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1、AnInvitationtoRandomSchrodingeroperators¨WernerKirschInstitutfurMathematik¨Ruhr-UniversitatBochum¨D-44780Bochum,Germanyemail:werner.kirsch@rub.deAbstractarXiv:0709.3707v1[math-ph]24Sep2007ThisreviewisanextendedversionofmyminicourseattheEtatsdelarecherche:′Op′erateursdeSchr¨odingeral′
2、eatoiresattheUniversit′eParis13inJune2002,asummerschoolorganizedbyFr′ed′ericKlopp.TheselecturenotestrytogivesomeofthebasicsofrandomSchr¨odingeropera-tors.Theyaremeantfornonspecialistsandrequireonlyminorpreviousknowledgeaboutfunctionalanalysisandprobabilitytheory.Neverthelessthissurve
3、yincludescompleteproofsofLifshitztailsandAndersonlocalization.Copyrightbytheauthor.Copyingforacademicpurposesispermitted.13CONTENTS1.Preface52.Introduction:WhyrandomSchr¨odingeroperators?72.1.Thesettingofquantummechanics72.2.RandomPotentials72.3.Theonebodyapproximation103.Setup:TheAn
4、dersonmodel133.1.DiscreteSchr¨odingeroperators133.2.Spectralcalculus153.3.Somemorefunctionalanalysis183.4.Randompotentials194.Ergodicityproperties234.1.Ergodicstochasticprocesses234.2.Ergodicoperators255.Thedensityofstates295.1.De?nitionandexistence295.2.Boundaryconditions355.3.Thege
5、ometricresolventequation405.4.Analternativeapproachtothedensityofstates415.5.TheWegnerestimate436.Lifshitztails516.1.StatementoftheResult516.2.Upperbound526.3.Lowerbound567.Thespectrumanditsphysicalinterpretation617.1.GeneralizedEigenfunctionsandthespectrum617.2.Themeasuretheoretical
6、decompositionofthespectrum657.3.Physicalmeaningofthespectraldecomposition678.Andersonlocalization758.1.Whatphysicistsknow758.2.Whatmathematiciansprove768.3.Localizationresults778.4.FurtherResults789.TheGreen'sfunctionandthespectrum819.1.GeneralizedeigenfunctionsandthedecayoftheGreen'
7、sfunction819.2.Frommultiscaleanalysistoabsenceofa.c.spectrum839.3.Theresultsofmultiscaleanalysis859.4.Aniterationprocedure869.5.Frommultiscaleanalysistopurepointspectrum8910.Multiscaleanalysis9310.1.Strategy93410.2.Analyticestimate-?rsttry9510.3.Analyticestimate-secondtry10010.4.Prob
8、abilisticest
