Exact L 2 series solution of the Dirac-Coulomb problem for all energies

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1 Exact L series solutio of the Dirac-Coulomb problem for all eergies A. D. Alhaidari Physics Departmet, Kig Fahd Uiversity of Petroleum & Mierals, Box 547, Dhahra 36, Saudi Arabia haidari@mailaps.org We obtai exact solutio of the Dirac equatio with the Coulomb potetial as a ifiite series of square itegrable fuctios. This solutio is for all eergies, the discrete as well as the cotiuous. The spior basis elemets are writte i terms of the cofluet hypergeometric fuctios ad chose such that the matrix represetatio of the Dirac- Coulomb operator is tridiagoal. The wave equatio results i a three-term recursio relatio for the expasio coefficiets of the wavefuctio which is solved i terms of the Meixer-Pollaczek polyomials. PACS umbers: 3.65.Pm, 3.65.Ge, 3.65.Nk, 3.65.Fd Keywords: Dirac equatio, Coulomb potetial, tridiagoal represetatios, recursio relatios, Pollaczek polyomials, scatterig, relativistic spectrum. I. INTRODUCTION Sometimes, it is argued that exact solutios of the wave equatio are by some (debatable) defiitios trivial. Nevertheless, exact solutios are importat because of the coceptual uderstadig of physics that ca oly be brought about by the aalysis of such solutios. I fact, exactly solvable problems are valuable meas for checkig ad improvig models ad umerical methods beig itroduced for solvig complicated physical problems. Furthermore, i some limitig cases or for some special circumstaces they may costitute aalytic solutios of realistic problems or approximatios thereof. I orelativistic quatum mechaics, the search for exact solutios of the wave equatio was carried out over the years by may authors where several classes of these solvable potetials are accouted for ad tabulated (see, for example, the refereces cited i []). Most of the kow exactly solvable problems fall withi distict classes of what is referred to as shape ivariat potetials. Supersymmetric quatum mechaics [], potetial algebras [3], ad poit caoical trasformatios [4] are three methods amog may which are used i the search for exact solutios of the wave equatio. These developmets were exteded to other classes of coditioally exactly [5] ad quasi exactly [6] solvable problems where all or, respectively, part of the eergy spectrum is kow. Recetly, the relativistic extesio of some of these formulatios was carried out where several relativistic problems where formulated ad solved exactly [7]. I all of these developmets, the objective is to fid solutios of the eigevalue wave equatio H χ = E χ, where H is the Hamiltoia ad E is the eergy which is either discrete (for boud states) or cotiuous (for scatterig states). I most cases, especially i the search for algebraic or umerical solutios, the wave fuctio χ is expaded i terms of discrete square itegrable basis { ψ } as χ( r, E) = = f ( E ) ψ ( r ), where r is the cofiguratio space coordiate. The basis fuctios must be compatible with the domai of the Hamiltoia ad should satisfy the boudary coditios. Typically the choice of basis is limited to those that carry diagoal

