Spectral Theory and Mathematical Physics

259 Pages·2016·2.288 MB·English

by  Marius Mantoiu, Georgi Raikov, Rafael Tiedra de Aldecoa (eds.)

Operator Theory
Advances and Applications
254
Marius Mantoiu
Georgi Raikov
Rafael Tiedra de Aldecoa
Editors
Spectral
Theory and
Mathematical
Physics

Operator Theory: Advances and Applications
Volume 254

Founded in 1979 by Israel Gohberg

Editors:
Joseph A. Ball (Blacksburg, VA, USA)
Harry Dym (Rehovot, Israel)
Marinus A. Kaashoek (Amsterdam, The Netherlands)
Heinz Langer (Wien, Austria)
Christiane Tretter (Bern, Switzerland)

Associate Editors: Honorary and Advisory Editorial Board:
Vadim Adamyan (Odessa, Ukraine) Lewis A. Coburn (Buffalo, NY, USA)
Wolfgang Arendt (Ulm, Germany) Ciprian Foias (College Station, TX, USA)
Albrecht Böttcher (Chemnitz, Germany) J.William Helton (San Diego, CA, USA)
B. Malcolm Brown (Cardiff, UK) Thomas Kailath (Stanford, CA, USA)
Raul Curto (Iowa, IA, USA) Peter Lancaster (Calgary, Canada)
Fritz Gesztesy (Columbia, MO, USA) Peter D. Lax (New York, NY, USA)
Pavel Kurasov (Stockholm, Sweden) Donald Sarason (Berkeley, CA, USA)
Vern Paulsen (Houston, TX, USA) Bernd Silbermann (Chemnitz, Germany)
Mihai Putinar (Santa Barbara, CA, USA) Harold Widom (Santa Cruz, CA, USA)
Ilya M. Spitkovsky (Williamsburg, VA, USA)

Subseries
Linear Operators and Linear Systems
Subseries editors:
Daniel Alpay (Beer Sheva, Israel)
Birgit Jacob (Wuppertal, Germany)
André C.M. Ran (Amsterdam, The Netherlands)

Subseries
Advances in Partial Differential Equations
Subseries editors:
Bert-Wolfgang Schulze (Potsdam, Germany)
Michael Demuth (Clausthal, Germany)
Jerome A. Goldstein (Memphis, TN, USA)
Nobuyuki Tose (Yokohama, Japan)
Ingo Witt (Göttingen, Germany)
Moreinformationaboutthisseriesathttp://www.springer.com/series/4850
Marius Mantoiu • Georgi Raikov
Rafael Tiedra de Aldecoa
Editors
Spectral Theory and
Mathematical Physics
Editors
Marius Mantoiu Georgi Raikov
Facultad de Ciencias Facultad de Matemáticas
Universidad de Chile Pontificia Universidad Católica de Chile
Santiago, Chile Santiago, Chile
Rafael Tiedra de Aldecoa
Facultad de Matemáticas
Pontificia Universidad Católica de Chile
Santiago, Chile
ISSN 0255-0156 ISSN22 96-4878 (electronic)
Operator Theory: Advances and Applications
ISBN 978-3-319-29990-7 ISBN 978-3-319-29992-1 (eBook)
DOI 10.1007/978-3-319-29992-1

Library of Congress Control Number: 2016945039

Mathematics Subject Classification (2010): 35J10, 35P25, 47B80, 81Q10, 81T10, 82B44

© Springer International Publishing Switzerland 2016
This work is subject to copyright. All rights are reserved 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, express or implied, with respect to the material contained herein or for any errors or
omissions that may have been made.

