schrodinger semigroups

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1、BULLETIN(NewSeries)OFTHEAMERICANMATHEMATICALSOCIETYVolume7,Number3,November1982SCHR?DINGERSEMIGROUPSBYBARRYSIMONABSTRACT.LetH=-L+VbeageneralSchr?dingeroperatoronR"(v~>1),whereAistheLaplacedifferentialoperatorandVisapotentialfunctiononwhichweassumeminimalhypothesesofgrowt

2、handregularity,andinparticularallowVwhichareunboundedbelow.Wegiveageneralsurveyofthepropertiesofe~tH,t>0,andrelatedmappingsgivenintermsofsolutionsofinitialvalueproblemsforthedifferentialequationdu/dt+Hu=0.AmongthesubjectstreatedareL^-propertiesofthesemaps,ex?istenceofcont

3、inuousintegralkernelsforthem,andregularitypropertiesofeigenfunctions,includingHarnack'sinequality.CONTENTSA.IntroductionAl.OverviewA2.TheclassKvA3.LiteratureonlargerclassesB.L^-propertiesBI.L^-smoothingofsemigroupsB2.SobolevestimatesB3.ContinuityandderivativeestimatesB4.L

4、ocalizationB5.GrowthofL^-semigroupnormsast->ooB6.WeightedL2-spacesB7.Integralkernels:GeneralpotentialsB8.Integralkernels:SomespecialoperatorsforsomespecialpotentialsB9.TraceidealpropertiesBIO.ContinuityinVBl1.HypercontractivesemigroupsandallthatB12.Someremarksonthecasewhe

5、nHisunboundedbelowB13.ThemagneticcaseC.EigenfunctionsCI.Harnack'sinequalityandsubsolutionestimatesC2.Localestimatesonv

6、ry81-02,35-02;Secondary47F05,35P05.?1982AmericanMathematicalSociety0273-0979/82/0000-0350/121.00447448BARRYSIMONC6.ThelocalspectraldensityanditsclassicallimitC7.TheintegrateddensityofstatesC8.Allegretto-PiepenbrinktheoryC9.UniquecontinuationA.INTRODUCTIONAl.Overview.ByaSc

7、hr?dingeroperator,wemeanapartialdifferentialoperatoronR"oftheform(Al)H=H0+V;i/0=-iA;V=V(x)whereAisthe^-dimensionalLaplaceoperatorA=2vJ=ld2/dxf(thereasonfortheconvention-^Aratherthan-Awillbecomeclearlater).ThenamecomesfromtheformofSchr?dinger'sequationwhich,inunitswithh=m=

8、1reads(A2)idxp/dt=Hxp.HisthustheHamiltonianoperatorofanonrelativisticparticle;H0isthekineticener