2 represetatios of the Hamiltoia. That is, oe looks for a L basis set { } H ψ = E ψ givig the discrete spectrum { } ψ such = E of H. The cotiuous spectrum that is obtaied from the aalysis of a ifiite sum of these complete basis fuctios. Trucatig this sum, for umerical reasos, may create problems such as the presece of uphysical states or fictitious resoaces i the spectrum. I this article we relax the restrictio of a diagoal represetatio of the Hamiltoia by searchig for square itegrable bases that could support a tridiagoal matrix represetatio of the wave operator. That is, the actio of the wave operator o the elemets of the basis is allowed to take the geeral form ( H E) ψ ψ ψ ψ such that ψ H E ψ = ( a x) δ bδ b δ (.) m, m, m, m where x ad the coefficiets { a, b} = are real ad, i geeral, fuctios of the eergy E, the agular mometum, ad the potetial parameters. Therefore, the matrix represetatio of the wave equatio, which is obtaied by expadig χ as f ψ m m m i ( H E) χ = ad projectig o the left by ψ, results i the followig three-term recursio relatio xf = af b f bf (.) Cosequetly, the problem traslates ito fidig solutios of the recursio relatio for the expasio coefficiets of the wavefuctio. I most cases this recursio is solved easily ad directly by correspodece with those for well kow orthogoal polyomials. A example of a problem which is already solved usig this approach is the orelativistic Coulomb problem where the expasio coefficiets of the wavefuctio are writte i terms of the Pollaczek polyomials [8]. It is obvious that the solutio of (.) is obtaied modulo a overall factor which is a fuctio of x but, otherwise, idepedet of. The uiqueess of the solutio is achieved by the requiremet that the wavefuctio χ( r, E) be eergy ormalizable. It should be oted that the solutio of the problem as give by Eq. (.) above is obtaied for all E, the discrete as well as the cotiuous, costraied oly by the reality ad boudedess of the tridiagoal represetatio. Moreover, the represetatio equatio (.) clearly shows that the discrete spectrum is easily obtaied by diagoalizatio which requires that: b =, ad a x= (.3) I Sec. II, we set up the three dimesioal Dirac equatio for a charged spior iteractig with the electromagetic four-potetial ( A, A ). Spherical symmetry is imposed ad we cosider the special case where the space compoet of the electromagetic potetial vaishes (i.e., A = ). The time compoet, o the other had, is take as the Coulomb potetial. As a result, the problem is reduced to solvig the radial Dirac equatio. A global uitary trasformatio is applied to this equatio to separate the variables such that the resultig secod order differetial equatio for the radial spior compoets becomes Schrödiger-like. This results i a simple ad straightforward correspodece to the well-kow orelativistic Schrödiger-Coulomb problem. The correspodece will be used i Sec. III as a guide for costructig a square

3 itegrable basis for the solutio space of the Dirac-Coulomb problem. I this costructio we impose the requiremet that the matrix represetatio of the Dirac operator be tridiagoal. The result is a three-term recursio relatio for the expasio coefficiets of the spior wavefuctio which is solvable for all eergies, the discrete as well as the cotiuous. The recursio relatio is writte i a form that makes its solutio easily attaiable by simple ad direct compariso with that of the Meixer-Pollaczek polyomials [9]. We coclude with a short discussio i Sec. IV. II. FORMULATION OF THE DIRAC-COULOMB PROBLEM Dirac equatio is a relativistically covariat first order differetial equatio i four dimesioal space-time for a spior wavefuctio χ. For a free structureless particle it µ reads ( i µ mc) χ =, where m is the rest mass of the particle ad c the speed of light. The summatio covetio over repeated idices is used. That is, µ 3 µ µ = µ. =. µ = ct µ. { } 3 µ= are four costat square matrices µ ν µ ν ν µ µν satisfyig the aticommutatio relatio {, } = = G, where G is the metric of Mikowski space-time which is equal to diag(,,, ). These are eve dimesioal matrices with a miimum dimesio of four correspodig to spi ½ represetatio of the Loretz space-time symmetry group. A four-dimesioal matrix represetatio that satisfies this relatio is chose as follows: I σ =, = (.) I σ where I is the uit matrix ad σ are the three hermitia Pauli matrices. I the atomic uits ( = m = e = ), the Compto wavelegth = mc = c is the relativistic µ parameter ad the Dirac equatio reads ( i µ ) χ =, where χ is a four-compoet spior. Next, we let the Dirac spior be charged ad coupled to the four compoet electromagetic potetial A = µ ( A, A). Gauge ivariat couplig, which is accomplished by the miimal substitutio i e µ µ c Aµ, trasforms the free Dirac equatio to µ i ( µ iaµ ) χ = which, whe writte i details, reads as follows i χ iα α A A = β χ Hχ (.) t ( ) where H is the Hamiltoia i uite of mc =, α ad β are the hermitia matrices σ I α = =, β = = (.3) σ I Substitutig these i Eq. (.) gives the followig matrix represetatio of the Dirac Hamiltoia A iσ σ A H = (.4) iσ σ A A Thus the eigevalue wave equatio reads ( H ε ) χ =, where ε is the relativistic eergy which is real, dimesioless ad measured i uits of. 3

