mathematical methods in quantum mechanics

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1、MathematicalMethodsinQuantumMechanicsWithApplicationstoSchr¨odingerOperatorsGeraldTeschlGeraldTeschlFakult¨atf¨urMathematikNordbergstraße15Universit¨atWien1090Wien,AustriaE-mail:Gerald.Teschl@univie.ac.atURL:http://www.mat.univie.ac.at/˜gerald/2000Mathematicssubjectclassification.81-01,81Qxx,46-0

2、1Abstract.Thismanuscriptprovidesaself-containedintroductiontomath-ematicalmethodsinquantummechanics(spectraltheory)withapplicationstoSchr¨odingeroperators.Thefirstpartcoversmathematicalfoundationsofquantummechanicsfromself-adjointness,thespectraltheorem,quantumdynamics(includingStone’sandtheRAGEt

3、heorem)toperturbationtheoryforself-adjointoperators.ThesecondpartstartswithadetailedstudyofthefreeSchr¨odingerop-eratorrespectivelyposition,momentumandangularmomentumoperators.ThenwedevelopWeyl-TitchmarshtheoryforSturm-Liouvilleoperatorsandapplyittosphericallysymmetricproblems,inparticulartotheh

4、ydrogenatom.Nextweinvestigateself-adjointnessofatomicSchr¨odingeroperatorsandtheiressentialspectrum,inparticulartheHVZtheorem.Finallywehavealookatscatteringtheoryandproveasymptoticcompletenessintheshortrangecase.Keywordsandphrases.Schr¨odingeroperators,quantummechanics,un-boundedoperators,spectr

5、altheory.TypesetbyAMS-LATEXandMakeindex.Version:July16,2007Copyrightc1999-2007byGeraldTeschlContentsPrefaceviiPart0.PreliminariesChapter0.AfirstlookatBanachandHilbertspaces3§0.1.Warmup:Metricandtopologicalspaces3§0.2.TheBanachspaceofcontinuousfunctions10§0.3.ThegeometryofHilbertspaces15§0.4.Compl

6、eteness19§0.5.Boundedoperators20§0.6.LebesgueLpspaces22§0.7.Appendix:Theuniformboundednessprinciple28Part1.MathematicalFoundationsofQuantumMechanicsChapter1.Hilbertspaces33§1.1.Hilbertspaces33§1.2.Orthonormalbases35§1.3.TheprojectiontheoremandtheRieszlemma39§1.4.Orthogonalsumsandtensorproducts40

7、§1.5.TheC∗algebraofboundedlinearoperators42§1.6.Weakandstrongconvergence44§1.7.Appendix:TheStone–Weierstraßtheorem46Chapter2.Self-adjointnessandspectrum49iiiivContents§2.1.Somequantummechanics49§2.2.Self-adjointoperators52§2