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CAMBRIDGE STUDIES IN ADVANCED MATHEMATICS 200 EditorialBoard J. BERTOIN, B. BOLLOBA´S, W. FULTON, B. KRA, I. MOERDIJK, C. PRAEGER, P. SARNAK, B. SIMON, B. TOTARO SCHRO¨DINGEROPERATORS:EIGENVALUES ANDLIEB–THIRRINGINEQUALITIES TheanalysisofeigenvaluesofLaplaceandSchro¨dingeroperatorsisanimportantand classical topic in mathematical physics with many applications. This book presents a thorough introduction to the area, suitable for masters and graduate students, and includes an ample amount of background material on the spectral theory of linear operatorsinHilbertspacesandonSobolevspacetheory. OfparticularinterestisafamilyofinequalitiesbyLiebandThirringoneigenvalues ofSchro¨dingeroperators,whichtheyusedintheirproofofstabilityofmatter.Theﬁnal partofthisbookisdevotedtotheactiveresearchonsharpconstantsintheseinequalities andcontainsstate-of-the-artresults,servingasareferenceforexpertsandasastarting pointforfurtherresearch. RupertL.Frank holdsachairinappliedmathematicsatLMUMunichandisdoing researchprimarilyinanalysisandmathematicalphysics.Heisaninvitedspeakeratthe 2022InternationalCongressofMathematics. Ari Laptev is Professor at Imperial College London. His research interests include differentaspectsofspectraltheoryandfunctionalinequalities.Heisamemberofthe RoyalSwedishAcademyofScience,aFellowofEurAScandamemberofAcademia Europaea. TimoWeidl isProfessorattheUniversityofStuttgart.Heworksonspectraltheory andmathematicalphysics. Published online by Cambridge University Press CAMBRIDGE STUDIES IN ADVANCED MATHEMATICS EditorialBoard J.Bertoin,B.Bolloba´s,W.Fulton,B.Kra,I.Moerdijk,C.Praeger,P.Sarnak,B.Simon,B.Totaro AllthetitleslistedbelowcanbeobtainedfromgoodbooksellersorfromCambridgeUniversityPress.Fora completeserieslisting,visitwww.cambridge.org/mathematics. AlreadyPublished 162C.J.Bishop&Y.PeresFractalsinProbabilityandAnalysis 163A.BovierGaussianProcessesonTrees 164P.SchneiderGaloisRepresentationsand(ϕ,(cid:3))-Modules 165P.Gille&T.SzamuelyCentralSimpleAlgebrasandGaloisCohomology(2ndEdition) 166D.Li&H.QueffelecIntroductiontoBanachSpaces,I 167D.Li&H.QueffelecIntroductiontoBanachSpaces,II 168J.Carlson,S.Mu¨ller-Stach&C.PetersPeriodMappingsandPeriodDomains(2ndEdition) 169J.M.LandsbergGeometryandComplexityTheory 170J.S.MilneAlgebraicGroups 171J.Gough&J.KupschQuantumFieldsandProcesses 172T.Ceccherini-Silberstein,F.Scarabotti&F.TolliDiscreteHarmonicAnalysis 173P.GarrettModernAnalysisofAutomorphicFormsbyExample,I 174P.GarrettModernAnalysisofAutomorphicFormsbyExample,II 175G.NavarroCharacterTheoryandtheMcKayConjecture 176P.Fleig,H.P.A.Gustafsson,A.Kleinschmidt&D.PerssonEisensteinSeriesandAutomorphic Representations 177E.PetersonFormalGeometryandBordismOperators 178A.OgusLecturesonLogarithmicAlgebraicGeometry 179N.NikolskiHardySpaces 180D.-C.CisinskiHigherCategoriesandHomotopicalAlgebra 181A.Agrachev,D.Barilari&U.BoscainAComprehensiveIntroductiontoSub-RiemannianGeometry 182N.NikolskiToeplitzMatricesandOperators 183A.YekutieliDerivedCategories 184C.DemeterFourierRestriction,DecouplingandApplications 185D.Barnes&C.RoitzheimFoundationsofStableHomotopyTheory 186V.Vasyunin&A.VolbergTheBellmanFunctionTechniqueinHarmonicAnalysis 187M.Geck&G.MalleTheCharacterTheoryofFiniteGroupsofLieType 188B.RichterCategoryTheoryforHomotopyTheory 189R.Willett&G.YuHigherIndexTheory 190A.BobrowskiGeneratorsofMarkovChains 191D.Cao,S.Peng&S.YanSingularlyPerturbedMethodsforNonlinearEllipticProblems 192E.KowalskiAnIntroductiontoProbabilisticNumberTheory 193V.GorinLecturesonRandomLozengeTilings 194E.Riehl&D.VerityElementsof∞-CategoryTheory 195H.KrauseHomologicalTheoryofRepresentations 196F.Durand&D.PerrinDimensionGroupsandDynamicalSystems 197A.ShefferPolynomialMethodsandIncidenceTheory 198T.Dobson,A.Malnicˇ&D.MarusˇicˇSymmetryinGraphs 199K.S.Kedlayap-adicDifferentialEquations 200R.L.Frank,A.Laptev&T.WeidlSchro¨dingerOperators:EigenvaluesandLieb–ThirringInequalities 201J.vanNeervenFunctionalAnalysis Published online by Cambridge University Press Schro¨dinger Operators: Eigenvalues and Lieb–Thirring Inequalities RUPERT L. FRANK Ludwig-Maximilians-Universita¨tMu¨nchen ARI LAPTEV ImperialCollegeofScience,TechnologyandMedicine,London TIMO WEIDL Universita¨tStuttgart Published online by Cambridge University Press UniversityPrintingHouse,CambridgeCB28BS,UnitedKingdom OneLibertyPlaza,20thFloor,NewYork,NY10006,USA 477WilliamstownRoad,PortMelbourne,VIC3207,Australia 