4 Now, we choose A = ad impose spherical symmetry by takig A = V() r. I this case, the agular variables could be separated ad we ca write the spior wavefuctio as [] j igr [ ( ) r] ϕ m χ = j (.5) [ f( r) r] σ rˆ ϕm where f ad g are real radial square-itegrable fuctios, ˆr is the radial uit vector, ad the agular wave-fuctio for the two-compoet spior is writte as m/ ± m / Y j ϕ () ˆ m r =, for j = ± ½ (.6) m / ± m / Y m± / Y () rˆ is the spherical harmoic fuctio ad m stads for the itegers i the rage j, j,..., j ad should ot be cofused with the mass. Spherical symmetry gives iσ ( r ) χ( rr, ˆ) = ( κ) χ( rr, ˆ), where κ is the spi-orbit quatum umber defied as κ =± ( j ½) =±, ±,... for = j ± ½. Usig this we obtai the followig useful relatios j df κ j ( σ )( σ rfr ˆ) () ϕ () ˆ m r = F ϕ m dr r (.7) j df κ j ( σ ) Fr () ϕ () ˆ ( ˆ m r = F σ r) ϕ m dr r Employig these i the wave equatio ( H ε ) χ = results i the followig matrix equatio for the two radial spior compoets V( r) ε κ d () ( r gr dr ) κ d = (.8) ( r ) V( r) ε dr f() r Takig V(r) as the Coulomb potetial Z r gives Z r ε κ d ( r g dr ) κ d ( ) Z = (.9) r dr r ε f where Z is the charge of the particle i our chose uits ( = m = e = ). It should be oted that i these uits the role of the fie structure costat α is played by the Compto wavelegth. The uits ( = c = ) where the fie structure costat α is used as the relativistic parameter are suitable for the electromagetic iteractio. The uits that we are adoptig here, where the relativistic parameter is, are suitable for dealig with a larger class of problems. The coectio betwee these two uits is i the relatio α Z = Z, where Z is the usual dimesioless charge i uits of e. Now has the dimesio of legth. Therefore, i these uits Z has the dimesio of iverse legth. Equatio (.9) results i two coupled first order differetial equatios for the two radial spior compoets f ad g. Elimiatig the lower compoet i favor of the upper gives a secod order differetial equatio. This equatio is ot Schrödiger-like (i.e., it cotais first order derivatives). To obtai a Schrödiger-like equatio we proceed as i follows. A global uitary trasformatio U ( η) = exp( ησ ) is applied to the radial Dirac equatio (.9), where η is a real costat parameter ad σ is the Pauli matrix 4

5 ( i i ). The Schrödiger-like requiremet dictates that the parameter η satisfies the costrait si( η) = Z κ, where π η π depedig o the sigs of Z ad κ. Equatio (.9) is ow trasformed ito the followig Z Z d () r κ ε r ( κ r φ dr ) Z d = (.) ( κ r dr ) κ ε θ () r where = κ ( Z κ) ad η η φ cos si g = U χ = θ η η (.) si cos f It is to be oted that the agular parameter of the uitary trasformatio U ( η ) was itetioally split as η ad ot collected ito a sigle agle, say ϕ. This is suggested by ivestigatig the costrait si( ϕ) = Z κ i the orelativistic limit ( ) where we should have si( ϕ) ϕ = Z κ. It also makes it obvious that i the orelativistic limit the trasformatio becomes the idetity (i.e., ot eeded). Equatio (.) gives the lower spior compoet i terms of the upper as follows Z d θ = φ (.) κ ε κ r dr for ε κ. Whereas, the resultig Schrödiger-like wave equatio for the upper compoet becomes d ( ) Zε ε φ( r) = (.3) dr r r Comparig this equatio with that of the well-kow orelativistic Coulomb problem d ( ) Z E ( r) Φ = (.4) dr r r gives, by correspodece, the followig map betwee the parameters of the two problems: Z Zε, E ( ε ), { (.5) The top (bottom) choice of the map correspods to positive (egative) values of κ, respectively. It should be oted that the map produced by the compariso of Eq. (.3) to Eq. (.4) is a correspodece map betwee the parameters of the two problems ad ot a equality of the parameters. That is we obtai, for example, the correspodece map but ot the equality =. I fact, is ot a iteger while, of course, is. Usig the parameter map (.5) i the well-kow orelativistic eergy spectrum, E = Z ( ), gives the followig relativistic spectrum for boud states Z ε =± Λ ( ) (.6) where =,,,... ad either Λ ( ) = or Λ ( ) = depedig o whether κ is positive or egative, respectively. Oe ca easily verify that i the orelativistic limit (, ε E ), the orelativistic spectrum is recovered. The upper radial 5

