Mathematical Physics, Spectral Theory and Stochastic Analysis

344 Pages·2013·3.037 MB·English

by  Michael Demuth, Werner Kirsch (eds.)

Operator Theory
Advances and Applications
232
Michael Demuth
Werner Kirsch
Editors
Mathematical
Physics,
Spectral Theory
and Stochastic
Analysis
A Advances in
P Partial
D Differential
E Equations

Operator Theory: Advances and Applications
Volume 232

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 (Vienna, Austria)
Christiane Tretter (Bern, Switzerland)

Associate Editors: Honorary and Advisory Editorial Board:
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Albrecht Böttcher (Chemnitz, Germany) Ciprian Foias (College Station, TX, USA)
B. Malcolm Brown (Cardiff, UK) J.William Helton (San Diego, CA, USA)
Raul Curto (Iowa, IA, USA) Thomas Kailath (Stanford, CA, USA)
Fritz Gesztesy (Columbia, MO, USA) Peter Lancaster (Calgary, Canada)
Pavel Kurasov (Stockholm, Sweden) Peter D. Lax (New York, NY, USA)
Leonid E. Lerer (Haifa, Israel) Donald Sarason (Berkeley, CA, USA)
Vern Paulsen (Houston, TX, USA) Bernd Silbermann (Chemnitz, Germany)
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Leiba Rodman (Williamsburg, VA, USA)
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Subseries
Linear Operators and Linear Systems
Subseries editors:
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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)
Michael Demuth
Werner Kirsch
Editors
Mathematical Physics,
Spectral Theory and
Stochastic Analysis
A Advances in
P Partial
D Differential
E Equations
Editors
Michael Demuth Werner Kirsch
Institut für Mathematik Fakultät für Mathematik und Informatik
TU Clausthal Fernuniversität Hagen
Clausthal-Zellerfeld Hagen
Germany Germany

ISBN 978-3-0348-0590-2 ISBN 978-3-0348-0591-9 (eBook)
DOI 10.1007/978-3-0348-0591-9
Springer Basel Heidelberg New York Dordrecht London
Library of Congress Control Number: 2013938259
Mathematics Subject Classification (2010): 35Pxx, 47XX, 60Hxx, 81Qxx

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Contents
Preface .................................................................. vii
M.S. Ashbaugh, F. Gesztesy, M. Mitrea, R. Shterenberg and G. Teschl
A Survey on the Krein–vonNeumann Extension, the Corresponding
Abstract Buckling Problem, and Weyl-type Spectral Asymptotics
for Perturbed Krein Laplacians in Nonsmooth Domains ............. 1
M. Demuth, M. Hansmann and G. Katriel
Eigenvalues of Non-selfadjoint Operators: A Comparison
of Two Approaches ................................................. 107
P. Exner
Solvable Models of Resonances and Decays .......................... 165
M. Gebert and P. Mu¨ller
Localization for Random Block Operators ........................... 229
T. Ichinose
Magnetic Relativistic Schr¨odinger Operators and Imaginary-time
Path Integrals ...................................................... 247
Y. Pinchover
Some Aspects of Large Time Behavior of the Heat Kernel:
An Overview with Perspectives ..................................... 299