314–321,3rdFloor,Plot3,SplendorForum,JasolaDistrictCentre, NewDelhi–110025,India 103PenangRoad,#05–06/07,VisioncrestCommercial,Singapore238467 CambridgeUniversityPressispartoftheUniversityofCambridge. ItfurtherstheUniversity’smissionbydisseminatingknowledgeinthepursuitof education,learning,andresearchatthehighestinternationallevels ofexcellence. www.cambridge.org Informationonthistitle:www.cambridge.org/9781009218467 DOI:10.1017/9781009218436 ©RupertL.Frank,AriLaptev,andTimoWeidl2023 Thispublicationisincopyright.Subjecttostatutoryexception andtotheprovisionsofrelevantcollectivelicensingagreements, noreproductionofanypartmaytakeplacewithoutthewritten permissionofCambridgeUniversityPress. Firstpublished2023 AcataloguerecordforthispublicationisavailablefromtheBritishLibrary. ISBN978-1-009-21846-7Hardback CambridgeUniversityPresshasnoresponsibilityforthepersistenceoraccuracyof URLsforexternalorthird-partyinternetwebsitesreferredtointhispublication anddoesnotguaranteethatanycontentonsuchwebsitesis,orwillremain, accurateorappropriate. Published online by Cambridge University Press To Semra,SamiandSima To Marilyn,Maria,Vanya,KatyaandEugenia To Galia,AdiandAlex Published online by Cambridge University Press Published online by Cambridge University Press Contents Preface pagexi Overview 1 PARTONE BACKGROUNDMATERIAL 5 1 ElementsofOperatorTheory 7 1.1 Hilbertspaces,self-adjointoperatorsandthespectral theorem 8 1.2 Semiboundedoperatorsandforms,andthevariational principle 29 1.3 Comments 64 2 ElementsofSobolevSpaceTheory 66 2.1 Weakderivatives 67 2.2 Sobolevspaces 85 2.3 Compactembeddings 96 2.4 Sobolevinequalitiesonthewholespace 103 2.5 FriedrichsandPoincaréinequalities 112 2.6 Hardyinequalities 127 2.7 HomogeneousSobolevspaces 142 2.8 Theextensionproperty 146 2.9 Comments 157 PARTTWO THELAPLACEANDSCHRÖDINGER OPERATORS 165 3 TheLaplacianonaDomain 167 3.1 TheDirichletandNeumannLaplacians 170 3.2 Weyl’sasymptoticformulafortheDirichletLaplacian 179 vii Published online by Cambridge University Press viii Contents 3.3 Weyl’sasymptoticformulafortheNeumannLaplacian 192 3.4 Pólya’sinequalityfortilingdomains 197 3.5 Lower bounds for the eigenvalues of the Dirichlet Laplacian 199 3.6 Upper bounds for the eigenvalues of the Neumann Laplacian 209 3.7 Phasespaceinterpretation 212 3.8 Appendix:TheLaplacianinsphericalcoordinates 219 3.9 Comments 232 4 TheSchrödingerOperator 244 4.1 DeﬁnitionoftheSchrödingeroperator 247 4.2 Explicitlysolvableexamples 251 4.3 BasicspectralpropertiesofSchrödingeroperators 265 4.4 Weyl’sasymptoticformulaforSchrödingeroperators 279 4.5 TheCwikel–Lieb–Rozenbluminequality 282 4.6 TheLieb–Thirringinequality 292 4.7 Extendinginequalitiesandasymptotics 298 4.8 ReversedLieb–Thirringinequality 303 4.9 Comments 308 PARTTHREE SHARPCONSTANTSIN LIEB–THIRRINGINEQUALITIES 319 5 SharpLieb–ThirringInequalities 321 5.1 BasicfactsaboutLieb–Thirringconstants 323 5.2 Lieb–Thirringinequalitiesforspecialclassesofpotentials 335 5.3 Thesharpboundforγ = 1 inonedimension 349 2 5.4 Thesharpboundforγ = 3 inonedimension 353 2 5.5 Traceformulasforone-dimensionalSchrödingeroperators 362 5.6 Comments 380 6 SharpLieb–ThirringInequalitiesinHigherDimensions 385 6.1 Schrödingeroperatorswithmatrix-valuedpotentials 388 6.2 The Lieb–Thirring inequality with the semiclassical constant 395 6.3 Thesharpboundinthematrix-valuedcaseforγ = 3 400 2 6.4 Traceformulasinthematrixcase 411 6.5 Comments 414 7 MoreonSharpLieb–ThirringInequalities 417 7.1 Monotonicitywithrespecttothesemiclassicalparameter 418 7.2 Boundsforradialpotentials 426 Published online by Cambridge University Press Contents ix 7.3 Moreontheone-particleconstants 434 7.4 ThedualLieb–Thirringinequality 436 7.5 Comments 448 8 MoreontheLieb–ThirringConstants 451 8.1 MoreonLieb–Thirringinequalitiesinthematrix-valued case 452 8.2 Thesumofthesquarerootsoftheeigenvalues 456 8.3 Thesumoftheeigenvalues 457 8.4 SummaryonconstantsinLieb–Thirringinequalities 463 8.5 Comments 468 References 471 Index 504 Published online by Cambridge University Press

Schrödinger Operators: Eigenvalues and Lieb–Thirring Inequalities PDF

523 Pages·2022·3.241 MB·English

by Rupert L. Frank, Ari Laptev, Timo Weidl

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Author:	Rupert L. Frank, Ari Laptev, Timo Weidl
Publication Year:	2022
ISBN:	9781009218467
Pages:	523
Language:	English
File Size:	3.241
Format:	PDF
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