6 compoet of the spior wavefuctio is obtaied usig the same parameter map (.5) i the orelativistic wavefuctio λ ()~( r r) e r Φ λ L ( λ r) (.7) Z. These fidigs will be used i the followig sectio as a guide to writig dow the L spior basis that supports a tridiagoal matrix represetatio of the Dirac-Coulomb operator ( H ε ) i Eq. (.). where λ = ( ) III. TRIDIAGONAL REPRESENTATION OF THE SOLUTION SPACE The orelativistic wavefuctio (.7) ad the parameter map (.5) suggest that a square itegrable basis for the upper radial spior compoet, which satisfies the boudary coditios, could be writte as ωγ ( ) ξ ωr ν φ() r = ( ωr) e L ( ) ( ) ωr (3.) Γ ν where ω is a positive basis scale parameter, ξ > ad ν >. The kietic balace relatio (.) suggests that the lower compoet of the spior basis is related to the upper as µ ζ d θ ~ φ (3.) r dr where the parameters µ ad ζ are real ad will be determied as we proceed. Substitutig (3.) ito (3.) ad usig the differetial ad recursio properties of the Laguerre polyomials (show i the Appedix) we obtai ωγ ( ) Γ ( ν ) θ () r = τ ( ωr) e µ ζ d ( r dr ) ξ ωr ν ν ν ω( ξ ζ ν) L( ωr) ( ω µ )( ν) L ( ωr) ( ω µ )( ) L ( ωr) = τ φ ( r) (3.3) where τ is aother real dimesioless parameter. Square itegrability requires that either () we impose the more striget requiremet that ξ >, or () choose the parameter ζ such that the sum of the Laguerre polyomials i square brackets becomes proportioal to ωr. The first alterative is suitable for κ > sice the parameter map (.5) ad the orelativistic wavefuctio (.7) suggest that ξ = as we will fid out shortly. However, for κ < the same parameter map gives ξ = which violates the requiremet ξ > for κ = ad possibly for other values of egative κ if the electric charge Z becomes large eough. Cosequetly, the secod choice will be used for κ <. A. Solutio for κ > Now for κ > ad with ξ >, we ca simplify the expressio (3.3) for θ ( r ) by elimiatig the first term iside the square brackets without affectig square itegrability. That is, we take ζ = ν ξ which results i the followig expressio for the lower spior compoet 6

7 µ νξ d ( r dr ) ωγ ( ) ω µ Γ ( ν ) ω µ ξ ωr ν ν θ( r) = τ( ω µ ) ( ωr) e ( ν) L ( ωr) ( ) L ( ωr) = τ φ ( r) { } φ I this spior basis ψ ( θ ) = operator i (.) reads ( ) ( H = κ Z r κ τ ) ( κ ) ( ) r (3.4) =, the matrix represetatio of the Dirac-Coulomb ψ εψ ε φ φ φ φ ε θ θ m m m m r θ φ θ φ ξ ν θ φ θ φ m m m m (3.5) where we have used itegratio by parts i writig φ dr θ m = φ dr θ m sice the product φ( r) θ m( r) vaishes at the boudaries r = ad r. Now, we require that this represetatio be tridiagoal. That is, ψ H ε ψm = for all m. For the first two terms o the right side of Eq. (3.5) to comply with this requiremet we must have ξ = ν. Moreover, the two terms iside the last square brackets o the right side of the equatio destroy the tridiagoal structure. Thus, the multiplyig factor ( ξ ν) must vaish. Cosequetly, the matrix represetatio of the Dirac-Coulomb operator for κ > is tridiagoal oly if ξ = ad ν =. Thus, the requiremet that ξ > is preserved for all κ >. Therefore, for positive κ, the two compoets of the radial spior basis become ωγ ( ) ωr φ () r = ( ωr) e L ( ) ( ) ωr (3.6a) Γ ωγ ( ) ωr θ () r = τ( ω µ ) ( ωr) e Γ ( ) (3.6b) ω µ ( ) L ( ωr) ω µ ( ) L ( ωr) Substitutig these ito (3.5) ad usig the orthogoality ad recurrece relatios of the Laguerre polyomials (show i the Appedix) we obtai the followig elemets of the symmetric tridiagoal matrix represetatio of the Dirac-Coulomb operator ( ) ( ) ( )( ) H ε =, κ ε τ ω µ κ ε τ τµ ( κ ) (3.7) Z 4 ( ) 4 ω τ µω κ ε τ τω( κ ) ( ) ( ), ( )( ) H ε = κ ε τ ω µ κ ε τ τµ ( κ ) (3.8) If we defie the followig quatities: p = τω ( κ ε τ ), q = τω ( κ ) (3.9) The, the matrix represetatio of the wave equatio ( H ε) χ =, where χ = f ψ m m m, results i the followig three-term recursio relatio for the expasio coefficiets of the wave-fuctio ( ) ρ Ω f ρ ( ) f ρ ( )( ) f = (3.) where κε µ q µ µ ωz q = ±, Ω= (3.) ρ ± p ω p ( ω) ω p d d 7

8 Rewritig (3.) i terms of the polyomials P ( ε ) = Γ ( ) Γ ( ) f ( ε ), we obtai the followig recursio relatio ρ ( Ω ) ρ ρ P ( ) P ( ) P = (3.) We compare this with the recursio relatio satisfied by the Meixer-Pollaczek λ polyomial P ( x, ϕ) [9] that reads λ λ λ ( [ λ)cosϕ xsi ϕ] P ( λ) P ( ) P = (3.3) where, λ > ad < ϕ < π. Thus, λ =, cosϕ = ρ ρ, x =Ω ρ ρ ad we ca write Γ ( ) ρ f( ε ) ( ) (,cos ( ) ρ ρ ) ρ = P Ω (3.4) Γ which is defied up to a multiplicative factor that depeds o ε but is idepedet of. The Meixer-Pollaczek polyomial could be writte i terms of the hypergeometric fuctio as λ Γ ( λ) i i ( x, ) e ϕ P ϕ = ( ) ( ) F(, λ ix; λ; e ϕ ) Γ Γ λ (3.5) The orthogoality relatio associated with these polyomials is as follows Γ ( ) Γ ( ) ρ ( x, ϕ) P ( x, ϕ) P m( x, ϕ) dx= δm (3.6) λ λ λ λ λ λ ( ϕπ) x where ρ ( x, ϕ) = (si ϕ) e Γ ( λ ix). Therefore, the exact L series solutio π of the Dirac-Coulomb problem for κ > could be writte as Γ ( ) χ(, r ε) = A () ε P,cos ( ) Ω ρ ρ ( ρ ρ ) ψ() r Γ (3.7) = where A ( ε ) ρ ( x, ϕ) ( dx dε) ( ) = is a ormalizatio factor that makes χ eergyormalizable, while the two compoets of the radial spior basis elemet ψ ( r ) those give by Eqs. (3.6). Further aalysis of these solutios, such as obtaiig the discrete spectrum, is tractable oly if the kietic balace relatio (.) is strictly imposed o the basis elemets. That is, relatio (3.) should be idetical to (.) which requires that µ = Z κ, τ = ( ε κ) (3.8) for ε κ. I this case, the tridiagoal represetatio of the Dirac-Coulomb operator simplifies to ( ) { ( ) ( ) } H ε = ( ) ( ), 4 4 Z Z κ ε ω κ κ ε ω ε κ ε (3.7 ) ( ) ( ) { ( ), 4 4 ( ) } H Z ε = κ ε ω κ κ ε (3.8 ) while the parameters ρ ± ad Ω i the solutio (3.7) read as follows ε ρ± = 4 ±, Ω= 4 ( Z ω) ε (3.9) ω λ The argumets of the Meixer-Pollaczek polyomial P ( x, ϕ), i this case, read as follows: are 8

