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1、APruferMethodforCalculatingEigenvaluesofSelf-AdjointSystemsofOrdinaryDi erentialEquations.Part1.LeonGreenbergDepartmentofMathematicsUniversityofMarylandCollegePark,Maryland20742AbstractAgeneralizationofthePrufermethod(forSturm-Liouvilleproblems)isdevel-opedforapproximatingtheeigenvaluesoflinear,se
2、lf-adjoint,elliptic,2-pointthboundaryvalueproblemsofarbitraryorder.Theneigenvaluecanbeapprox-imatedwithoutconsiderationofothereigenvalues,andanposteriorierroresti-mateisprovided.Themethodcanbeadaptedtoproblemswheretheeigenvalueoccursnonlinearlyinthedi erentialequationandmayoccurintheboundarycondit
3、ions.ThemethodisbasedonaformulaforthespectralfunctionN(),whichequalsthenumberofeigenvalueslessthan.InPart1,variationalprinciplesforeigenvaluesareappliedto ndthe rstoftwoformulasforN().Keywords:Eigenvalue,spectralfunction,self-adjoint,elliptic,boundarycondi-tion,energyform,Rayleighquotient,Prufe
4、rmethod,Hamiltoniansystem,symplecticstructure,Lagrangianplane,unitarymatrix,phaseangle,Sturmcomparisontheorem.AMS(MOS)classi cation:65L15TableofContentsPart1Contents1Introduction.12TheClassicalPruferMethod.43TheHamiltonianSystem.74Self-AdjointnessandSymplecticStructure.95LagrangianPlanes.126PhaseA
5、ngles.157TheEnergyForm.188TheSpectralFunctionN().229NonseparatedBoundaryConditions351Introduction.Twostandardapproachestothenumericalapproximationofeigenvaluesfor2-pointboundaryvalueproblemsarediscretizationandshooting.Discretizationmethods(suchas nitedi erencesand niteelements)involvesubstantial
6、arithmeticandthestorageoflargematrices.Moreover,theaccuracyquicklydeterioratesforthehighereigenvalues.Shootingmethodsrequirelessstorageandarithmetic,butusuallytheydonotdeterminethemodeoftheeigenvalue.(Thuswehavenowayofknowingifwehavefoundthe rst,thirdorseventeentheigenvalue.)ForSturm-Liouvilleprob
7、lems,thesedicultiesareavoidedbythePrufermethod,whichisashootingmethodbasedonoscillation.IthasbeenimplementedbyP.Bailey,M.GordonandL.Shampine[4]intheSLEIGNCode,andbyJ.D.Pryce[17]intheNAGlibrarycodeDO2KDF.Inthispa
