Schrödinger Operators: Eigenvalues and Lieb–Thirring Inequalities

523 Pages·2022·3.241 MB·English

by  Rupert L. Frank, Ari Laptev, Timo Weidl

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
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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
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Informationonthistitle:www.cambridge.org/9781009218467
DOI:10.1017/9781009218436
©RupertL.Frank,AriLaptev,andTimoWeidl2023
Thispublicationisincopyright.Subjecttostatutoryexception
andtotheprovisionsofrelevantcollectivelicensingagreements,
noreproductionofanypartmaytakeplacewithoutthewritten
permissionofCambridgeUniversityPress.
Firstpublished2023
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ISBN978-1-009-21846-7Hardback
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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
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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
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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
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