Determining Spectra in Quantum Theory

223 Pages·2005·1.281 MB·English

by  Michael Demuth, Maddaly Krishna (auth.)

Progress in Mathematical Physics
Volume44
Editors-in-Chief
AnneBoutetdeMonvel,Universite´ ParisVIIDenisDiderot
GeraldKaiser,CenterforSignalsandWaves,Austin,TX
Editorial Board
C.Berenstein,UniversityofMaryland,CollegePark
SirM.Berry,UniversityofBristol
P.Blanchard,Universita¨tBielefeld
M.Eastwood,UniversityofAdelaide
A.S.Fokas,UniversityofCambridge
D.Sternheimer,Universite´ deBourgogne,Dijon
C.Tracy,UniversityofCalifornia,Davis
Michael Demuth
.
M Krishna
Determining Spectra
in Quantum Theory
Birkha¨user
Boston • Basel • Berlin
MichaelDemuth MaddalyKrishna
TechnischeUniversita¨tClausthal InstituteofMathematicalSciences
Institutfu¨rMathematik CITCampus—Taramani
D-38678Clausthal-Zellerfeld Chennai600113
Germany India
[email protected] [email protected]
MathematicsSubjectClassicifications(2000):31B15,31C15,35P05,35P25,35S30,47A10,47A11,
47A35,47A40,47A60,60K40,81Q10,81Q15,81U05,81U10
LibraryofCongressCataloging-in-PublicationData
Demuth,Michael,1946-
Determiningspectrainquantumtheory/MichaelDemuth,MaddalyKrisha.
p.cm.–(Progressinmathematicalphysics;v.44)
Includesbibliographicalreferencesandindex.
ISBN-13:978-0-8176-4366-9(alk.paper)
ISBN-10:0-8176-4366-4(alk.paper)
1.Potentialtheory(Mathematics).2.Scattering(Mathematics)3.Spectraltheory
(Mathematics)4.Operatortheory.I.Krishna,M.(Maddaly)II.Title.III.Series.
QA404.7.D462005
515’.7222–dc22 2005048054
ISBN-100-8176-4366-4 eISBN0-8176-4439-3 Printedonacid-freepaper.
ISBN-13978-0-8176-4366-9
(cid:1)c2005Birkha¨userBoston
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Preface
The main objective of this book is to give a collection of criteria available in
the spectral theory of selfadjoint operators, and to identify the spectrum and
its components in the Lebesgue decomposition. Many of these criteria were
published in several articles in different journals. We collected them, added
someandgavesomeoverviewthatcanserveasaplatformforfurtherresearch
activities.
Spectral theory of Schro¨dinger type operators has a long history; however
the most widely used methods were limited in number. For any selfadjoint
operator A on a separable Hilbert space the spectrum is identified by looking
atthetotalspectralmeasureassociatedwithit;oftenstudyingsuchameasure
meant looking at some transform of the measure. The transforms(cid:1)were of the
form(cid:1)f,ψ(A)f(cid:2)whichisexpressible,bythespectraltheorem,as ψ(x)dµ(x)
for some finite measure µ. The two most widely used functions ψ were the
exponential function ψ(x) = esx and the inverse function ψ(x) = (x−z)−1.
These functions are “usable” in the sense that they can be manipulated with
respect to addition of operators, which is what one considers most often in
the spectral theory of Schro¨dinger type operators.
Starting with this basic structure we look at the transforms of measures
from which we can recover the measures and their components in Chapter 1.
In Chapter 2 we repeat the standard spectral theory of selfadjoint oper-
ators. The spectral theorem is given also in the Hahn–Hellinger form. Both
Chapter 1 and Chapter 2 also serve to introduce a series of definitions and
notations, as they prepare the background which is necessary for the criteria
in Chapter 3.
SomecriteriaforspectralcomponentsaresummarisedinChapter3,which
is the central part of this book. They are based on Borel and Fourier trans-
form,oneigenfunctionandcommutatormethods,andonresultsinscattering
theory.
The criteria in Chapter 3 are very general and can be used for any semi-
bounded selfadjoint operator. Nevertheless, we want to illustrate the power
of the criteria. In doing so we investigate a series of applications. Hence we
vi Preface
introduce in Chapter 4 a collection of operators that are of interest in this
context. We do not intend to give the most general example or an exhaustive
list of operators. We selected those examples of interest to us.
