ESITheErwinSchrodingerInternational InstituteforMathematicalPhysics Pasteurgasse6/7 A-1090Wien,Austria

Abstract

hisapositivenumber,andvisanalmosteverywherenitemeasurablefunction periodicwithaperiodh0.Thisfunctioncanbecomplexvalued.Weprove (H)(x)=(x+h)+(x);x2R; (x?h)+v(x)(x); (1.2) (1.1) anordinarydierentialequationwithperiodiccoecients,onecallsitssolution translationbytheperiod: aBlochsolutionifitisinvariantuptoaconstantfactorwithrespecttothe ThecentralpointintheproofisrelatedtothenotionofBlochsolutions.For Theorem1.1.Equation(1.1)hasnosolutionsfromL2(R). Forequation(1.1) udependingh-periodicallyonx,u(x+h)=u(x): iscalledaBlochsolutionifitsatises(1.3)withacoecient (x+h0)=u(x);x2R: (1.3) TheworkwaspartiallysupportedbygrantINTAS-93-1815. 1 TypesetbyAMS-TEX dimensionalmodulovertheringofh-periodicfunctions.Theideaoftheproof istoshowthatifthereisanL2(R)-solutionof(1.1),thenthisequationhasalsoa idealeadstoanimmediateproofoftheanalogoustheoremfortheone-dimensional dierentialequationwithperiodiccoecients.Inthecaseunderconsidirationthe Thisdenitionisnaturalsincethesetofsolutionsofequation(1.1)isatwo- coecientufromthedenitionofBlochsolutionsdependsonx,andtheproof BlochsolutionbelongingtoL2(R),andtocheckthatthisisimpossible.Thesame andfromtheknowntheoremsonthestructureofthediscretespectrumoftheer- oftheproblemintermsofthecorrespondingproblemsontheinvariantlattices, mentofthetheoremcanbeeasilyderivedfromthedirect-integraldecomposition becomesalittlemorecomplicated. completelyuselesssinceitisquitedirectandelementary. godicoperators,see[PF].Buteveninthiscasethepresentproof,probably,isnot Inthecasewherevis,forexample,aboundedreal-valuedfunction,thestate

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