Unbounded Self-adjoint Operators on Hilbert Space

435 Pages·2012·2.507 MB·English

by  Konrad Schmüdgen (auth.)

265
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Konrad Schmüdgen
Unbounded
Self-adjoint Operators
on Hilbert Space
KonradSchmüdgen
Dept.ofMathematics
UniversityofLeipzig
Leipzig,Germany
ISSN0072-5285 GraduateTextsinMathematics
ISBN978-94-007-4752-4 ISBN978-94-007-4753-1(eBook)
DOI10.1007/978-94-007-4753-1
SpringerDordrechtHeidelbergNewYorkLondon
LibraryofCongressControlNumber:2012942602
MathematicsSubjectClassification: 47B25,81Q10(Primary),47A,47E05,47F05,35J(Secondary)
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To ELISA
Preface and Overview
This book is designed as an advanced text on unbounded self-adjoint operators in
Hilbertspaceandtheirspectraltheory,withanemphasisonapplicationsinmathe-
maticalphysicsandvariousfieldsofmathematics.Thoughinseveralsectionsother
classes of unbounded operators (normal operators, symmetric operators, accretive
operators, sectorial operators) appear in a natural way and are developed, the leit-
motifistheclassofunboundedself-adjointoperators.
Beforeweturntotheaimsandthecontentsofthisbook,webrieflyexplainthe
two main notions occurring therein. Suppose that H is a Hilbert space with scalar
product (cid:2)·,·(cid:3) and T is a linear operator on H defined on a dense linear subspace
D(T).ThenT issaidtobeitsymmetricif
(cid:2)Tx,y(cid:3)=(cid:2)x,Ty(cid:3) forx,y∈D(T). (0.1)
TheoperatorT iscalledself-adjointifitissymmetricandifthefollowingproperty
issatisfied:Supposethaty∈Handthereexistsavectoru∈Hsuchthat(cid:2)Tx,y(cid:3)=
(cid:2)x,u(cid:3)forallx∈D(T).Theny liesinD(T).(SinceD(T)isassumedtobedense
inH,itfollowsthenthatu=Ty.)
UsuallyitiseasytoverifythatEq.(0.1)holds,sothatthecorrespondingopera-
torissymmetric.Forinstance,if Ω isanopenboundedsubsetofRd andT isthe
LaplacianΔonD(T)=C∞(Ω)intheHilbertspaceL2(Ω),asimpleintegration-
0
by-partscomputationyields(0.1).Ifthesymmetricoperatorisbounded,itscontin-
uous extension to H is self-adjoint. However, for unbounded operators, it is often
difficulttoproveornottrue(asinthecaseT =Δ)thatasymmetricoperatorisself-
adjoint.Differentialoperatorsandmostoperatorsoccurringinmathematicalphysics
arenotbounded.Dealingwithunboundedoperatorsleadsnotonlytomanytechni-
calsubtleties;itoftenrequiresthedevelopmentofnewmethodsortheinventionof
newconcepts.
Self-adjoint operators are fundamental objects in mathematics and in quantum
physics.Thespectra(cid:2)ltheoremstatesthatanyself-adjointoperatorT hasanintegral
representationT = λdE(λ)withrespecttosom(cid:2)euniquespectralmeasureE.This
gives the possibilityto definefunctions f(T)= f(λ)dE(λ) of the operator and
to develop a functional calculus as a powerful tool for applications. The spectrum
vii
viii PrefaceandOverview
of a self-adjoint operator is always a subset of the reals. In quantum physics it is
postulatedthateachobservableisgivenbyaself-adjointoperatorT.Thespectrum
ofT isthenthesetofpossiblemeasuredvaluesoftheobservable,andforanyunit
vectorx∈HandsubsetM⊆R,thenumber(cid:2)E(M)x,x(cid:3)istheprobabilitythatthe
measured value in the state x lies in the set M. If T is the Hamilton operator, the
one-parameterunitarygroupt→eitT describesthequantumdynamics.
All this requires the operator T to be self-adjoint. For general symmetric oper-
ators T,thespectrumisnolon(cid:2)gerasubsetofthereals,anditisimpossibletoget
anintegralrepresentationT = λdE(λ)ortodefinetheexponentiationeitT.That
is,thedistinctionbetweensymmetricoperatorsandself-adjointoperatorsiscrucial!
However,manysymmetricoperatorsthat are not self-adjointcan beextendedto a
self-adjointoperatoractingonthesameHilbertspace.
Themainaimsofthisbookarethefollowing:
• toprovideadetailedstudyofunboundedself-adjointoperatorsandtheirproper-
ties,
• todevelopmethodsforprovingtheself-adjointnessofsymmetricoperators,
• tostudyanddescribeself-adjointextensionsofsymmetricoperators.