In Chapter 4 the unperturbed and perturbed operators are introduced.
HereitisshownthatforthechosenoperatorsthecriteriaofChapter3canbe
used. Finally in Chapter 5 we apply the criteria and give a series of spectral
theroreticresultsfortheperturbedoperatorsintroducedabove,i.e.,weensure
that the operators satisfy the assumptions stated in the criteria.
In order to maintain a fluent description, we present references and com-
ments in the notes at the end of each chapter. If we do not prove a result in
the text, we give a detailed reference for the reader to find a proof (or work
out a proof on the lines given in the reference). We also give some additional
informationonsomeresultsinthenotes.Wetrytogiveproofsincaseswhere
we think that they improve the understanding of the text. While most of the
results are from the literature, there are occasional new results not available
in the literature and some of our proofs are improved or even new.
In this book there is a bias towards operators with absolutely continuous
spectrumbothforthedeterministicandrandomcases;thisbiasisintentional
and reflects our own interests.
Some theorems appear with the names of the authors and some may not.
We tried as much as possible to attach a name; however it is possible that we
unintentionally missed out in some cases.
There are a few books available on the subject of random Schro¨dinger op-
eratorsinadditiontoseveralreviewarticles.ThesearethebooksofCyconet
al. [54], Carmona–Lacroix [39], Figotin–Pastur [83], Stollmann [184] and the
most recent one of Bourgain [30]. The reviews of Simon [177] and Bellissard
[21] contain material not in the above books. There are more books deal-
ing with spectral theory of deterministic Schro¨dinger operators that include
theclassicalAchiezer–Glazman[1],Reed–Simon[159,156,158,157]volumes,
Weidmann[189]andthemostrecentonesofAmrein–AnneBoutetdeMonvel–
Georgescu [10] and Hislop–Sigal [92]. The mathematical theory of scatter-
inghasAmrein–Jauch-Sinha[12],Reed–Simon[158],Baumgartel–Wollenberg
[20], Sigal [172], Pearson [154], Yafaev [192] and D´erezenskii–G´erard [72] to
cite a few.
Thereseemstobealotmorematerial thatisstill notfound in books. For
example there seems to be scope for writing a book each on the density of
states, the random magnetic fields, and one on transport in random media,
and so on. Since the results are large in number, we decided to present just
a few illustrative results and probably do some disservice to a large number
of researchers by not including their work in this book; we apologise to all of
them.
Inthepreparationofthisbookwegreatlybenefittedfromdiscussionswith
Werner Kirsch, Peter Hislop, Walter Craig, Arne Jensen, Stefan Bo¨cker, Ivan
Vese’lic, Walter Renger, Andres Noll, Sven Eder, Eckhard Giere at various
times.WethankMichaelBarowhowaskindenoughtocarefullyreadthrough
Preface vii
the manuscript which helped us greatly in the preparation. We thank Anne
Boutet de Monvel for readily agreeing to consider this book in the Progress
in Mathematical Physics series.
We thank Ina Ku¨ster-Zippel and Ambika Vanchinathan for typesetting
the manuscript and the staff at Birkha¨user especially Ann Kostant, Regina
Gorenshteyn, and Elizabeth Loew, for their help in the publication process.
We acknowledge financial support from Deutsche Forschungsgemeinschaft
andIndianNationalScienceAcademy.Wearegratefultothekindhospitality
providedbytheInstitutesofMathematicsatTU-Clausthal,Ruhr-Universita¨t-
BochumandtheInstituteofMathematicalSciences,Chennaiatvarioustimes
that made it possible for us to meet each other and work on this book.