Printed on acid-free paper

This book is published under the trade name Birkhäuser.
The registered company is Springer International Publishing AG (www.birkhauser-science.com)
Contents
Preface .................................................................. vii
J. Asch, O. Bourget, V.H. Cort´es and C. Fern´andez
Lower Bounds for Sojourn Time in a Simple
Shape Resonance Model ............................................ 1
J.-M. Barbaroux, J. Faupin and J.-C. Guillot
Spectral Properties for Hamiltonians of Weak Interactions ........... 11
V. Bonnaillie-No¨el, M. Dauge, N. Popoff and N. Raymond
Magnetic Laplacian in Sharp Three-dimensional Cones .............. 37
V. Bruneau and D. Sambou
Spectral Clusters for Magnetic Exterior Problems ................... 57
A. Carey, F. Gesztesy, G. Levitina and F. Sukochev
The Spectral Shift Function and the Witten Index .................. 71
F. Clivaz
Stahl’s Theorem (aka BMV Conjecture):
Insights and Intuition on its Proof .................................. 107
D. Ranjan Dolai
Some Estimates Regarding Integrated Density of States for Random
Schro¨dinger Operator with Decaying Random Potentials ........... 119
D. Ha¨fner
Boundary Values of Resolvents of Self-adjoint Operators in
Krein Spaces and Applications to the Klein–Gordon Equation ....... 133
S. Richard
Levinson’s Theorem: An Index Theorem in Scattering Theory ....... 149
vi Contents
D. Sambou
Counting Function of Magnetic Eigenvalues for
Non-definite Sign Perturbations ..................................... 205
M. T¨aufer, M. Tautenhahn and I. Veseli´c
Harmonic Analysis and Random Schro¨dinger Operators ............. 223
Preface
ThisvolumecontainstheproceedingsoftheconferenceSpectralTheory and Math-
ematical Physics, which took place at Pontificia Universidad Cato´lica de Chile
(PUC), Santiago, in November 2014. The main purpose of this conference was to
bring together a number of established specialists in spectral theory and math-
ematical physics, as well as some young and beginning researchers in this field
in order to connect people from different schools and generations, give them the
opportunity to exchange ideas, and try to attract more young mathematicians to
this fascinating area of research.
TheconferenceSpectralTheoryandMathematicalPhysicsandthepreceding
course onrandomSchro¨dingeroperatorsgivenby Werner Kirschand Ivan Veseli´c
were organized within the framework of the International Spectral Network, fi-
nanced in the Chilean side by Iniciativa Cient´ıfica Milenio (ICM) of the Chilean
Ministry of Economy. All the organizers belong to the Millennium Nucleus in
Mathematical Physics RC120002. They gratefully acknowledge the financial sup-
port ofICM throughthe projectNetworking2013,as well as supportof the other
conferencesponsors:theVice-RectoryofResearchandtheFacultyofMathematics
of PUC, the International Association of Mathematical Physics, and the Chilean
Science Foundation FONDECYT. Special gratitude is due to the administrative
staff ofthe Faculty of Mathematics of PUC for logistic help in organizationofthe
conference.
We would like to thank the conference mini-course lecturers Serge Richard
(Nagoya) and Fedor Sukochev (Sydney); our invited speakers: Jean-Marie Bar-
baroux(Toulon),VirginieBonnaillie-No¨el(Rennes),VincentBruneau(Bordeaux),
HoriaCornean(Aalborg),Rafaeldel Rio(Mexico),ErdalEmsiz(Santiago),Clau-
dio Ferna´ndez (Santiago), Dietrich H¨afner (Grenoble), Werner Kirsch (Hagen),
Hynek Kovaˇr´ık (Brescia), David Krejˇciˇr´ık (Prague), Alexander Nazarov (St. Pe-
tersburg),RolandoRebolledo(Santiago),ThomasSørensen(Munich),MatˇejTuˇsek
(Prague), and Ivan Veseli´c (Chemnitz); and the students who gave talks: Harold
Bustos(Santiago),FabienClivaz(Zu¨rich),DhritiDolai(Chennai),Toma´sLungen-
strass (Santiago), Joseph Mehringer (Munich), Daniel Parra (Lyon), and Hanne
van den Bosch (Santiago).
This volume contains survey articles as well as original results presented at
the conference. Most of the articles are dedicated to some of the following topics:
viii Preface
• Ergodic Quantum Hamiltonians
• Magnetic Schr¨odinger Operators
• Quantum Field Theory
• Scattering Theory
• Semiclassical and Microlocal Analysis
• Spectral Shift Function and Quantum Resonances
As editors, we are grateful to the authors who contributed to this book, and
to all the anonymousreferees for their professionaland time-consuming work.We
would also like to thank Liliya Simeonova for handling manuscripts and referee
reports and for technical assistance in the preparation of the volume.
The proceedings Spectral Analysis of Quantum Hamiltonians of the confer-
ence held in Santiago in 2010, were published in the Birkha¨user series Operator
Theory Advances and Applications in 2012. The present publication is the second
of what we hope to be a series of proceedings of regular conferences on spectral
theory held in Santiago de Chile.
Marius Mantoiu
Georgi Raikov
Rafael Tiedra de Aldecoa
OperatorTheory:
Advances andApplications,Vol.254,1–9
(cid:2)c 2016SpringerInternational Publishing
Lower Bounds for Sojourn Time in
a Simple Shape Resonance Model
J. Asch, O. Bourget, V.H. Cort´es and C. Fern´andez
Abstract. We consider a simple model for shape resonance in the spirit of
Gamov and prove that the sojourn time diverges as the square root of the
heightofthebarrier.ThisresultillustratesthepowerofLavine’slowerbound
theory.
Mathematics SubjectClassification (2010). 35P99, 81Q15.
Keywords. Quantum resonances, lifetime estimates.
1. Introduction
Shaperesonanceshavebeenstudiedsincetheyouthofquantummechanicsinorder
to explainradioactivedecay, see, e.g., [4]. To be specific, consider the Schr¨odinger
operator
d2
H =− +V ,
a dx2 a
onthe positivehalf-axiswithDirichletboundaryconditionatzeroandapotential
bumpV ofheighta>0,compactlysupportedawayfromtheoriginandanenergy
a
E <a.
Gamov observed that an initial state ϕ of energy E, supported betw√een the
Dirichletwallandthepotential,decaysintimewitharateproportionalto a−E
(√see [4]). In the present contribution, we show that a lower bound proportional to
a−E holdsforthesojourntimeofsuchstatesforaspecificfamilyofreal-valued
potentials (Va)a≥0 (see Theorem 2.1). We achieve this by using Lavine’s lower
bound theory, which is recalled in Section 2. The proof is developed in Section 3.
ThefirstauthorwassupportedbyECOS-ConicytC10E01,EPlanet.
The second, third and fourth author were supported by Proyecto Fondecyt No. 1120786, No.
1141120 AnilloConicytPIA-ACT1112.