On the analytic solution of the pairing problem on

arXiv:nucl-th/0206070v128Jun2002Ontheanalyticsolutionofthepairingproblem:onepairinmanylevelsM.Barbaro1,R.Cenni2,A.Molinari1andM.R.Quaglia21DipartimentodiFisicaTeorica—Universit`adiTorinoIstitutoNazionalediFisicaNucleare—Sez.diTorinoTorino—Italy2DipartimentodiFisica—Universit`adiGenovaIstitutoNazionalediFisicaNucleare—Sez.diGenovaGenova—ItalyAbstractWesearchforapproximate,butanalyticsolutionsofthepairingproblemforonepairofnucleonsinmanylevelsofapotentialwell.Forthecollectiveenergyageneralformula,independentofthedetailsofthesingleparticlespectrum,isgiveninboththestrongandweakcouplingregimes.Nextthedisplacementsofthesolutionstrappedinbetweenthesingleparticlelevelswithrespecttotheunperturbedenergiesareexplored:theirdependenceuponasuitablydefinedquan-tumnumberisfoundtoundergoatransitionbetweentwodifferentregimes.PACS:24.10.Cn;21.60.-nKeywords:Pairinginteraction1IntroductionInthispaperwedealwiththeproblemofthepairingHamiltonianforonepairofnucleonslivinginasetoflevelsofapotentialwell.1Thereareseveralmotivationsforthisstudy.First,tofindapproximate,butanalytic,solutionsfortheeigenvaluesandtheeigenfunctions,whichwebelievetobenotonlyinterestingperse,butalsousefulforpavingthewaytothegeneralproblemofninteractingpairs(seeref.[1]).Second,tocon-necttheenergyofthecollectivemode,bothinthestrongandintheweakcouplingregime,toglobalfeaturesofthesingleparticlelevelsspectrumlikethevariance,theskewness,thekurtosis,etc.Third,tounraveltheremark-ablepatterndisplayedbythesolutionstrappedinbetweenthesingleparticlelevels,alreadyhintedatinref.[2].Indeedbyconnectingthetrappedsolu-tionstoaquantumnumberλ,itisfoundthattheirbehaviourversusλisnotonlysmooth,butdisplaysatransitionbetweentwodifferentregimes.Interestingly,thistransitionmaybeononesiderelatedtoasortofsumruleobeyedbythetrappedeigenvalues(stemmingfromtheVi`eteconditionsforthesolutionsofanalgebraicequation)andontheothertothebasicnatureofthepairinginteraction.Thepaperisorganisedasfollows:insection2thegeneralformalismispresentedandthebasicequationsarededucedintheframeworkoftheGrassmannvariables;insection3thecollectivesolutionisaddressedinboththestrongandweakcouplingregimes;insection4thetrappedsolutionsandtheirregularitiesarestudiedwithintheharmonicoscillatorandrelatedmodelsforthesingleparticlepotentialwell.2GeneralformalismConsiderashellofanaveragepotentialwell,whoseshapeneedsnottobespecified,withLnon-degeneratesingleparticlelevelsofenergyeνandangularmomentumjν,theassociatedmultiplicitybeing2Ων=2jν+1(1≤ν≤L).Letidenticalfermions(e.g.,neutrons)livinginthissetoflevelsinteractthroughthepairingforce.TheHamiltonianofthesystemˆH=ˆH0+ˆHPsplitsthenintoasingleparticleˆH0=LXν=1eνjνXmν=−jνˆa†jνmνˆajνmν(1)2andintoapairinginteractionˆHP=−GLXμ,ν=1ˆA†μˆAν(2)term.In(2)ˆAμ=jμXmμ=1/2(−1)jμ−mμˆajμ,−mμˆajμmμ,(3)ˆajm,ˆa†jmbeingthenucleon’sdestructionandcreationoperators.Theoper-atorsˆAμandˆA†μdestroyandcreatepairshavingtotalangularmomentumJ=0intheleveljμ.Aswell-known,theproblemoftheHamiltonian(1,2)hasbeenaddressed[1]byfirstdiagonalisingtheassociatedbosonicHamiltonianandthenbyaccountingforthePauliprinciple.Herewesearchforcompact,possiblyaccurate,expressionsfortheeigenvaluesandeigenvectorsofˆHandtounravelhiddencorrelationsamongthesolutions.WestartbyrecastingtheeigenvalueequationintheBargmann-Fockrep-resentationofthehamiltonianformalism,wheretheodd(anti-commuting)Grassmannvariablesλjm,λ∗jmreplacethefermionicoperatorsˆajm,ˆa†jm.Forthispurposeweintroducetheevenvariablesϕjm≡(−1)j−mλj−mλjm,(4)intermsofwhichtheoperatorsˆAμbecomeˆAμ−→Φμ=jμXmμ=1/2ϕjμmμ(5)andtheHamiltonian(better,thenormalkernelof)H=LXν=1eνjνXmν=−jνλ∗jνmνλjνmν−GLXμ,ν=1Φ∗μΦν.(6)Theindexofnil-potencyofthecollectiveGrassmannvariableΦμ(tobereferredtoass-quasi-boson)isΩμ,i.e.,(Φμ)n=0fornΩμ.(7)3Wenowsearchforeigenstatesofnpairsoffermionsinthes-quasibosonssubspaceasproductsofnfactors,namelyψn(Φ∗)=nYk=1B∗k(8)whereB∗k=LXν=1β(k)νΦ∗ν,(9)isasuperpositionofs-quasibosonsplacedinalltheavailablelevels.ForthisscopeitisconvenienttostartfromtheeffectiveHamiltonianHeff(ϕ∗,ϕ)=LXν=12eνjνXmν=1/2ϕ∗jνmνϕjνmν−GLXμ,ν=1Φ∗μΦν,(10)coincidentwith(6)inthes-quasibosonssubspacespannedbythestates(8).Indeedwhiletermslikeλ∗λcountthenumberofparticles,ϕ∗ϕ≡λ∗λ∗λλcountsthenumberofpairs.TheeigenvalueequationthenreadsHψn(Φ∗)=Zdλ′dλ∗′Heff(ϕ∗,ϕ′)expLXμ=1jμXmμ=−jμλ∗jμmμλ′jμmμ×exp−LXμ=1jμXmμ=−jμλ∗′jμmμλ′jμmμψn(Φ∗′)=Enψn(Φ∗),(11)where[dλ∗′dλ′]≡LYν=1jνYmν=−jνdλ∗′jν,mνdλ′jν,mν.(12)Byexpandingtheexponentialsin(11),onlytheevenpowers,henceonlytheϕvariables,survive.Thus(11)canberewrittenasZ[dϕ∗′dϕ′]Heff(ϕ∗,ϕ′)M(ϕ∗+ϕ∗′,ϕ′)ψn(Φ∗′)=Enψn(Φ∗),(13)4whereM(ϕ∗,ϕ)≡expLXμ=1jμXmμ=1/2ϕ∗jμmμϕjμmμ.(14)Theintegralsovertheϕ’s,relevantfordealingwitheq.(11),arelistedin[3,4].OnegetsEn=nXk=1ηk,β(k)μ=12eμ−ηk,(15)theηkbeingthesolutionsofthenon-linearsystemLXμ=1Ωμ2eμ−ηk−nXl=1l6=k1ηl−ηk=1G.(16)Inthispaperweconfineourselvestothecaseofasinglepaironly.Then(16)reducestoasingleequationandthewavefunctionreadsψ1(Φ∗)=LXν=1βνΦ∗ν.(17)SincetheactionofHon(17)isHψ1(Φ∗)=LXν=12eνLXμ=1βμjνXmν=1/2ϕ∗jνmνZ[dϕ∗′dϕ′]ϕ′jνmνM(ϕ∗+ϕ∗′,ϕ′)Φ∗′μ−GLXμ,ν=1