L 2 series solution of the relativistic Dirac-Morse problem for all energies

Transcription

1 L series solutio of the relativistic Dirac-Morse problem for all eergies A. D. Alhaidari Physics Departmet, Kig Fahd Uiversity of Petroleum & Mierals, Box 547, Dhahra 36, Saudi Arabia We obtai aalytic solutios for the oe-dimesioal Dirac equatio with the Morse potetial as a ifiite series of square itegrable fuctios. These solutios are for all eergies, the discrete as well as the cotiuous. The elemets of the spior basis are writte i terms of the cofluet hypergeometric fuctios. They are chose such that the matrix represetatio of the Dirac-Morse operator for cotiuous spectrum (i.e., for scatterig eergies larger tha the rest mass) is tridiagoal. Cosequetly, the wave equatio results i a three-term recursio relatio for the expasio coefficiets of the wavefuctio. The solutio of this recursio relatio is obtaied i terms of the cotiuous dual Hah orthogoal polyomials. O the other had, for the discrete spectrum (i.e., for boud states with eergies less tha the rest mass) the spior wave fuctios result i a diagoal matrix represetatio for the Dirac-Morse Hamiltoia. PACS umbers: 3.65.Pm, 3.65.Ge, 3.65.Nk, 3.65.Fd Keywords: Dirac equatio, Morse potetial, tridiagoal represetatios, 3-term recursio relatios, cotiuous dual Hah polyomials, scatterig, relativistic spectrum. I. INTRODUCTION Most of the kow exactly solvable orelativistic problems fall withi distict classes of what is referred to as shape ivariat potetials []. Supersymmetric quatum mechaics, potetial algebras, path itegratio, ad poit caoical trasformatios are four methods amog may which are used i the search for exact solutios of the wave equatio. These formulatios were exteded to other classes of coditioally exactly [] ad quasi exactly [3] solvable problems where all or, respectively, part of the eergy spectrum is kow. Recetly, the relativistic extesio of some of these developmets was carried out where several relativistic problems are formulated ad solved exactly [4]. 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). The solutio of this equatio for a geeral physical system is ofte very difficult to obtai. However, for simple systems or for those with high degree of symmetry, a aalytic solutio is feasible. O the other had, i a large class of problems that model realistic physical systems, the Hamiltoia H could be writte as the sum of two compoets: H = H + V. The referece Hamiltoia H is ofte simpler ad carries a high degree of symmetry while the potetial compoet V is ot. However, it is usually edowed with either oe of two properties. Its cotributio is either very small compared to H or is limited to a fiite regio i cofiguratio or fuctio space. Perturbatio techiques are used to give a umerical evaluatio of its cotributio i the former case, while algebraic methods are used i the latter. Thus, the aalytic problem is limited to fidig the solutio of the referece H problem, ( H E) χ =. To obtai a approximate solutio to the full problem i the case where the potetial V is short rage, we could use

2 oe of two alterative methods. I oe, the cotributio of V is limited to a fiite regio i cofiguratio space. That is, the full Hamiltoia is approximated by H + V() r r < R H, (.a) H r R where R is the effective rage of the potetial. I the other method, the matrix represetatio of the Hamiltoia, i a suitable basis { } ( H ) + Vm, m< N m Hm ( H ), m N ψ =, is approximated by, (.b) m for some adequately large iteger N. That is, the potetial is cofied to a fiite regio i fuctio space where it is approximated by its matrix represetatio i a fiite subset of N. This latter approach to the solutio is called the J-matrix method [5]. the basis, { ψ } = It is edowed with formal ad computatioal similarities to the former approach the celebrated R-matrix method [6]. They both use square itegrable basis fuctios to carry out the calculatio. The utilizatio of L methods i scatterig calculatios uderwet several major developmets startig with the stabilizatio techique of Hazi ad Taylor [7]. It was the followed by the developmet of the L -Fredholm method [8] i which scatterig iformatio is extracted from the discrete eigevalues of the full Hamiltoia ad those of the referece Hamiltoia i a fiite L -basis { ψ } N =. This method is aalogous to discretizig the cotiuous spectrum of the system by cofiig it to a box i cofiguratio space. The J-matrix method was the ext substatial developmet i this formulatio i which the cotributio of the referece Hamiltoia is take fully ito accout. O the other had, there is a limited class of short rage potetials where a full aalytic solutio to the problem is possible. I orelativistic quatum mechaics oe such class that has a symmetry which is associated with the dyamical group SO(,) icludes the Morse, Pöschl-Teller, Eckart, etc. potetials. Due to the higher degree of symmetry of such problems, it is frequetly possible to fid a special basis for the solutio space that could support a tridiagoal matrix represetatio for the wave operator H E. This property makes the solutio of the problem easily attaiable as will ψ = be explaied ext. Let { } be such a basis, which is complete, square itegrable ad compatible with the domai of the Hamiltoia. Therefore, the actio of the wave operator o the elemets of the basis takes the geeral form ( H E) ψ ψ +, where we ca also write ψ H E ψ = ( a y) δ + bδ + b δ, (.) ψ + ψ + m, m, m, m+ where y 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 ψ m m i ( H E) χ = ad projectig o the left by ψ, results i the followig three-term recursio relatio yf = af + b f + bf + (.3) 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 m f

