Mathematical Methods in Quantum ...

Mathematical Methods in Quantum Mechanics - G. Teschl, Angielskie techniczne

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MathematicalMethods
inQuantumMechanics
WithApplicationstoSchr¨odingerOperators
GeraldTeschl
GeraldTeschl
Fakult¨atf¨urMathematik
Nordbergstraße15
Universit¨atWien
1090Wien,Austria
E-mail:
URL:
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
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Version:April19,2006
Copyrightc1999-2005byGeraldTeschl
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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