9 ε ( ) ( ) ε cos ϕ = 4 4 ω ω, x= Zε ε (3.) The rage < ϕ < π implies that the solutio obtaied above i (3.7) is valid for ε >. That is, the solutio (3.7) is for eergies larger tha the rest mass mc which correspods to scatterig states. Solutios for ε < correspod to boud states. To obtai these solutios ad their discrete eergy spectrum, we impose the diagoalizatio requiremet (.3). This requiremet traslates, i the case of the recursio relatio (3.), ito the followig coditios ρ =, Ω= ( ) (3.) givig Z ε = ε =± Zε ( ), ω = ω =, =,,,... (3.) which agrees with the well-kow relativistic boud states eergy spectrum for the Dirac-Coulomb problem whe κ >. The correspodig spior wavefuctios are ω Γ ( ) Γ ( ) ( ) ( ) r r r ω e L ( r) φ = ω ω (3.3a) ω Z κ ω Γ ( ) ε κ Γ ( ) θ () r = ( ω r) e ( ) ω Z κ ω r ( ) L ( ωr) ( ) L ( ωr) ω Z κ Takig the orelativistic limit (, (3.3b) ε E ) i (3.9) gives ρ ± = 8E ω ± ad Ω = 4 Z ω resultig i the three-term recursio relatio for the orelativistic Coulomb problem which was solved by Yamai ad Reihardt [8]. I the followig subsectio we obtai the solutio of the Dirac-Coulomb problem for κ <. B. Solutio for κ < I this case ad as stated below Eq. (3.3) we choose the parameter ζ i (3.) such that the resultig sum of the Laguerre polyomials iside the square brackets i (3.3) becomes proportioal to ωr. Usig the properties of the Laguerre polyomials i the Appedix oe ca show that ν ν ν ν ν ων L( x) ( ω µ )( ν ) L ( x) ( ω µ )( ) L ( x) = x ( ω µ ) L( x) ωl ( x) This meas that choosig ζ = ξ i (3.3) results i a square itegrable lower spior compoet without the eed for a stroger costrait o the real parameter ξ other tha ξ >. Cosequetly, we obtai the followig expressio for the lower compoet of the spior basis ωγ ( ) ξ ωr ν ω µ ν θ( r) = τ( ω µ ) ( ωr) e L ( ) ( ωr) L ( r) ν ω µ ω Γ (3.4) µ ξ = τ d φ ( r) ( r dr ) I this basis, the matrix represetatio of the Dirac-Coulomb operator i (.) reads 9

10 H = ( κ ) Z r ( κ τ ) ( κ ) ( ) r r ψ εψm ε φ φm φ φm ε θ θm (3.5) θ φm θm φ ξ θ φm θm φ The same argumets, which were preseted below Eq. (3.5), apply to this represetatio as well givig, however, ξ = ad ν =. Thus, the resultig two compoets of the spior basis for κ < are ωγ ( ) Γ( ) φ = ω ω (3.6a) ωr () r ( r) e L ( r) ωγ ( ) ω µ Γ( ) ω µ θ r ( r) τ ( ω µ ) ( ω ω r) e L ( ω = r) L ( ω r) (3.6b) Substitutig these ito (3.5), we obtai the followig elemets of the symmetric tridiagoal matrix represetatio of the Dirac-Coulomb operator for κ < ( ) ( ), ( )( ) H ε = κ ε τ ω µ κ ε τ τµ ( κ ) (3.7) Z 4 4 ω τ µω κ ε τ τω κ ( ) ( ) ( ) ( ), ( )( ) H ε = κ ε τ ω µ κ ε τ τµ ( κ ) (3.8) Comparig these with the correspodig formulas (3.7) ad (3.8) for κ >, shows that the oly differece is i the -depedet factors. I these factors, ad oly i these factors, the replacemet takes effect. Cosequetly, the matrix represetatio of the wave equatio results i the followig three-term recursio relatio for the expasio coefficiets of the wavefuctio ( ) ρ Ω f ρ ( ) f ρ ( )( ) f = (3.9) where the parameters ρ ± ad Ω are exactly those defied i (3.) above. Pursuig the same developmet carried out i the case κ >, we obtai the followig solutio for κ < ad for ε > = Γ ( ) Γ( ) ( ( )) χ(, r ε) = A () ε P Ω ρ ρ,cos ρ ρ ψ() r (3.3) Imposig the kietic balace relatio amog the compoets of the spior basis (3.6) results i the same parameter assigmets i (3.8). It also gives the followig diagoalizatio coditios for obtaiig the discrete eergy spectrum: ρ =, Ω= ( ) (3.3) resultig i Z ( ) ε = ε =± Zε, ω = ω =, =,,,... (3.3) which complemet the results obtaied i (3.) above for κ >. The correspodig spior wave-fuctios read as follows: ω Γ ( ) Γ( ) ω r φ () r = ( ω r) e L ( ω r) (3.33a) θ ω Z κ ω Γ ( ) ω r () r = ( ω ) ( ) r e ε κ Γ Z κ ω L ( ω r) L ( ) Z ω r ω κ (3.33b)

