geometrically-induced corrections. Recently it has been observed that this method is very efficient in the proofs of isoperimetric inequalities for the lowest eigenvalue of the Robin Laplacian on bounded [24, 25] and exterior [27, 28] domains with an ‘attractive’ boundary condition. In the present paper we adapt this approach for the case of a bounded domain and its exterior coupled via the transmission boundary condition (2) of δ  type.

1.4 Organisation of the Paper In Sect. 2 we recall the known spectral properties of Hω,' that are needed in this paper. Section 3 is devoted to the spectral analysis of Hω,C with the interaction supported on a circle C. The method of parallel coordinates is briefly outlined in Sect. 4. Theorem 1 is proven in Sect. 5. The paper is concluded by Sect. 6 containing a discussion of the obtained results and their possible extensions.

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2 The Spectral Problem for the δ  -Interaction Supported on a Closed Contour Recall that we consider a bounded, simply connected, C 2 -smooth domain *+ ⊂ R2 with the boundary ' = ∂*+ and with the complement *− := R2 \ *+ . Recall also that for a function u ∈ L2 (R2 ), we set u± := u|*± . At the same time, the (attractive) coupling strength ω is a fixed positive number. We are interested in the spectral properties of the self-adjoint operator Hω,' in L2 (R2 ) introduced via the first representation theorem [32, Thm. VI 2.1] as associated with the closed, densely defined, symmetric, and semi-bounded quadratic form hω,' defined in (1); see [4, Sec. 3.2] and also [6, Sec. 3.3 and Prop. 3.15]. We would like to warn the reader that in the majority of the papers on δ  interactions not ω itself, but its inverse β := ω−1 is called the strength of the interaction. This tradition goes back to papers on point δ  -interaction on the real line; see [33] and the references therein. Preserving this tradition for δ  -interactions on hypersurfaces can be physically motivated, but leads to a technical mathematical inconvenience, which we would like to avoid. Let us add a few words about the explicit characterisation of the operator Hω,' . The domain of Hω,' consists of functions u ∈ H 1 (R2 \ '), which satisfy $u± ∈ L2 (*± ) in the sense of distributions and the δ  -type boundary condition (2) on ' in the sense of traces. Moreover, for any u ∈ dom Hω,' we have Hω,' u = (−$u+ ) ⊕ (−$u− ). The reader may consult [4, Sec. 3.2 and Thm. 3.3], where it is shown that the operator characterised above is indeed the self-adjoint operator representing the quadratic form hω,' in (1). It is worth mentioning that C 2 -smoothness of ' is not needed to define the operator Hω,' , but it is important for the method of parallel coordinates used in the proof of Theorem 1. The lowest spectral point of Hω,' can be characterised by the min-max principle [34, Sec. XIII.1] as follows λω1 (') =

hω,' [u] . 2 u∈H 1 (R2 \') u 2 2 L (R ) inf

(3)

u#=0

It is not surprising that the operator Hω,' has a non-empty essential spectrum. In fact, one can show that Hω,' is a compact perturbation in the sense of resolvent differences of the free Laplacian on R2 and thus the essential spectrum coincides with the positive semi-axis. Using the characteristic function of *+ as a test function for (3) one gets that the negative discrete spectrum of Hω,' is non-empty. More specifically, we have the following statement. Proposition 1 For all ω > 0, the following hold. (i) The essential spectrum of Hω,' is characterized as follows σess (Hω,' ) = [0, ∞). (ii) The negative discrete spectrum of Hω,' is non-empty.

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A proof of (i) in the above proposition can be found in [4, Thm. 4.2 (ii)], see also [6, Thm. 3.14 (i)]. A proof of (ii) is contained in [4, Thm. 4.4]. Some further properties of the discrete spectrum of Hω,' are investigated in or follow from [6, 7]. Note that by [6, Thm. 3.14 (ii)] the negative discrete spectrum of Hω,' is finite for C ∞ smooth ' and it can be shown in a similar way that the discrete spectrum persists to be finite for C 2 -smooth '. Taking that the spectral threshold of Hω,' is a negative discrete eigenvalue into account, we can slightly modify the characterisation of λω1 (') given in (3) as follows: λω1 (') =

inf u∈H 1 (R2 \')

hω,' [u]