The Heun equation and the Calogero-Moser-Sutherlan

arXiv:math/0508093v1[math.CA]4Aug2005THEHEUNEQUATIONANDTHECALOGERO-MOSER-SUTHERLANDSYSTEMV:GENERALIZEDDARBOUXTRANSFORMATIONSKOUICHITAKEMURAAbstract.WeobtainisomonodromictransformationsforHeun’sequationbygen-eralizingDarbouxtransformation,andwefindpairsandtripletsofHeun’sequationwhichhavethesamemonodromystructure.BycomposinggeneralizedDarbouxtransformations,weestablishanewconstructionofthecommutingoperatorwhichensuresfinite-gapproperty.Asanapplication,weproveconjecturesinpartIII.1.IntroductionItwasshownin[9]thatsomepairsofSchr¨odingeroperatorsareisomonodromic.SetH1=−d2dx2+6℘(x),(1.1)H2=−d2dx2+2℘(x)+2℘(x+ω1)+2℘(x+ω2),(1.2)where℘(x)istheWeierstrass℘-functionwithperiods(2ω1,2ω3)andω2=−ω1−ω3.NowweconsidereigenfunctionsofH1(resp.H2)withtheeigenvalueE.SetΞ1(x,E)=9℘(x)2+3E℘(x)+E2−9g2/4,Ξ2(x,E)=(E−3e3)℘(x)+(E−3e2)℘(x+ω1)+(E−3e1)℘(x+ω2)+E2−3g2/2,Q(E)=(E2−3g2)(E−3e1)(E−3e2)(E−3e3),Λk(x,E)=pΞk(x,E)expZp−Q(E)dxΞk(x,E),(k=1,2),whereei=℘(ωi)(i=1,2,3)andg2=−4(e1e2+e2e3+e3e1).ThenitwasshownthatthefunctionsΛ1(x,E)andΛ1(−x,E)(resp.Λ2(x,E)andΛ2(−x,E))areeigen-functionsofH1(resp.H2)withtheeigenvalueE,andtheysatisfyΛk(±(x+2ωi),E)=Λk(x,E)exp∓12ZE√3g2−6˜Eηi+(2˜E2−3g2)ωiq−Q(˜E)d˜E,fork=1,2andi=1,2,3,whereηi=ζ(ωi)andζ(x)istheWeierstrasszetafunction.HencethemonodromyofeigenfunctionsofH1withtheeigenvalueEcoincideswiththatofeigenfunctionsofH2.Inthispaper,weinvestigatethisphenomenabyDarbouxtransformationandgen-eralizedDarbouxtransformation.Letφ0(x)beaneigenfunctionoftheoperator1991MathematicsSubjectClassification.33E10,34M35,82B23.12KOUICHITAKEMURAH=−d2/dx2+q(x)withaneigenvalueE0,i.e.−d2dx2+q(x)φ0(x)=E0φ0(x).Forthiscase,thepotentialq(x)iswrittenasq(x)=(φ′0(x)/φ0(x))′+(φ′0(x)/φ0(x))2+E0.SetL=d/dx−φ′0(x)/φ0(x)and˜H=−d2/dx2+q(x)−2(φ′0(x)/φ0(x))′.Thenwehave˜HL=LH.Hence,ifφ(x)isaneigenfunctionoftheoperatorHwiththeeigenvalueE,thenLφ(x)isaneigenfunctionoftheoperator˜HwiththeeigenvalueE.ThistransformationiscalledtheDarbouxtransformation.WegeneralizetheoperatorLtobethedifferentialoperatorofhigherorder,andwecallitthegeneralizedDarbouxtransformation.TheSchr¨odingeroperatorweconsiderinthispaperistheHamiltonianoftheBC1Inozemtsevmodel,whichiswrittenas(1.3)H(l0,l1,l2,l3)=−d2dx2+3Xi=0li(li+1)℘(x+ωi),whereω0=0.ThepotentialofthisoperatoriscalledtheTreibich-Verdierpotential,becauseTreibichandVerdier[13]foundandshowedthat,ifli∈Z≥0foralli∈{0,1,2,3},thenitisanalgebro-geometricfinite-gappotential.Forfurtherresultsonthissubject,see[2,5,6,8,12].Thealgebro-geometricfinite-gappropertycausethepossibilityforcalculationofeigenfunctionandmonodromyoftheoperatorH(l0,l1,l2,l3).Letf(x)beaneigenfunctionoftheoperatorH(l0,l1,l2,l3)withtheeigenvalueE,namely,(1.4)−d2dx2+3Xi=0li(li+1)℘(x+ωi)!f(x)=Ef(x).ThenthisequationisanellipticrepresentationofHeun’sequation.HereHeun’sequationisthestandardcanonicalformofaFuchsianequationwithfoursingularities(see[4]).Thus,solvingHeun’sequationisequivalenttostudyingeigenvaluesandeigenfunctionsoftheHamiltonianoftheBC1Inozemtsevmodel.Wenowdescribethemainresultofthispaper.Letαibeanumbersuchthatαi=−liorαi=li+1foreachi∈{0,1,2,3}.Setd=−P3i=0αi/2andassumed∈Z≥0.ThenthereexistsadifferentialoperatorLoforderd+1whichsatisfiesH(α0+d,α1+d,α2+d,α3+d)L=LH(l0,l1,l2,l3).NotethatKhareandSukhatme[3]essentiallyestablishedthisresultforthecased=0,thatisthecaseoforiginalDarbouxtransformation.Itfollowsimmediatelythat,ifφ(x)isaneigenfunctionoftheoperatorH(l0,l1,l2,l3)withaneigenvalueE,thenLφ(x)isaneigenfunctionoftheoperatorH(α0+d,α1+d,α2+d,α3+d)withtheeigenvalueE.SinceallcoefficientsoftheoperatorLwithrespecttothedifferential(d/dx)k(k=0,...,d+1)isshowntobedoubly-periodic,theoperatorLpreservesthedataofmonodromy.HencetheoperatorsH(l0,l1,l2,l3)andH(α0+d,α1+d,α2+d,α3+d)areisomonodromic,andisospectral,becauseboundaryconditionforspectralproblemischaracterizedbymonodromy.Notethattheconditiond∈Z≥0correspondstoquasi-solvabilityoftheoperatorH(l0,l1,l2,l3).HEUNEQUATIONV3Forthecasethatl0,l1,l2,l3areallintegers,thereexistsanoperatorH(˜l0,˜l1,˜l2,˜l3)suchthatthepairH(l0,l1,l2,l3)andH(˜l0,˜l1,˜l2,˜l3)isconnectedbyisomonodromictransforma-tion.Insomecases,theyareself-dual.Forexample,theoperatorH1(=H(2,0,0,0))inEq.(1.1)isconnectedtotheoperatorH2(=H(1,1,1,0))inEq.(1.2)bythetransforma-tionL=d/dx−℘′(x)/(2(℘(x)−e1))−℘′(x)/(2(℘(x)−e2)),i.e.wehaveH(1,1,1,0)L=LH(2,0,0,0).Forthecasethatl0+1/2,l1+1/2,l2+1/2,l3+1/2areallintegers,thereexiststwooperatorsH(l(1)0,l(1)1,l(1)2,l(1)3)andH(l(2)0,l(2)1,l(2)2,l(2)3)suchthatthetripletH(l0,l1,l2,l3),H(l(1)0,l(1)1,l(1)2,l(1)3)andH(l(2)0,l(2)1,l(2)2,l(2)3)isconnectedbyisomonodromictransformations.Inthepaper[8],finite-gappropertyoftheoperatorH(l0,l1,l2,l3)forthecasel0,l1,l2,l3∈Z≥0isstudied(seealso[13,12,2,5]).Especially,adifferentialoperatorAofoddorderwhichcommuteswithH(l0,l1,l2,l3)isconstructed.Inthispaper,weproposeanewmethodforconstructionofthecommutingoperatorbycomposingfourgeneral-izedDarbouxtransformations.NotethateachgeneralizedDarbouxtransformationiswrittenexplicitly.ToshowthatthecommutingoperatorconstructedbycomposingfourgeneralizedDarbouxtransformationscoincid