Mathematical_Methods_in_Quantum_Mechanics_-_G._Teschl.pdf

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Mathematical Methods in Quantum Mechanics
MathematicalMethods
inQuantumMechanics
WithApplicationstoSchr¨odingerOperators
GeraldTeschl
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GeraldTeschl
Fakult¨atf¨urMathematik
Nordbergstraße15
Universit¨atWien
1090Wien,Austria
2000Mathematicssubjectclassification.81-01,81Qxx,46-01
Abstract.Thismanuscriptprovidesaself-containedintroductiontomath-
ematicalmethodsinquantummechanics(spectraltheory)withapplications
toSchr¨odingeroperators.Thefirstpartcoversmathematicalfoundations
ofquantummechanicsfromself-adjointness,thespectraltheorem,quantum
dynamics(includingStone’sandtheRAGEtheorem)toperturbationtheory
forself-adjointoperators.
ThesecondpartstartswithadetailedstudyofthefreeSchr¨odingerop-
eratorrespectivelyposition,momentumandangularmomentumoperators.
ThenwedevelopWeyl-TitchmarshtheoryforSturm-Liouvilleoperatorsand
applyittosphericallysymmetricproblems,inparticulartothehydrogen
atom.Nextweinvestigateself-adjointnessofatomicSchr¨odingeroperators
andtheiressentialspectrum,inparticulartheHVZtheorem.Finallywe
havealookatscatteringtheoryandproveasymptoticcompletenessinthe
shortrangecase.
Keywordsandphrases.Schr¨odingeroperators,quantummechanics,un-
boundedoperators,spectraltheory.
TypesetbyA M S-L A T E XandMakeindex.
Version:April19,2006
Copyrightc1999-2005byGeraldTeschl
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Contents
Preface
vii
Part0.Preliminaries
Chapter0.AfirstlookatBanachandHilbertspaces 3
§0.1.Warmup:Metricandtopologicalspaces 3
§0.2.TheBanachspaceofcontinuousfunctions 10
§0.3.ThegeometryofHilbertspaces 14
§0.4.Completeness 19
§0.5.Boundedoperators 20
§0.6.LebesgueL p spaces 22
§0.7.Appendix:Theuniformboundednessprinciple 27
Part1.MathematicalFoundationsofQuantumMechanics
Chapter1.Hilbertspaces 31
§1.1.Hilbertspaces 31
§1.2.Orthonormalbases 33
§1.3.TheprojectiontheoremandtheRieszlemma 36
§1.4.Orthogonalsumsandtensorproducts 38
§1.5.TheC algebraofboundedlinearoperators 40
§1.6.Weakandstrongconvergence 41
§1.7.Appendix:TheStone–Weierstraßtheorem 44
Chapter2.Self-adjointnessandspectrum 47
iii
 
iv Contents
§2.1.Somequantummechanics 47
§2.2.Self-adjointoperators 50
§2.3.Resolventsandspectra 61
§2.4.Orthogonalsumsofoperators 67
§2.5.Self-adjointextensions 68
§2.6.Appendix:Absolutelycontinuousfunctions 72
Chapter3.Thespectraltheorem 75
§3.1.Thespectraltheorem 75
§3.2.MoreonBorelmeasures 85
§3.3.Spectraltypes 89
§3.4.Appendix:TheHerglotztheorem 91
Chapter4.Applicationsofthespectraltheorem 97
§4.1. Integralformulas 97
§4.2.Commutingoperators 100
§4.3.Themin-maxtheorem 103
§4.4.Estimatingeigenspaces 104
§4.5.Tensorproductsofoperators 105
Chapter5.Quantumdynamics 107
§5.1.ThetimeevolutionandStone’stheorem 107
§5.2.TheRAGEtheorem 110
§5.3.TheTrotterproductformula 115
Chapter6.Perturbationtheoryforself-adjointoperators 117
§6.1.RelativelyboundedoperatorsandtheKato–Rellichtheorem 117
§6.2.Moreoncompactoperators 119
§6.3.Hilbert–Schmidtandtraceclassoperators 122
§6.4.RelativelycompactoperatorsandWeyl’stheorem 128
§6.5.Strongandnormresolventconvergence 131
Part2.Schr¨odingerOperators
Chapter7.ThefreeSchr¨odingeroperator 139
§7.1.TheFouriertransform 139
§7.2.ThefreeSchr¨odingeroperator 142
§7.3.Thetimeevolutioninthefreecase 144
§7.4.TheresolventandGreen’sfunction 145

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