mathematical methods in quantum mechanics - g. teschl

《mathematical methods in quantum mechanics - g. teschl》由会员上传分享，免费在线阅读，更多相关内容在学术论文-天天文库。

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

2、jectclassification.81-01,81Qxx,46-01Abstract.Thismanuscriptprovidesaself-containedintroductiontomath-ematicalmethodsinquantummechanics(spectraltheory)withapplicationstoSchr¨odingeroperators.Thefirstpartcoversmathematicalfoundationsofquantummechanicsfromself-adjoin

3、tness,thespectraltheorem,quantumdynamics(includingStone’sandtheRAGEtheorem)toperturbationtheoryforself-adjointoperators.ThesecondpartstartswithadetailedstudyofthefreeSchr¨odingerop-eratorrespectivelyposition,momentumandangularmomentumoperators.ThenwedevelopWeyl-

4、TitchmarshtheoryforSturm-Liouvilleoperatorsandapplyittosphericallysymmetricproblems,inparticulartothehydrogenatom.Nextweinvestigateself-adjointnessofatomicSchr¨odingeroperatorsandtheiressentialspectrum,inparticulartheHVZtheorem.Finallywehavealookatscatteringtheo

5、ryandproveasymptoticcompletenessintheshortrangecase.Keywordsandphrases.Schr¨odingeroperators,quantummechanics,un-boundedoperators,spectraltheory.TypesetbyAMS-LATEXandMakeindex.Version:April19,2006Copyrightc1999-2005byGeraldTeschlContentsPrefaceviiPart0.Prelimina

6、riesChapter0.AfirstlookatBanachandHilbertspaces3§0.1.Warmup:Metricandtopologicalspaces3§0.2.TheBanachspaceofcontinuousfunctions10§0.3.ThegeometryofHilbertspaces14§0.4.Completeness19§0.5.Boundedoperators20§0.6.LebesgueLpspaces22§0.7.Appendix:Theuniformboundednessp

7、rinciple27Part1.MathematicalFoundationsofQuantumMechanicsChapter1.Hilbertspaces31§1.1.Hilbertspaces31§1.2.Orthonormalbases33§1.3.TheprojectiontheoremandtheRieszlemma36§1.4.Orthogonalsumsandtensorproducts38§1.5.TheC∗algebraofboundedlinearoperators40§1.6.Weakandst

8、rongconvergence41§1.7.Appendix:TheStone–Weierstraßtheorem44Chapter2.Self-adjointnessandspectrum47iiiivContents§2.1.Somequantummechanics47§2.2.Self-adjointoperators50§