Indefinite Sturm-Liouville operators with the sing

arXiv:math/0612173v1[math.SP]6Dec2006IndefiniteSturm-LiouvilleoperatorswiththesingularcriticalpointzeroIllyaM.Karabash∗andAlekseyS.Kostenko†AbstractWepresentanewnecessaryconditionforsimilarityofindefiniteSturm-Liouvilleoperatorstoself-adjointoperators.ThisconditionisformulatedintermsofWeyl-Titchmarshm-functions.Alsoweobtainnecessaryconditionsforregularityofthecriticalpoints0and∞ofJ-nonnegativeSturm-Liouvilleoperators.Usingthisresult,weconstructseveralexamplesofoperatorswiththesingularcriticalpointzero.Inparticular,itisshownthat0isasingularcriticalpointoftheoperator−(sgnx)(3|x|+1)−4/3d2dx2actingintheHilbertspaceL2(R,(3|x|+1)−4/3dx)andthereforethisoperatorisnotsimilartoaself-adjointone.AlsoweconstructaJ-nonnegativeSturm-Liouvilleoperatoroftype(sgnx)(−d2/dx2+q(x))withthesameproperties.1IntroductionInthispaper,weareinterestedinSturm-Liouvilleequations−y′′(x)+q(x)y(x)=λr(x)y(x),x∈R,(1.1)withanindefiniteweightr.Morespecifically,westudythespectralpropertiesoftheassociatednon-self-adjointoperatorA:=1r−d2dx2+q(1.2)actingintheweightedHilbertspaceL2(R,|r(x)|dx)(anexplicitdefinitionoftheoperatorAisgiveninthenextsection).HeretheweightrandthepotentialqarerealandlocallyLebesgueintegrablefunctionsonR(q,r∈L1loc(R)),andxr(x)0forallx∈R\{0}.Thusrchangessignat0.Thespectralproblem−y′′(x)+q(x)y(x)=λ|r(x)|y(x),x∈R,(1.3)withthepositiveweight|r|isusuallytreatedinthecontextoftheHilbertspaceL2(R,|r(x)|dx)withthescalarproduct(f,g)=RRfg|r|dx.Undertheassumptionthat(1.3)isinthelimitpointcaseat−∞and+∞,theoperatorLassociatedwith(1.3)isself-adjointinL2(R,|r(x)|dx)andtheoperator∗Permanentaddress:DepartmentofPartialDifferentialEquations,InstituteofAppliedMathematicsandMechan-icsofNASofUkraine,R.Luxemburgstr.,74,Donetsk83114,UKRAINE([e-mail])†Permanentaddress:DepartmentofNonlinearAnalysis,InstituteofAppliedMathematicsandMechanicsofNASofUkraine,R.Luxemburgstr.,74,Donetsk83114,UKRAINE([e-mail])12Aassociatedwith(1.1)isJ-self-adjoint.ThelettermeansthatAisself-adjointwithrespecttotheindefiniteinnerproduct[f,g]:=(Jf,g)=ZRfgrdx,wheretheoperatorJisdefinedbyJ:f(x)7→(sgnx)f(x).Obviously,theoperatorsAandLareconnectedbytheequalityA=JL.NoticealsothattheoperatorAisnon-self-adjoint(intheHilbertspaceL2(R,|r(x)|dx)).ThemainobjectofthepresentpaperisthesimilarityoftheoperatorAtoaself-adjointoperator.LetusrecallthattwoclosedoperatorsT1andT2inaHilbertspaceHarecalledsimilarifthereexistaboundedoperatorSwiththeboundedinverseS−1inHsuchthatSdom(T1)=dom(T2)andT2=ST1S−1.ThesimilarityofthecorrespondingJ-self-adjointoperatorstoaself-adjointoperatorisessentialforthesolutionofforward-backwardboundaryvalueproblems,whichariseincertainphysicalmodels,particularlyintransportandscatteringtheory(see[4,24,16,20,21]),andinthetheoryofrandomprocesses(see[37]andreferencestherein).IftheoperatorLisnonnegative,L≥0,onecanstudythesimilarityproblemfortheoperatorAinthecontextofthespectraltheoryofJ-nonnegativeoperators[33](necessarynotionsandfactsarecontainedinSection2.3).If,inaddition,theresolventsetoftheoperatorAisnonempty,ρ(A)6=∅,thentheoperatorApossessesthefollowingproperties:(i)thespectrumofAisreal,σ(A)⊂R;(ii)ifλ6=0isaneigenvalueofA,thenitissemisimple(i.e.,ker(A−λ)=ker(A−λ)2);(iii)if0isaneigenvalueofA,thenitsRieszindex≤2,i.e.,kerA2=kerA3(generally,0maybeanonsemisimpleeigenvalue).Moreover,AadmitsaspectralfunctionEA(Δ).ThepropertiesofEA(Δ)aresimilartothepropertiesofaspectralfunctionofaself-adjointoperator.Themaindifferenceistheoccurrenceofcriticalpoints.SignificantlydifferentbehaviorofthespectralfunctionEA(Δ)occursatsingularcriticalpointinanyneighborhoodofwhichthespectralfunctionisunbounded.Thecriticalpoints,whicharenotsingular,arecalledregular.Itshouldbestressedthatonly0and∞maybecriticalpointsforJ-nonnegativeoperators.UndertheadditionalassumptionkerA=kerA2,thefollowingassertionsareequivalent:(i)Aissimilartoaself-adjointoperatorinH;(ii)0and∞areregularcriticalpointsofA.In[4],BealsshowedthattheeigenfunctionsofregularSturm-Liouvilleproblemsofthetype(1.1)formaRieszbasisifr(x)behaveslikeapowerofxat0.ImprovedversionsofBeals’conditionhavebeenprovidedin[5,12,38,39,40,41].In[5,12],singulardifferentialoperatorswithindefiniteweightshavebeenconsideredandtheregularityofthecriticalpoint∞wasprovenforawideclassofweightfunctions.TheexistenceofindefiniteSturm-Liouvilleoperatorswiththesingularcriticalpoint∞wasestablishedin[41],andcorrespondingexampleswereconstructedin[1,13,38].Thequestionofnonsingularityof0ismuchharder.Itwasshownin[6,10,14,17,22,23,27,28,29]that0isaregularcriticalpointforseveralmodelclassesofdifferentialoperators.In[23]severalnecessarysimilarityconditionsintermsofWeylfunctionswereobtainedalso.Thefollowingproblemnaturallyarisesinthiscontext:3Problem1.WhetherthereareanyJ-nonnegativeSturm-LiouvilleoperatorsAwiththesingularcriticalpoint0.Itwillbeshowninthepresentpaperthatthoseoperatorsdoexist.Thepaperisorganizedasfollows.InSection2wesummarizenecessarydefinitionsandstatementsfromthespectraltheoryofSturm-LiouvilleoperatorsandfromthespectraltheoryofJ-nonnegativeoperators.Themainresultsofthepaperare