11 IV. DISCUSSION We would like to coclude with some commets. First, it is worthwhile otig that the eergy spectrum i (3.) ad (3.3) shows that the lowest positive eergy state is ε = ε = κ = ( Z κ) for κ =,,... The highest egative eergy state, o the other had, is ε = ε = κ for κ =,,... These two are o-degeerate states, while all others are. This is so because, ε = for =,,,... ad for ε κ> κ< all κ. The spior wavefuctio associated with the lowest positive eergy state is obtaied from (3.33) as Z κ Zr κ ψ () r = ( Zr κ ) e Γ ( ) (4.) Z where, for boud states, Z is egative. Obtaiig the spior wavefuctio associated with the highest egative eergy state is more subtle. This is due to the fact that the kietic balace relatio (.) does ot hold for this state (ad oly this state) sice ε = κ. Oe has to redo the developmet i subsectio III.B for this case (where, ε = κ ad = ) without the costrait (3.8) but with arbitrary µ ad τ. The secod commet we wat to make has to do with the type ad umber of solutios of the recursio relatio (3.) or (3.9). Typically, there are two solutios to such three-term recursio relatio. This could be uderstood by otig that the orthogoal polyomials that satisfy the recursio relatio are at the same time solutios of a secod order differetial equatio. I other words, there is a correspodece betwee three-term recursio relatios ad secod order differetial equatios for a give set of iitial relatios or boudary coditios, respectively. The solutios obtaied above i (3.7) ad (3.3) could be termed regular solutios. These correspod to solutios of the recursio relatio i terms of polyomials of the first kid. Polyomials of the secod kid satisfy the same recursio relatio (for ) but with a differet iitial relatio (for = ). These correspod to irregular solutios, or i a more precise term regularized solutios, sice they are regular at the origi of cofiguratio space while behavig asymptotically as the irregular solutio. I what remais we cosider the recursio relatio (3.) or, equivaletly (3.). However, the same aalysis could as well be carried out for the recursio relatio (3.9). For a very large positive iteger N the recursio relatio (3.) could be rewritte as zpn ( z) PN ( z) PN ( z) =. Defiig P ˆ ( z) PN ( z), we could write it as zp ˆ ˆ ˆ ( z) P ( z) P ( z) = (4.) where z = ρ ρ = cosϕ. This is the recursio relatio of the Chebyshev polyomials. For large, they are oscillatory (i.e., they behave like sie s ad cosie s). The two ± idepedet oscillatory solutios of (4.), which we will desigate by P ( z), differ by a phase. The origi of this phase differece could be traced back to the iitial relatio ( = ) of the recursio (3.). Thus, the iitial relatio must have two differet forms. This differece propagates through the recursio to the asymptotic solutios. Oe of these iitial relatios is homogeeous ad correspods to the regular solutio, which was obtaied above. The other is ihomogeeous ad correspods to the regularized solutio. They could be writte as:

12 ρ ( Ω ) ρ ρ P P = (4.3a) ρ ( Ω ) ρ ρ P P = W (4.3b) where W = W( x, ϕ) ad is related to the Wroskia of the two solutios. For scatterig ± problems, the phase shift is obtaied by the aalysis of the two solutios P ( z). Such aalysis is typical of algebraic scatterig methods i orelativistic quatum mechaics. A clear example is foud i the J-matrix method of scatterig []. APPENDIX: PROPERTIES OF THE LAGUERRE POLYNOMIALS The followig are useful formulas ad relatios satisfied by the geeralized ν orthogoal Laguerre polyomials L ( x) that are relevat to the developmets carried out i this work. They are foud o most textbooks o orthogoal polyomials []. We list them here for ease of referece. The differetial equatio: d d ν x ( ν x) L ( x) = (A.) dx dx where ν > ad =,,,... Expressio i terms of the cofluet hypergeometric fuctio: ν ( ) Γ ( ν ) L x = ( ) ( ) ( ; ; ) F ν x (A.) Γ Γ ν The three-term recursio relatio: ν ν ν ν xl = ( ν ) L ( ν) L ( ) L (A.3) Other recurrece relatios: ν ν ν xl = ( ν ) L ( ) L (A.4) ν ν ν L = L L (A.5) Differetial formula: d ν ν ν x L = L ( ν ) L (A.6) dx Orthogoality relatio: Γ ( ) Γ ( ) ρ ( xl ) ( xl ) m( xdx ) = δm (A.7) ν ν ν ν ν ν x where ρ ( x) = xe.

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