Inverse Scattering, the Coupling Constant Spectrum

arXiv:hep-th/0111067v17Nov2001RU01-14-BInverseScattering,theCouplingConstantSpectrum,andtheRiemannHypothesisN.N.KhuriDepartmentofPhysicsTheRockefellerUniversity,NewYork,NewYork10021AbstractItiswellknownthatthes-waveJostfunctionforapotential,λV,isanentirefunctionofλwithaninfinitenumberofzerosextendingtoinfinity.ForarepulsiveV,andatzeroenergy,thesezerosofthe”couplingconstant”,λ,willallberealandnegative,λn(0)0.Byrescalingλ,suchthatλn−1/4,andchangingvariablestos,withλ=s(s−1),itfollowsthatasafunctionofstheJostfunctionhasonlyzerosonthelinesn=12+iγn.ThusfindingarepulsiveVwhosecouplingconstantspectrumcoincideswiththeRiemannzeroswillestablishtheRiemannhypothesis,butthiswillbeaverydifficultandunguidedsearch.Inthispaperwemakeasignificantenlargementoftheclassofpotentialsneededforageneralizationoftheaboveidea.Wealsomakethisnewclassamenabletoconstructionviainversescatteringmethods.Weshowthatalloneneedsisaoneparameterclassofpotentials,U(s;x),whichareanalyticinthestrip,0≤Res≤1,ImsTo,andinadditionhaveanasymptoticexpansioninpowersof[s(s−1)]−1,i.e.U(s;x)=Vo(x)+gV1(x)+g2V2(x)+...+O(gN),withg=[s(s−1)]−1.ThepotentialsVn(x)arerealandsummable.UndersuitableconditionsontheV′nsandtheO(gN)termweshowthatthecondition,R∞o|fo(x)|2V1(x)dx6=0,wherefoisthezeroenergyandg=0JostfunctionforU,issufficienttoguaranteethatthezerosgnarerealandhencesn=12+iγn,forγn≥To.StartingwithajudiciouslychosenJostfunction,M(s,k),whichisconstructedsuchthatM(s,0)isRiemann’sξ(s)function,wehaveusedinversescatteringmeth-odstoactuallyconstructaU(s;x)withtheaboveproperties.Bynecessitywehadtogeneralizeinversemethodstodealwithcomplexpotentialsandanon-unitaryS-matrix.Thiswehavedoneatleastforthespecialcasesunderconsideration.Forourspecificexample,R∞o|fo(x)|2V1(x)dx=0,andhencewegetnorestric-tiononImgnorResn.Thereasonsforthevanishingoftheaboveintegralaregiven,andtheygiveushintsonwhatoneneedstoproceedfurther.Theproblemofdealingwithsmallbutnon-zeroenergiesisalsodiscussed.1I.IntroductionManyphysicistshavebeenintriguedbytheRiemannconjectureonthezerosofthezetafunction.Themainreasonforthisistherealizationthatthevalidityofthehypothesiscouldbeestablishedifonefindsaself-adjointoperatorwhoseeigenvaluesaretheimaginarypartsofthenon-trivialzeros.ThehopeisthatthisoperatorcouldbetheHamiltonianforsomequantummechanicalsystem.ResultsbyDyson1,andMontgomery2firstmadethesituationmorepromising.Thepairdistributionbetweenneighboringzerosseemedtoagreewiththatobtainedfortheeigenvaluesofalargerandomhermitianmatrix.ButlaternumericalworkshowedcorrelationsbetweendistantspacingsdonotagreewiththoseofarandomHermitianmatrix.ThesearchforsuchaHamiltonianinphysicalproblemshaseludedallefforts.Berry3hassuggestedthedesiredHamiltoniancouldresultfromquantizingsomechaoticsystemwithouttimereversalsymmetry.ThisseemstobeinbetteragreementwithnumericalworkonthecorrelationsoftheRiemannzeros,butoneisstillfarfromevenamodelorexample.Itisusefultoexplorenewideas.Ourchoiceforthispaperisanidea,duetoChadan4.InthisapproachonetriestorelatethezerosoftheRiemannzetafunctiontothe”couplingconstantspectrum”ofthezeroenergy,S-wave,scatteringproblemforrepulsivepotentials.Wesketchthisideabriefly.2TheSchrodingerequationonx∈[0,∞)is−d2fdx2(λ;k;x)+λV(x)f(λ;k;x)=k2f(λ;k;x),(1.1)wherekisthewavenumber,λaparameterphysicistscallthecouplingconstant,V(x)isarealpotentialsatisfyinganintegrabilityconditionasinEq.(2.2)below,andfistheJostsolutiondeterminedbyaboundaryconditionatinfinity,(e−ikxf)→1asx→+∞.TheJostfunction,M(λ;k),isdefinedbylimx→0f(λ;k;x)=M(λ;k).ItiswellknownthatMisalsotheFredholmdeterminantoftheLippmann-SchwingerscatteringintegralequationforS-waves.Bothf(λ;k;x)andM(λ;k)areforanyfixedx≥0,analyticintheproductofthehalfplane,Imk0,andanylargeboundedregionintheλplane.Infactitisknownthatforanyfixedk,Imk≥0,M(λ;k)isentireinλandoffiniteorder.ThusM(λ;k)hasaninfinitenumberofzeros,λn(k),withλn(k)→∞asn→∞.StartingwithEq.(1.1),anditscomplexconjugatewithk=iτ,τ0,andsettingλ=λn(iτ)weobtain[Imλn(iτ)]Z∞o|f(λn(iτ);iτ;x)|2V(x)dx=0.(1.2)FortheclassofpotentialswedealwithV=O(e−mx)asx→∞.Thuswecantake3thelimitτ→0,andget[Imλn(0)]Z∞o|f(λn(0);0;x)|2V(x)dx=0.(1.3)Henceforrepulsivepotentials,V(x)≥0,allthezerosλn(0)arereal.Foranyτ,τ0,thesameistrueforallλn(iτ).Butλn(iτ)mustbenegativesincethepotential[λn(iτ)V]willhaveaboundstateatE=−τ2,andthatcouldnothappenifV≥0andλn(iτ)0.Hencebycontinuity,λn(0),foralln,isrealandnegative.5Thezeroenergycouplingconstantspectrum,λn(0),liesonthenegativereallineforV≥0.Chadan’sideaisverysimple.Heintroducesanewvariable,s,anddefinesλ≡s(s−1).(1.4)ThusonecanwriteM(λ,0)=M(s(s−1);0)≡χ(s).(1.5)Itiseasytoseenowthatfor|Ims|1,thezeroes,sn,ofχ(s)areallsuchthatsn=12+iγn;λn(0)≡sn(sn−1).(1.6)Theproblemisactuallysomewhatsimplifiedbynotingthatfirstwedonotneedtheconditionλn0aslongaswerestrictourselvestothestrip0≤Res≤1,andIms1.Second,itissufficienttoprovethattheintegralinEq.(1.3)doesnotvanish.Thus,onedoesnotneedafullyrepulsivepotentialfortheRiemannproblem.4Onemightcommentthatitisverydifficulttofindapotentialwithλn(0)=sn(sn−1)andsn=12±iγn,snbeingtheRiemannzeros.Butitisprobablyasdifficultasfind