M. Demuth M. Krishna
Clausthal Chennai
August 2004
Contents
Preface ........................................................ v
1 Measures and Transforms.................................. 1
1.1 Measures ............................................... 1
1.2 Fourier Transform ....................................... 5
1.3 The Wavelet Transform .................................. 7
1.4 Borel Transform......................................... 16
1.5 Gesztesy–Krein–Simon ξ Function ......................... 24
1.6 Notes .................................................. 25
2 Selfadjointness and Spectrum.............................. 29
2.1 Selfadjointness .......................................... 29
2.1.1 Linear Operators and Their Inverses ................. 29
2.1.2 Closed Operators.................................. 30
2.1.3 Adjoint and Selfadjoint Operators ................... 32
2.1.4 Sums of Linear Operators .......................... 34
2.1.5 Sesquilinear Forms ................................ 35
2.2 Spectrum and Resolvent Sets ............................. 37
2.3 Spectral Theorem ....................................... 40
2.4 Spectral Measures and Spectrum .......................... 43
2.5 Spectral Theorem in the Hahn–Hellinger Form .............. 45
2.6 Components of the Spectrum ............................. 49
2.7 Characterization of the States in Spectral Subspaces ......... 53
2.8 Notes .................................................. 56
3 Criteria for Identifying the Spectrum...................... 59
3.1 Borel Transform ........................................ 59
3.2 Fourier Transform ....................................... 68
3.3 Wavelet Transform ...................................... 69
3.4 Eigenfunctions .......................................... 70
3.5 Commutators .......................................... 72
x Contents
3.6 Criteria Using Scattering Theory .......................... 80
3.6.1 Wave Operators................................... 81
3.6.2 Stability of the Absolutely Continuous Spectra........ 95
3.7 Notes ..................................................104
4 Operators of Interest ......................................111
4.1 Unperturbed Operators ..................................111
4.1.1 Laplacians........................................112
4.1.2 Unperturbed Semigroups and Their Kernels ..........119
4.1.3 Associated Processes...............................120
4.1.4 Regular Dirichlet Forms, Capacities and Equilibrium
Potentials ........................................121
4.2 Perturbed Operators.....................................125
4.2.1 Deterministic Potentials............................125
4.2.2 Random Potentials ................................133
4.2.3 Singular Perturbations .............................135
4.3 Notes ..................................................142
5 Applications ...............................................153
5.1 Borel Transforms........................................153
5.1.1 Kotani Theory ....................................153
5.1.2 Aizenman–Molchanov Method ......................160
5.1.3 Bethe Lattice .....................................172
5.1.4 Jaksi´c–Last Theorem ..............................181
5.2 Scattering ..............................................183
5.2.1 Decaying Random Potentials........................183
5.2.2 Obstacles and Potentials ...........................187
5.3 Notes ..................................................196
References.....................................................203
Index..........................................................215
Determining Spectra
in Quantum Theory
1
Measures and Transforms
1.1 Measures
In the following we consider a positive complete Borel measure on R to be
a countably additive set function µ from the Borel subsets BR to R+, with
the additional property that all subsets of measure zero sets are measurable.
In addition the measure µ is said to be σ-finite if there exists a countable
collection of sets {K }, R = ∪K with µ(K ) < ∞ for all i. It is said to be
i i i
finite if µ(R) < ∞ and it is said to be a probability measure if µ(R) = 1. A
finite complex measure is the sum µ1−µ2+i(µ3−µ4) of four finite positive
measures µi,i=1,2,3,4. When µ3 and µ4 are zero, such a complex measure
isalsocalledasignedmeasure.ForanypositiveBorelmeasure,itstopological
support is defined as the smallest closed subset of R whose complement has
measure zero. We use the following definitions concerning measures: In what
follows by a measure it is always meant to be a positive measure in the above
sense unless stated otherwise.
Definition 1.1.1. Let µ and ν be σ-finite Borel measures.
• µ is said to be supported on a set S if µ(R\S)=0.
• µ and ν are said to be mutually singular if there is a Borel set S such that
µ(S)=0 and ν(R\S)=0.
• µ is absolutely continuous with respect to ν if ν(S) = 0 =⇒ µ(S) = 0
and they are said to be equivalent if µ(S)=0 ⇐⇒ ν(S)=0.
Remark 1.1.2.Thesupportofameasureshouldsomehowcapturethesmall-
estanduniquesetonwhichthemeasureisconcentrated.However,exceptfor
atomic measures it is not possible to find such sets. The above definition of
support is so general that R itself is a support of any measure!
One of the ways to pin down a set where a measure is concentrated is to
compare it with another measure, for example the Lebesgue measure. Such
comparisons give a notion called minimal support. The minimal support of a
measure µ can be defined in this sense to be the set S such that µ(R\S)=0