3 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 Morse problem which was tackled successfully by J. T. Broad [9] ad P. C. Ojha []. The orelativistic Coulomb problem, despite beig log rage, was also solved by Yamai ad Reihardt usig the same approach i which the expasio coefficiets of the wavefuctio are writte i terms of the Pollaczek polyomials []. The relativistic extesio of the Coulomb problem has recetly bee carried out by this author where the expasio coefficiets of the spior wavefuctio are writte i terms of the Meixer-Pollaczek polyomials []. I this article the same approach will also be employed i fidig exact aalytic solutio of the relativistic Dirac-Morse problem for all eergies. It is obvious that the solutio of (.3) is obtaied modulo a overall factor which is a fuctio of y but, otherwise, idepedet of. The uiqueess of the solutio is achieved by the requiremet that, for example, the wavefuctio be eergy ormalizable. It should also be oted that the solutio of the problem as give by Eq. (.3) above is obtaied for all E, the discrete as well as the cotiuous, costraied oly by the reality ad boudedess of the represetatio. Moreover, the matrix represetatio of the wave equatio, Eq. (.), clearly shows that the discrete spectrum is easily obtaied by diagoalizatio which requires that: b =, ad a y = (.4) I Sec. II, we set up the oe dimesioal Dirac equatio for a two-compoet spior coupled o-miimally to the potetial ( A, A ). The space ad time compoets of the potetial are take to be of the same type. A global uitary trasformatio is applied to the Dirac equatio to separate the variables such that the resultig secod order differetial equatio for the spior compoets becomes Schrödiger-like. This results i a simple ad straightforward correspodece betwee the relativistic ad orelativistic problems. The correspodece will be used i Sec. III as a guide for costructig a square itegrable basis for the solutio space of the Dirac-Morse 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. The recursio relatio is writte i a form that makes its solutio easily attaiable by simple ad direct compariso with that of the cotiuous dual Hah orthogoal polyomials [3]. We coclude with a short discussio i Sec. IV. II. THE DIRAC-MORSE PROBLEM µ The oe-dimesioal Dirac equatio for a free particle reads ( i γ µ mc) χ =, where m is the rest mass of the particle, c the speed of light, ad χ is the two-compoet spior wavefuctio. The summatio covetio over repeated idices is used. That is, µ γ = µ γ + γ = γ γ ct + x, where γ ad γ are two costat square matrices µ ν µ ν ν µ µν satisfyig the aticommutatio relatio { γ, γ } = γ γ + γ γ = G ad G is the space-time metric which is equal to diag( +, ). A two-dimesioal matrix represetatio γ σ 3 γ = iσ =. I the that satisfies this relatio is chose as = = ( ) ad ( ) 3

4 atomic uits = m =, the Compto wavelegth, = mc = c, is the relativistic µ parameter ad Dirac equatio reads ( iγ µ ) χ =. Next, we let the Dirac spior be coupled to the two compoet potetial Aµ = ( A, A). Gauge ivariat couplig, which is accomplished by the miimal substitutio µ µ + i Aµ, trasforms the free Dirac equatio ito µ iγ ( µ + i Aµ ) χ =, (.) which, whe writte i details, reads as follows i i A A χ = σ + σ + + σ χ, (.) t ( x 3) where σ = ( ). For time idepedet potetials, this equatio gives the followig matrix represetatio of the Dirac Hamiltoia (i uits of mc = ) + A A d i dx H = (.3) d A i + A dx Thus the eigevalue wave equatio reads ( H ε ) χ =, where ε is the relativistic eergy which is real, dimesioless ad measured i uits of. Equatio (.) is ivariat uder the local gauge trasformatio i Λ Aµ Aµ + µ Λ, χ e χ, (.4) where Λ (, tx) is a real space-time scalar fuctio. Cosequetly, the off diagoal term A i the Hamiltoia (.3) could be elimiated ( gauged away ) by a suitable choice of the gauge field Λ(x) i the trasformatio (.4). However, our choice of couplig will be o-miimal, which is obtaied by the replacemet A ± i A, respectively. That is the Hamiltoia (.3) is replaced by the followig + A i A d i dx H = (.5) d i A i + A dx Next, we take the two compoets of the potetial, A ad A, to be of the same fuctioal form by writig A = V( x) ad A = V( x), where ξ is a real positive ξ ig ( x) parameter with iverse legth dimesio. Writig χ as ( f ( x )) gives us the followig twocompoet wave equatio + V ε V d ( g dx ) ξ V d = (.6) ( + ) + V ε ξ dx f This matrix equatio results i two coupled first order differetial equatios for the two spior compoets. Elimiatig oe compoet i favor of the other gives a secod order differetial equatio. This will ot be Schrödiger-like. That is, it cotais first order derivatives. Obtaiig a Schrödiger-like wave equatio is desirable because it results i a substatial reductio of the efforts eeded for gettig the solutio. It puts at our disposal a variety of well established techiques to be employed i the aalysis ad 4

5 solutio of the problem. These techiques have bee well developed over the years by may researchers i dealig with the Sturm-Liouville problem ad the Schrödiger equatio. Oe such advatage, which will become clear shortly, is the resultig map betwee the parameters of the relativistic ad orelativistic problems. This parameter map could be used i obtaiig, for example, the relativistic eergy spectrum i a simple ad straight-forward maer from the kow orelativistic spectrum. To obtai a Schrödiger-like equatio we apply a global uitary trasformatio i U ( η) = exp( ησ ) to the Dirac equatio (.6), where η is a real costat parameter ad i σ =. The Schrödiger-like requiremet results i the costrait that si( η) = ( i ) ± ξ, where π π < η <+. It is to be oted that the agular parameter of the uitary trasformatio was itetioally split as η ad ot collected ito a sigle agle, say ϕ. This is suggested by ivestigatig the costrait si( ϕ) = ± ξ i the orelativistic limit ( ) where we should have si( ϕ) ϕ = ± ξ. It also makes it obvious that i the orelativistic limit the trasformatio becomes the idetity (i.e., ot eeded). The uitary trasformatio together with the parameter costrait map Eq. (.6) ito the followig oe + C ε + ( ± ) V C d ( ) ( ξ + V φ x ξ dx ) C d =, (.7) ( ξ + V + ) C ε + ( ) V ξ dx φ ( x) where C = cos( η) = ( ξ) > ad + η η φ cos si g ψ = φ U = η η si cos f (.8) Equatio (.7) gives the followig ( kietic balace ) relatio betwee the two spior compoets ( x ) C V ( x ) d ± φ = ξ ± + φ ( x ) (.9) C ± ε ξ dx While, the resultig Schrödiger-like wave equatio becomes d C C dv ε + ( ) V + εv φ ± ( x) = dx ξ ξ (.) dx This equatio could be rewritte i a form which is more familiar i the laguage of supersymmetric quatum mechaics d dw ( ε C) + ( W ) ( x) dx φ ± = dx, (.) where ( ) C ξ W x = V( x) + ε. W ± W are two superparter potetials sharig the same ξ C eergy spectrum (i.e., they are isospectral ) except for the highest positive eergy state ad the lowest egative eergy state, where ε = ± C, respectively. These two states belog oly to W W [4]. ξ. Therefore, The orelativistic limit ( ) gives ε + E ad C Eq. (.9) shows that φ + is the larger of the two relativistic spior compoets (i.e., φ + is 5

6 the compoet that survives the orelativistic limit, whereas φ ~ φ + ). Cosequetly, if we favor the upper spior compoet the our choice of sig i the trasformatio parameter costrait is the top + sig. That is, we choose si( η) =+ ξ. For the Dirac-Morse problem the potetial fuctio V( x) = Ae ωx, where the parameters A ad ω are real ad ω >. Substitutig this potetial ito Eq. (.) gives the followig Schrödiger-like secod order differetial equatio for the upper spior compoet d CA ωx CA ωx ε + + ( ) e ( ω+ ξε C) e φ ( x) = dx ξ ξ (.) We compare this equatio with that of the orelativistic Schrödiger-Morse problem d ω x ωx + Be B ( ω + D) e E Φ ( x) =, (.3) dx where B ad D are real, ad B >. The compariso gives the followig correspodece betwee the parameters of the two problems: ξε C E ( ε ), B ± CA ξ, D (.4) ω ξε C The top (bottom) choice of map for B ad D correspods to positive (egative) values of A. Usig this parameter map ad the well-kow orelativistic eergy spectrum [5], E D ( ) ω = ω, we obtai the followig boud states relativistic spectrum ± C A ξω ± ω > = C + ± C + A < ε ( ), (.5) ξω( ) [ ω( ) ], where =,,,..., max for A >, =,,,..., max for A <, ad max is the maximum iteger which is less tha or equal to Cω. The two eergy spectra are ± related by ε = ε A< +. Therefore, all states are degeerate except two. Those are A> the highest positive, ad lowest egative eergy states correspodig to ε ± =± C, A> respectively. I fact, these are the two states that belog oly to the super-potetial W W i Eq. (.) above. The upper spior compoet could be obtaied from the orelativistic wavefuctio [6], α z α Φ ( x)~ z e L ( z), (.7) usig the same parameter map (.4), where B x z = ω e ω, D λ α = ω, ad L ( z) is the Laguerre polyomial [7]. Square itegrability requires that α > (i.e., D ω ). The above fidigs, which are valid oly for boud states, will be used i the followig sectio as a guide to writig dow a L spior bases for the solutio space of scatterig states. These bases will be chose such that the matrix represetatio of the Dirac-Morse operator ( H ε ) i Eq. (.7) is tridiagoal. 6

7 III. TRIDIAGONAL REPRESENTATIONS As suggested by the expressio of the orelativistic wavefuctio i (.7) ad CA ωx CA ωx the parameter map (.4), we write z =± e = e for ± A >. This maps the whole real lie ito its positive half. That is, x [ +, ] z [, ] ωξ +. I this coordiate otatio ad with the choice si( η) = + ξ, the Dirac-Morse equatio (.7) reads as follows: ξ C ε ( ωξ C) z ω z d ( ω ± z dz ) φ ξ ω z d =, (3.) ( ω ± + z dz ) C ε θ + φ φ =, ad the top (bottom) sig correspods to positive (egative) value of A. where ( θ ) ( φ ) A square itegrable basis fuctio (with respect to the measure dx = dz zω ) i this cofiguratio space which is compatible with the domai of the wave operator ad satisfies the boudary coditios could be writte as ωγ ( + ) ( ) ( ) α βz ν φ ( ) = βz e L βz, (3.) Γ + ν + where α ad β are real, positive, dimesioless parameters ad ν >. The orelativistic wavefuctio (.7) ad the parameter map (.4) support this choice of basis. The kietic balace relatio (.9) i the ew coordiate otatio reads as follows ξ θ = ω z d C ( ω ± + z ε dz ) φ, (3.3) + for ± A >. It suggests that the lower compoet of the spior basis is obtaiable from the upper by the followig geeral differetial relatio θ ~ d ( µ + ζz+ z ) φ, (3.4) dz where the dimesioless parameters µ ad ζ are real ad will be determied as we proceed. Substitutig (3.) ito (3.4) ad usig the differetial ad recursio properties of the Laguerre polyomials (show i the Appedix) we obtai ωγ ( + ) ( ) ( ) α βz { ( ) ν θ ( ) ( ) = τ βz e β µ + α ν+ + ζ + ν+ L βz Γ + ν + (3.5) ν ν ( β + ζ)( + ν) L ( βz) + ( β ζ)( + ) L ( βz) = βτ µ + ζ z+ z d φ + } ( dz ) φ where τ is aother real parameter. I this spior basis { ψ ( θ )} represetatio of the Dirac-Morse operator of (3.) reads ψ ( ) ( ) H εψm = C ε φ φm ( ωξ C) φ z φm C+ ε ω θ βτ θm + ( µω ξ) θ φm + θm φ + ω( ζ ) θ z φm + θm z φ 7 ωξ =, the matrix = (3.6) where we have used itegratio by parts i writig φ z θ dz m = φ z θ dz m sice the product φ ( z) θ ( z) vaishes at the boudaries z = ad z, ad m + d dθ m dθm z m = dz z dx = dz dz ω dz dφ + dφ d ω θ m dz θ dz mz dx φ dz z θ dz m φ θ φ φ = = = d d (3.7)

8 Now, we require that the represetatio (3.6) be tridiagoal. That is, ψ H ε ψm = for all m. For the first two terms o the right side of Eq. (3.6) to comply with this requiremet we must have α = 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 must vaish. That is, we must choose ζ = ± for ± A >. I additio, the term θ θ m results i a tridiagoal represetatio oly if ( β )( β + ) =, otherwise the multiplyig factor must vaish. The latter choice is ot acceptable sice ω β it requires that τ = which violates the kietic balace relatio (3.3) that will be C+ ε ecessary later o for further ivestigatios. Now, sice β must be positive, the we ed up with the oly possibility that β =. Cosequetly, we are left with three udetermied real parameters (α, µ, ad τ) ad with the followig two compoets of the spior basis ωγ ( + ) ( ) ( ) α z α φ ( ) = z e L z (3.8a) Γ + α α α ωγ ( + ) α z ( µ + α) L ( z) zl ( z), A> θ = τ ( z) e Γ ( + α ) (3.8b) α α ( µ + α) L ( z) zl ( z), A< Substitutig these ito (3.6) ad usig the orthogoality ad recurrece relatios of the Laguerre polyomials (show i the Appedix) we obtai, after some maipulatios, the followig elemets of the symmetric tridiagoal matrix represetatio of the Dirac- Morse operator ( H ε) = C ε ( ωξ C)( + α) ± 4 τ( ξ µω)( + α ± µ ), (3.9a) 4 τ ( C+ ε ω τ) + (α ± µ ) + ( α ± µ ) + ( ) α ( H ε) = ( + α ), (3.9b) ± ( ωξ C) ± τ ( µω ξ ) + 4 τ ( C+ ε ω τ ) ( + α ± µ ± ) where the top ad bottom sigs correspod to ± A >, respectively. If we defie the followig quatities: p( ε ) = 4τ ( C+ ε ω τ) ad q = τ ( µω ξ ), (3.) 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 wavefuctio q+ ωξ C µ q (α µ ) ( α µ ) ( α) ( ) C p α ε p f + ± + ± ± + + ± + p ( ) q+ ωξ C + α ± + α ± µ ± f (3.) ( p ) q+ ωξ C ( p ) ( + )( + α) + + α ± µ ± f = for. Rewritig (3.) i terms of the polyomial Q( ε ) = Γ ( + ) Γ ( + α) f( ε), gives the followig recursio relatio q+ ωξ C µ q (α µ ) ( α µ ) ( α) ( ) C p p α ε Q + ± + ± ± + ± + + p (3.) q+ ωξ C q+ ωξ C ± + α ± µ ± Q + α + + α ± µ ± Q = ( p ) ( )( p ) + + 8

9 for ± A >. We compare this with the recursio relatio satisfied by the cotiuous dual λ Hah orthogoal polyomials S ( y ; a, b) [3] that reads λ λ ys = ( + λ+ a)( + λ+ b) + ( + a+ b ) λ S (3.3) λ λ ( + a+ b ) S ( + λ+ a)( + λ+ bs ) + where y >. The compariso results i the followig values for the parameters of the polyomial q+ ωξ C b= µ +, C p y = ε µ ( µ + q p) ; A> p λ = a = α, (3.4) q+ ωξ C b = µ, C p y = ε µ ( µ q p) ; A < p Therefore, we ca write Γ ( + α ) α f ( ε) = S ( y ; α, b), (3.5) Γ ( + ) which is defied up to a multiplicative factor that depeds o ε but, otherwise, idepedet of. The cotiuous dual Hah polyomials could be writte i terms of the hypergeometric fuctio 3 F as, [3] λ, λ+ iy, λ iy S ( y ; a, b) = 3F( λ+ a, λ+ b ) (3.6) The orthogoality relatio associated with these polyomials is as follows where λ λ λ Γ ( + ) Γ ( + a+ b) ρ ( y) S( y ; a, b) Sm( y ; a, b) dy = δ Γ ( + λ+ a) Γ ( + λ+ b) m, (3.7) λ Γ ( λ+ iy) Γ ( a+ iy) Γ ( b+ iy) π Γ ( λ+ a) Γ ( λ+ b) Γ( iy) ρ ( y) =. Therefore, the ormalizable L series solutio of the Dirac-Morse problem could be writte as α Γ ( + α ) α χ( x, ε) = N ( ε) S y( ε) ; α, b( ε) ψ ( x), (3.8) = Γ ( + ) α α where N ( ε ) ρ ( y) ( dy dε) ( ) = is a ormalizatio factor that makes χ eergyormalizable, whereas the two compoets of the spior basis elemet ψ ( x ) are those give by Eqs. (3.8). Further aalysis of these solutios, such as obtaiig the discrete spectrum, is tractable oly if the kietic balace relatio (3.3) is strictly imposed o the basis elemets. That is, relatio (3.4) should be idetical to (3.3) which requires that ω µ = ξ ω ad τ =, (3.9) C+ ε where ε C. This gives q = ad p = ω C + ε resultig i the followig polyomial parameters: λ = a = α, y( ε ) = ( ε ) ω ad { ( ξ Cω ) ε A b( ε ) = > + ( ξ Cω) ε, A<. Thus, the eergy depedece of the parameters simplifies resultig i the followig expressio for the wavefuctio = ε ξε ( ω Cω ) Γ ( + α ) α Γ ( + ) χ( x, ε) N α = ( ε) S ; α, ψ ( x), (3.) for ± A >. If we take the orelativistic limit (, of this solutio we obtai ε + E, ad C ξ ) 9

10 ( ω ) χ α Γ ( + α ) α ( x, E) = N ( E) E NR ;, D S ( ) ( ) x ω α Γ + φ, (3.) = which agrees with the fidigs by J. T. Broad [9] ad P. C. Ojha []. The requiremet that the argumet y of S a ( y ; b, c ) be positive implies that the solutio (3.) is valid for ε >. I other words, the solutio (3.8) or (3.) is for eergies larger tha the rest mass mc correspodig to scatterig states. Solutios for ε < are for boud states ad correspod to y <. To obtai these solutios, which are differet from (3.), ad to calculate the discrete eergy spectrum, we impose the diagoalizatio requiremet (.4). I the case of the recursio relatio (3.) ad with the parameters assigmet (3.9), this requiremet traslates ito the followig coditios + α + b=, ad y = α, (3.) givig exactly the same eergy spectrum i Eq. (.5). Therefore, both the parameter correspodece map method (preseted ear the ed of Sec. II) ad the diagoalizatio method have idepedetly give the same results. I additio, we also obtai the followig as a cosequece of the diagoalizatio requiremet (3.): ( ξ Cω) ε, A> α = (3.3) ( ξ Cω) ε, A < where ξ Cω ad ξ Cω for ± A >, respectively. The boud states spior wavefuctios become ωγ ( + ) ( ) ( ) α z α φ ( ) = z e L z (3.4a) Γ + α α α ω ωγ ( + ) α ( ) ( ) ( ), z α + ξ ω L z zl z A> θ = ( z) e C+ ε ( ) Γ + α (3.4b) α α ( α + ξ ω) L ( z) zl ( z), A< IV. DISCUSSION We would like to coclude with some commets that have to do with the type ad umber of solutios of the recursio relatio (3.) or (3.). 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 (3.8) 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. For a large positive iteger N the recursio relatio (3.) could be rewritte as =, where ( ) uqn+ ( u) QN+ ( u) QN+ + ( u) we could write it as y N u =. Defiig Q ˆ ( u) Q ( u), N+

11 uq ˆ ˆ ˆ ( z) Q ( u) Q+ ( u) = (4.) This is the recursio relatio for 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 will be desigated by Q ( u), 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: + + y αα ( + b) Q + αα ( + b) Q = (4.a) y αα ( + b) Q + αα ( + b) Q = W, (4.b) where W = W ( y, b) 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 Q ( u). Such aalysis is typical of algebraic scatterig methods i orelativistic quatum mechaics. A clear example is foud i the J-matrix method of scatterig [5,]. ACKNOWLEDGMENT The author is grateful to M.E.H. Ismail for the help i idetifyig the solutio of the three-term recursio relatio (3.) as the cotiuous dual Hah orthogoal polyomials. 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 i most textbooks o orthogoal polyomials [7]. 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:

12 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. REFERENCES: [] See, for example, G. A. Natazo, Teor. Mat. Fiz. 38, 9 (979) [Theor. Math. Phys. 38, 46 (979)]; L. E. Gedeshtei, Zh. Eksp. Teor. Fiz. Pis ma Red. 38, 99 (983) [JETP Lett. 38, 356 (983)]; F. Cooper, J. N. Giocchi, ad A. Khare, Phys. Rev. D 36, 458 (987); R. Dutt, A. Khare, ad U. P. Sukhatme, Am. J. Phys. 56, 63 (988); 59, 73 (99); G. Lévai, J. Phys. A, 689 (989); 7, 389 (994) [] A. de Souza-Dutra, Phys. Rev. A 47, R435 (993); N. Nag, R. Roychoudhury, ad Y. P. Varshi, Phys. Rev. A 49, 598 (994); R. Dutt, A. Khare, ad Y. P. Varshi, J. Phys. A 8, L7 (995); C. Grosche, J. Phys. A, 8, 5889 (995); 9, 365 (996); G. Lévai ad P. Roy, Phys. Lett. A 7, 55 (998); G. Juker ad P. Roy, A. Phys. (N.Y.) 64, 7 (999); R. Roychoudhury, P. Roy, M. Zojil, ad G. Lévai, J. Math. Phys. 4, 996 () [3] A. V. Turbier, Commu. Math. Phys. 8, 467 (988); M. A. Shifma, It. J. Mod. Phys. A 4, 897 (989); R. Adhikari, R. Dutt, ad Y. Varshi, Phys. Lett. A 4, (989); J. Math. Phys. 3, 447 (99); R. Roychoudhury, Y. P. Varshi, ad M. Segupta, Phys. Rev. A 4, 84 (99); L. D. Salem ad R. Motemayor, Phys. Rev. A 43, 69 (99); M. W. Lucht ad P. D. Jarvis, Phys. Rev. A 47, 87 (993); A. G. Ushveridze, Quasi-exactly Solvable Models i Quatum Mechaics (IOP, Bristol, 994) [4] A. D. Alhaidari, Phys. Rev. Lett. 87, 45 (); 88, 899 (); J. Phys. A 34, 987 (); 35, 67 (); Guo Jia-You, Fag Xiag Zheg, ad Xu Fu- Xi, Phys. Rev. A 66, 65 (); J-Y Guo, J. Meg, ad F-X Xu, Chi. Phys. Lett., 6 (3); A. D. Alhaidari, It. J. Mod. Phys. A 8, 4955 (3) [5] E. J. Heller ad H. A. Yamai, Phys. Rev. A 9, (974); H. A. Yamai ad L. Fishma, J. Math. Phys. 6, 4 (975); A. D. Alhaidari, E. J. Heller, H. A. Yamai, ad M. S. Abdelmoem (eds.), J-matrix method ad its applicatios (Nova Sciece, New York, 4) [6] A. M. Lae ad R. G. Thomas, Rev. Mod. Phys. 3, 57 (958); A. M. Lae ad D. Robso, Phys. Rev. 78, 75 (969) [7] A. U. Hazi ad H. S. