Regular approximations of singular Sturm-Liouville

REGULARAPPROXIMATIONSOFSINGULARSTURM-LIOUVILLEPROBLEMSWITHLIMIT-CIRCLEENDPOINTSL.KONG,Q.KONG,H.WU,ANDA.ZETTLAbstract.Foranyself-adjointrealizationSofasingularSturm-Liouvilleequationonaninterval(a;b)withlimit-circleendpoints,weconstructafamilyofself-adjointrealizationsSr;r2(0;1);ofthisequationonsubintervals(ar;br)of(a;b)suchthateveryeigenvalueofSisthelimitofacontinuouseigenvaluebranchofthisfamily.Ofparticularinterestarethecaseswhenatleastoneendpointisoscillatoryortheleadingcoecientfunctionchangessign.Inthesecases,weshowthattheindexdeterminingeachcontinuouseigenvaluebranchhasaninnitenumberofjumpdiscontinuitiesandgiveanexplicitcharacterizationofthesediscontinuities.1.IntroductionGivenaself-adjointrealizationSofasingularSturm-Liouvilleequation,howcanitsspectrumbeapproximatedbyeigenvaluesofregularSturm-Liouvilleproblems(SLP's)?Thisquestionhasbeenstudiedbymanyauthors,forsomerecentresultssee[1,2,8,9,10].In[2],Bailey,Everitt,andZettlconstructedanalgorithmforcomputingtheeigenvaluesofsingularSLP'swithpositiveleadingcoecient,limitcircleendpoints,andseparatedboundarycondition(BC),seealso[10].In[1]theabovequestionwasstudiedforgeneralsingularself-adjointSLP'switheitherlimit-circleorlimit-pointendpointsandeitherseparatedorcoupledBC's.Convergencepropertieswereestablishedin[1]usingtheabstracttheoryofstrongresolventandnormresolvent,particularlytheHilbert-Schmidtnormresolvent,convergenceofself-adjointoperatorsinHilbertspaces.AlthoughtheseverygeneralandpowerfulabstractmethodsprovidesomeofthetheoreticalunderpinningsfortheFortrancodeSLEIGN2[3],theydonotyieldexplicitandconstructivealgorithmswhichcanbenumericallyimplemented.Inthispaperweinvestigatethegenerallimit-circleSLP'swithseparatedorcoupledBC'swheretheleadingcoecientmaychangesign.Weestablishatheoryforcontinuouseigenvaluebranchesofaone-parameterfamilyof\inducedrealizationsSr;r2(0;1),totheeigenvaluesofS.Thisallowsustoutilizeandextendtheresultsoncontinuouseigenvaluebranchesobtainedin[16]forregularSLP'stosingularSLP's.WeshowthatinthesimplestcasewhenthespectrumofSisboundedbelow,thereexistsr2(0;1)suchthatforeachn2N0:=f0;1;2;:::g,then-theigenvalueofSr,n(Sr)forr2(r;1),constituteacontinuouseigenvaluebranchton(S),andtheminimumvalueofsuchrisfound;inthecasewhenatleastoneendpointisoscillatory,theindexfunction~n(r)ofacontinuouseigenvaluebranch~n(r)(Sr)ton(S)hasaninnitenumberofjumpdiscontinuities.Thus,inthelattercase,itisimportanttogiveacarefulandsystematicanalysisofthejumpbehavioroftheindex~n(r)alongacontinuouseigenvaluebranch.Wegiveanexplicitcharacterizationofwheretheindex~n(r)changesandshowexactlyhowitchanges.Wealsoextendtheseresultstothemuchmorecomplicatedcasewheretheleadingcoecientfunctionpchangessign.Inthiscase,ourindexingschemeforeigenvaluesoftheassociatedregularproblemsisadoptedfromtherecentpaperofBindingandVolkmer[4]forseparatedBC'sandfrom[5]forcoupledBC's.Theresultsinthispapercanbeusedtoconstructalgorithmsforthenumerical1991MathematicsSubjectClassication.Primary34B24,34L15.Keywordsandphrases.Eigenvalueapproximation,Sturm-Liouvilleproblems,eigenvalueindexjumps.12L.KONG,Q.KONG,H.WU,ANDA.ZETTLcomputationofeigenvaluesofSLP'sfortheabovecases.However,inthispaper,wedonotpursuethenumericalimplementationofthesealgorithms.Thisworkwillbedoneinasubsequentpaper.Ourapproachiselementaryinthesensethatnoabstracttheoryofconvergenceofself-adjointoperatorsinHilbertspacesisused.ThemaintoolisatransformationwhichtransformstheSLPofasingularsecondorderscalarequationtoaboundaryvalueproblem(BVP)ofaregularrstordersystem.Also,acriticalroleisplayedbythejumpset,i.e.,thesetofregularself-adjointBC'swheretheindexedeigenvaluesasfunctionsoftheBC'shavediscontinuities.ForasystematicdiscussionofthejumpsetandthediscontinuousbehavioroftheindexforthecontinuouseigenvaluebranchesforregularSLP'swithp0,see[17].Thesingularproblemswithp0werestudiedin[18],andtheregularproblemswithpchangingsignwereinvestigatedin[5].Thispaperisorganizedasfollows:Section2containsabasicdiscussionoflimit-circleSLP's.ThemainresultsarestatedinSection3.TheproofstogetherwithsometechnicallemmasaregiveninSection4.2.Limit-circleSLP'sConsidertheSturm-Liouvilleequation(2.1)(py0)0+qy=wyonJ=(a;b);where1ab1;1=p;q;w2Lloc(J;R)andw0a.e.onJ;andLloc(J;R)isthesetofreal-valuedfunctionswhichareLebesgueintegrableonanycompactsubsetofJ.Here,eitherp0a.e.onJ;orpchangessignonJ,i.e.,thereexistsubsets(notnecessarilysubintervals)J1;J2ofJwithpositiveorinniteLebesguemeasuressuchthatp0onJ1andp0onJ2.ForanysubintervalJofJ,letL2(J;w)bethesetofcomplex-valuedmeasurablefunctionsfonJsuchthatRJjfj2w1.TheendpointaofJiscalledalimit-circleendpoint(orsimplyaisLC)ifallsolutionsofequation(2.1)areinL2((a;c);w)forsomec2J,anditiscalledalimit-pointendpoint(orsimplyaisLP)otherwise.Theendpointaissaidtobeoscillatory(orsimplyaisO)ifeveryreal-valuedsolutionhasaninnitenumberofzerosin(a;c)foranyc2J,anditisnonoscillatory(orsimplyaisNO)otherwise.LCOmeansLCandO,LCNOmeansLCandNO.Similardenitionsaremadefortheendpointb.Itiswell-knownthattheLCandLPclassicationisindependentofi