A particular focus and careful consideration is on the technical subtleties and
difficultiesthatarisewhendealingwithunboundedoperators.
Letusgiveanoverviewofthecontentsofthebook.PartIisconcernedwiththe
basicsofunboundedclosedoperatorsonaHilbertspace.Theseincludefundamental
general concepts such as regular points, defect numbers, spectrum and resolvent,
andclassesofoperatorssuchassymmetricandself-adjointoperators,accretiveand
sectorialoperators,andnormaloperators.
Ourfirstmaingoalisthetheoryofspectralintegralsandthespectraldecompo-
sitionofself-adjointandnormaloperators,whichistreatedindetailinPartII.We
use the bounded transform to reduce the case of unbounded operators to bounded
ones and derive the spectral theorem in great generality for finitely many strongly
commuting unbounded normal operators. The functional calculus for self-adjoint
operatorsdevelopedherewillbewillbeessentialfortheremainderofthebook.
PartIIIdealswithgeneratorsofone-parametergroupsandsemigroups,aswellas
withanumberofimportantandtechnicaltopicsincludingthepolardecomposition,
quasi-analyticandanalyticvectors,andtensorproductsofunboundedoperators.
Thesecondmainthemeofthebook,addressedinPartIV,isperturbationsofself-
adjointnessandofspectraofself-adjointoperators.TheKato–Rellichtheorem,the
invariance of the essential spectrum under compact perturbations, the Aronszajn–
DonoghuetheoryofrankoneperturbationsandKrein’sspectralshiftandtracefor-
mulaaretreatedtherein.Aguidingmotivationformanyresultsinthebook,andin
thispartinparticular,areapplicationstoSchrödingeroperatorsarisinginquantum
mechanics.
Part V contains a detailed and concise presentation of the theory of forms and
theirassociatedoperators.Thisisthethirdmainthemeofthebook.Herethecen-
tralresultsarethreerepresentationtheoremsforclosedforms,oneforlowersemi-
boundedHermitianformsandtwoothersforboundedcoerciveformsandforsec-
torialforms.OthertopicstreatedincludetheFriedrichsextension,theorderrelation
PrefaceandOverview ix
of self-adjoint operators, and the min–max principle. The results on forms are ap-
pliedtothestudyofdifferentialoperators.TheDirichletandNeumannLaplacians
onboundedopensubsetsofRd andWeyl’sasymptoticformulafortheeigenvalues
oftheDirichletLaplacianaredevelopedindetail.
The fourth major main theme of the book, featured in Part VI, is the self-
adjoint extension theory of symmetric operators. First, von Neumann’s theory of
self-adjoint extensions, and Krein’s theory and the Ando–Nishio theorem on pos-
itive self-adjoint extensions are investigated. The second chapter in Part VI gives
an extensive presentation of the theory of boundary triplets. The Krein–Naimark
resolventformulaandtheKrein–Birman–Vishiktheoryonpositiveself-adjointex-
tensionsaretreatedinthiscontext.ThetwolastchaptersofPartVIareconcerned
withtwoimportanttopicswhereself-adjointnessandself-adjointextensionsplaya
crucialrole.TheseareSturm–LiouvilleoperatorsandtheHamburgermomentprob-
lemontherealline.
Throughoutthe book applicationsto Schrödingeroperators and differential op-
eratorsareourguidingmotivation,andwhileanumberofspecialoperator-theoretic
resultsontheseoperatorsarepresented,itisworthstatingthatthisisnotaresearch
monographonsuchoperators.Again,theemphasisisonthegeneraltheoryofun-
bounded self-adjoint Hilbert space operators. Consequently, basic definitions and
factsonsuchtopicsasSobolevspacesarecollectedinanappendix;wheneverthey
areneededforapplicationstodifferentialoperators,theyaretakenforgranted.
This book is an outgrowth of courses on various topics around the theory of
unbounded self-adjoint operators and their applications, given for graduate and
Ph.D. students over the past several decades at the University of Leipzig. Some
of these covered advanced topics, where the material was mainly to be found in
researchpapersandmonographs,withanysuitableadvancedtextnotablymissing.
Mostchaptersofthisbookaredrawnfromtheselectures.Ihavetriedtokeepdif-
ferent parts of the book as independent as possible, with only one exception: The
functional calculus for self-adjoint operators developed in Sect. 5.3 is used as an
essentialtoolthroughout.
Thebookcontainsanumberofimportantsubjects(Krein’sspectralshift,bound-
arytriplets,thetheoryofpositiveself-adjointextensions,andothers)andtechnical
topics (the tensor product of unbounded operators, commutativity of self-adjoint
operators,theboundedtransform,Aronzajn–Donoghuetheory)whicharerarelyif
everpresentedintextbooks.Itisparticularlyhopedthatthematerialpresentedwill
be found to be useful for graduate students and young researchers in mathematics
andmathematicalphysics.
Advancedcoursesonunboundedself-adjointoperatorscanbebuiltonthisbook.
Oneshouldprobablystartwiththegeneraltheoryofclosedoperatorsbypresenting
the core material of Sects. 1.1, 1.2, 1.3, 2.1, 2.2, 2.3, 3.1, and 3.2. This could be
followedbyspectralintegralsandthespectraltheoremfor unboundedself-adjoint
operatorsbasedonselectedmaterialfromSects.4.1,4.2,4.3and5.2,5.3,avoiding
technicalsubtleties.Therearemanypossibilitiestocontinue.Onecouldchooserel-
ativelyboundedperturbationsandSchrödingeroperators(Chap.8),orpositiveform
andtheirapplications(Chap.10),orunitarygroups(Sect.6.1),orvonNeumann’s
x PrefaceandOverview
extension theory (Sects. 13.1, 13.2), or linear relations (Sect. 14.1) and boundary
triplets (Chap. 14). A large number of special topics treated in the book could be
usedasapartofanadvancedcourseoraseminar.
The prerequisites for this book are the basics in functional analysis and of the
theory of bounded Hilbert space operators as covered by a standard one semester
courseonfunctionalanalysis,togetherwithagoodworkingknowledgeofmeasure
theory. The applications on differential operators require some knowledge of or-
dinaryandpartialdifferentialequationsandofSobolevspaces.InChaps.9and16
afewselectedresultsfromcomplexanalysisarealsoneeded.Fortheconvenienceof
thereader,wehaveaddedsixappendices;onboundedoperatorsandclassesofcom-
pactoperators,onmeasuretheory,ontheFouriertransform,onSobolevspaces,on
absolutely continuous functions, and on Stieltjes transforms and Nevanlinna func-
tions. These collect a number of special results that are used at various places in
thetext.Fortheresults,herewehaveprovidedeitherprecisereferencestostandard
worksorcompleteproofs.
Afewgeneralnotationsthatarerepeatedlyusedarelistedafterthetableofcon-
tents.Amoredetailedsymbolindexcanbefoundattheendofthebook.Occasion-
ally,Ihaveusedeithersimplifiedoroverlappingnotations,andwhilethismightat
first sight seem careless, the meaning will be always clear from the context. Thus
the symbol x denotes a Hilbert space vector in one section and a real variable or
eventhefunctionf(x)=x inothers.
A special feature of the book is the inclusion of numerous examples which are
developedindetailandwhichareaccompaniedbyexercisesattheendsofthechap-
ters.Anumberofsimplestandardexamples(forinstance,multiplicationoperators
or differential operators −i d or − d2 on intervals with different boundary con-
dx dx2
ditions) are guiding examples and appear repeatedly throughout. They are used to
illustratevariousmethodsofconstructingself-adjointoperators,aswellasnewno-
tions evenwithintheadvancedchapters. Thereader mightalso consider someex-
amples as exercises with solutions and try to prove the statements therein first by
himself,comparingtheresultswiththegivenproofs.Oftenstatementsofexercises
provideadditionalinformationconcerningthetheory.Thereaderwhoisinterested
in acquiring an ability to work with unbounded operators is of course encouraged
toworkthroughasmanyoftheexamplesandexercisesaspossible.Ihavemarked
thesomewhatmoredifficultexerciseswithastar.Allstatedexercises(withthepos-
sibleexceptionofafewstarredproblems)arereallysolvablebystudents,ascanbe
attestedtobymyexperienceinteachingthismaterial.Thehintsaddedtoexercises
alwayscontainkeytricksorstepsforthesolutions.
In the course of teaching this subject and writing this book, I have benefited
from many excellent sources. I should mention the volumes by Reed and Simon
[RS1,RS2,RS4],Kato’smonograph[Ka],andthetexts(inalphabeticorder)[AG,
BSU, BEH, BS, D1, D2, DS, EE, RN, Yf, We]. I have made no attempt to give
precise credit for a result, an idea or a proof, though the few names of standard
theoremsarestatedinthebodyofthetext.Thenotesattheendofeachpartcontain
some(certainlyincomplete)informationaboutafewsources,importantpapersand
monographsinthisareaandhintsforadditionalreading.AlsoIhavelistedanumber