Preface
This volume contains survey articles on various aspects of operator theory and
partial differential operators. These papers are meant as self-contained introduc-
tions to specific fields written by experts for non specialists. They are accessible
for graduate students and young researchers but – we believe – they are also of
interest to scientists already familiar with the respective area.
Thetopicscoveredrangefromdifferentialoperatorsonabstractmanifoldsto
finite difference operatorson a lattice modeling some aspects of impure supercon-
ductors. All of them share a view towards applications in physics.
Theideatocollectthesecontributionsaroseduringaconferenceorganizedby
D. Mayer,I. Witt andone ofthe editors in Goslar (Germany)in September 2011.
But, insteadof collecting highly specializedarticles onmostrecentresearchasfor
conference proceedings our focus was on introductory aspects and readability of
up-to-date contributions.
During the Goslar conference mentioned we also had the opportunity to cel-
ebrate the 65th birthday of Michael Demuth. It gives me a great pleasure to wish
him here once more many happy years to come.
October 2012 Werner Kirsch
FernUniversit¨at in
Hagen (Germany)
OperatorTheory:
Advances andApplications,Vol.232,1–106
⃝c 2013SpringerBasel
A Survey on the Krein–von Neumann
Extension, the Corresponding Abstract
Buckling Problem, and Weyl-type Spectral
Asymptotics for Perturbed Krein Laplacians
in Nonsmooth Domains
Mark S. Ashbaugh, Fritz Gesztesy, Marius Mitrea,
Roman Shterenberg and Gerald Teschl
Dedicated with great pleasure to Michael Demuth
on the occasion of his 65th birthday
Abstract. In the first (and abstract) part of this survey we prove the uni-
tary equivalence of the inverse of the Krein–von Neumann extension (on the
orthogonalcomplementofitskernel)ofadenselydefined,closed,strictlypos-
itive operator, 𝑆 ≥ 𝜀𝐼 for some 𝜀 > 0 in a Hilbert space ℋ to an abstract
ℋ
bucklingproblem operator.
In the concrete case where 𝑆 = −Δ∣ in 𝐿2(Ω;𝑑𝑛𝑥) for Ω ⊂ ℝ𝑛
𝐶∞(Ω)
0
an open, bounded(and sufficiently regular) set, this recovers, as a particular
caseofageneral resultduetoG.Grubb,that theeigenvalueproblem forthe
Krein Laplacian 𝑆 (i.e., theKrein–von Neumann extension of 𝑆),
𝐾
𝑆 𝑣=𝜆𝑣, 𝜆∕=0,
𝐾
isinone-to-onecorrespondencewiththeproblemofthebucklingofaclamped
plate,
(−Δ)2𝑢=𝜆(−Δ)𝑢 in Ω, 𝜆∕=0, 𝑢∈𝐻2(Ω),
0
where 𝑢 and 𝑣 are related via thepair of formulas
𝑢=𝑆−1(−Δ)𝑣, 𝑣=𝜆−1(−Δ)𝑢,
𝐹
with 𝑆 the Friedrichsextension of 𝑆.
𝐹
BaseduponworkpartiallysupportedbytheUSNationalScienceFoundationunderGrantNos.
DMS-0400639andFRG-0456306andtheAustrianScienceFund(FWF)underGrantNo.Y330.
2 M.S. Ashbaugh et al.
ThisestablishestheKreinextensionasanaturalobjectinelasticitythe-
ory (inanalogy totheFriedrichsextension,which foundnaturalapplications
in quantummechanics, elasticity, etc.).
Inthesecond,andprincipalpartofthissurvey,westudyspectralprop-
ertiesfor𝐻 ,theKrein–vonNeumannextensionoftheperturbedLaplacian
𝐾,Ω
−Δ+𝑉 (in short, theperturbedKrein Laplacian) definedon 𝐶∞(Ω),where
0
𝑉 is measurable, bounded and nonnegative, in a bounded open set Ω ⊂ ℝ𝑛
belongingtoaclassofnonsmoothdomainswhichcontainsallconvexdomains,
along with all domains of class 𝐶1,𝑟, 𝑟>1/2. (Contrary to other uses of the
notion of “domain”, a domain in this survey denotes an open set without
any connectivity hypotheses. In addition, by a “smooth domain” we mean
a domain with a sufficiently smooth, typically, a 𝐶∞-smooth, boundary.) In
particular, in the aforementioned context we establish the Weyl asymptotic
formula
( )
#{𝑗 ∈ℕ∣𝜆 ≤𝜆}=(2𝜋)−𝑛𝑣 ∣Ω∣𝜆𝑛/2+𝑂 𝜆(𝑛−(1/2))/2 as 𝜆→∞,
𝐾,Ω,𝑗 𝑛
where 𝑣 =𝜋𝑛/2/Γ((𝑛/2)+1) denotes the volume of the unit ball in ℝ𝑛, ∣Ω
𝑛
denotes the volume of Ω, and 𝜆 , 𝑗 ∈ ℕ, are the non-zero eigenvalues of
𝐾,Ω,𝑗
𝐻 ,listedinincreasingorderaccordingtotheirmultiplicities.Weprovethis
𝐾,Ω
formula by showing that the perturbed Krein Laplacian (i.e., the Krein–von
Neumann extension of −Δ+𝑉 defined on 𝐶∞(Ω)) is spectrally equivalent
0
to the buckling of a clamped plate problem, and using an abstract result
of Kozlov from the mid 1980’s. Our work builds on that of Grubb in the
early1980’s,whohasconsideredsimilarissuesforellipticoperatorsinsmooth
domains, and shows that the question posed by Alonso and Simon in 1980
pertainingtothevalidityof theaboveWeylasymptoticformula continuesto
havean affirmative answer in this nonsmooth setting.
We also study certain exterior-type domains Ω = ℝ𝑛∖𝐾, 𝑛 ≥ 3, with
𝐾 ⊂ℝ𝑛compactandvanishingBesselcapacity𝐵 (𝐾)=0,toproveequality
2,2
of Friedrichs and Krein Laplacians in 𝐿2(Ω;𝑑𝑛𝑥), that is, −Δ∣ has a
𝐶∞(Ω)
0
uniquenonnegativeself-adjoint extension in 𝐿2(Ω;𝑑𝑛𝑥).
MathematicsSubjectClassification(2010). Primary35J25, 35J40, 35P15; sec-
ondary 35P05, 46E35, 47A10, 47F05.
Keywords.Lipschitzdomains,KreinLaplacian,eigenvalues,spectralanalysis,
Weylasymptotics, bucklingproblem.
Contents
1. Introduction ............................................................. 3
2. The abstract Krein–vonNeumann extension ............................. 15
3. The abstract Krein–vonNeumann extension and its connection
to an abstract buckling problem ......................................... 24
4. Trace theory in Lipschitz domains ....................................... 31
4.1. Dirichlet and Neumann traces in Lipschitz domains ................ 31
4.2. Perturbed Dirichlet and Neumann Laplacians ..................... 37