Unbounded Self-adjoint Operators on Hilbert Space PDF
Preview Unbounded Self-adjoint Operators on Hilbert Space
265 Graduate Texts in Mathematics Graduate Texts in Mathematics SeriesEditors: SheldonAxler SanFranciscoStateUniversity,SanFrancisco,CA,USA KennethRibet UniversityofCalifornia,Berkeley,CA,USA AdvisoryBoard: ColinAdams,WilliamsCollege,Williamstown,MA,USA AlejandroAdem,UniversityofBritishColumbia,Vancouver,BC,Canada RuthCharney,BrandeisUniversity,Waltham,MA,USA IreneM.Gamba,TheUniversityofTexasatAustin,Austin,TX,USA RogerE.Howe,YaleUniversity,NewHaven,CT,USA DavidJerison,MassachusettsInstituteofTechnology,Cambridge,MA,USA JeffreyC.Lagarias,UniversityofMichigan,AnnArbor,MI,USA JillPipher,BrownUniversity,Providence,RI,USA FadilSantosa,UniversityofMinnesota,Minneapolis,MN,USA AmieWilkinson,UniversityofChicago,Chicago,IL,USA GraduateTextsinMathematicsbridgethegapbetweenpassivestudyandcreative understanding, offering graduate-level introductions to advanced topics in mathe- matics.Thevolumesarecarefullywrittenasteachingaidsandhighlightcharacter- isticfeaturesofthetheory.Althoughthesebooksarefrequentlyusedastextbooks ingraduatecourses,theyarealsosuitableforindividualstudy. Forfurthervolumes: www.springer.com/series/136 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) ©SpringerScience+BusinessMediaDordrecht2012 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped.Exemptedfromthislegalreservationarebriefexcerptsinconnection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. 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Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) 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
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