Exact solution for a fermion in the background of a scalar i(3)

2q 1Ψ+=N+ξqe ξ/2Ln(ξ),(q≥1forn=0)(26)

q+1Ψ =N ξq+1e ξ/2L2n 1(ξ)(27)

E=±mc

k2wherewehaveusedLk0(ξ)=1andL 1(ξ)=0forallk.NotethatE≈mc

andEeff≈E mc2,isvalidaslongasn q,thusthenonrelativisticlimitofthetheorywouldbe,inalimitedsense,aregimeoflargequantumnumbers.Ontheotherhand,intheregimeofstrongcoupling,i.e.,forq 1,onehasEeff≈ mc2/2+nmc2/qandasthecouplingbecomesextremelystrongtheloweste ectiveeigenvaluesendupcloseto mc2/2,correspondingtoDiraceigenvaluesE,nearzero.Nowoneseesclearlythattheeigenvaluesaswellastheeigenfunctionsforazero-energysolution,incontrasttowhatisdeclared 2 q+n 2,n=0,1,2,...(28)

9

The problem of a fermion subject to a general scalar potential in a two-dimensional world is mapped into a Sturm-Liouville problem for nonzero eigenenergies. The searching for possible bounded solutions is done in the circumstance of power-law potentials.

inRef.[23],canbeobtainedasalimitcaseofanonzero-energysolution.Fig.2showsthebehaviorofthepositive-eigenenergiesasafunctionofq.Thee ectiveKratzerpotentialleadstoboundedsolutionsintherange+ mc2/18≤Eeff<0whenacriticalpotential(Veff)isthereandVeffhas aminimumvalueequalto mc2/6.Forq=1theminimumvalueofVeffis mc2/4andthee ectiveeigenvalueisgreaterthanorequalto mc2/8.For±q>1thee ectivepotentialVeffhastheminimumvalue mc2q/2(q 1)andthee ectiveeigenvaluesaregreaterthanorequalto mc2q2/2(q+1)2.Inthislastcasethefermiontendstoavoidtheorigin moreandmoreas 22qincreasesduetohigherandhighercentrifugalbarrierh¯q(q 1)/2mx,whichinturnthrusttheminimumofthee ectivepotentialfurther(this+ shiftingisproportionalto(q 1))andhigher(lower)forVeff(Veff)insuch± awaythatVeff→Veff.

Fig.3illustratethebehavioroftheprobabilitypositiondensity,|Ψ|2=|Ψ+|2+|Ψ |2,forthezero-modesolutionforq=1.Figs.4,5and6illustratethebehavioroftheupperandlowercomponentsoftheDiracspinor,|Ψ+|2and|Ψ |2,andtheprobabilitypositiondensity,|Ψ|2=|Ψ+|2+|Ψ |2,forthepositive-energysolutionsofthe rst-,second-andthird-excitedstatesforq=1.Theresultsfornegativeenergiesarethesameasfarasthecharge

conjugationdoesΨ+→Ψ +andΨ → Ψ Therelativenormalizationof

Ψ+andΨ isobtainedbysubstitutingthesolutionsdirectlyintotheoriginal rst-ordercoupledequations(3).TheresultisnotprettyforE=parisonofthese guresshowsclearlythat,exceptforthezero-eigenmode(n=0),|Ψ |iscomparableinimportanceto|Ψ+|and|Ψ |→|Ψ+|asnincreases,exceptforaminordi erenceneartheorigin.Itshouldnotbeforgotten,though,thatwehaverestrictedourdiscussiontothepositivehalfline.Forx≤0therolesofΨ+andΨ arereversed.

Inconclusion,wehavesucceedinsearchingforDiracboundedsolutionsforthescalarpotentialV= h¯cq/|x|.Thesatisfactorycompletionofthistaskfornonzeroeigenenergieshasbeenalleviatedbythemethodologyofe ectivepotentialswhichhastransmutedthequestionintoSturm-Liouvilleproblemswithe ectiveinverselylinearplusinverselyquadraticpotentialsforbothcomponentsoftheDiracspinor.Adiscreteandnondegeneratespectrumhasbeenfound.Forastrongenoughpotentialanondegeneratezero-eigenmodecorrespondingtothegroundstatecomestothescene.Onlyintheregimeofextremelystrongcouplingtherearedegeneratesolutions

10

The problem of a fermion subject to a general scalar potential in a two-dimensional world is mapped into a Sturm-Liouville problem for nonzero eigenenergies. The searching for possible bounded solutions is done in the circumstance of power-law potentials.

relatedtothezero-energysolutions,oneoftheeigenspinorrepresentingthefermionandtheotherrepresentingtheantifermion.Beyonditsintrinsicimportanceasanewsolutionforafundamentalequationinphysics,theproblemanalyzedinthispaperpresentsunusualresults.

Acknowledgments

TheauthorwishestothankM.B.Hottforusefuldiscussionsandforacriticalreadingofthepaper.ThisworkwassupportedinpartbymeansoffundsprovidedbyCNPqandFAPESP.

11

The problem of a fermion subject to a general scalar potential in a two-dimensional world is mapped into a Sturm-Liouville problem for nonzero eigenenergies. The searching for possible bounded solutions is done in the circumstance of power-law potentials.

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The problem of a fermion subject to a general scalar potential in a two-dimensional world is mapped into a Sturm-Liouville problem for nonzero eigenenergies. The searching for possible bounded solutions is done in the circumstance of power-law potentials.

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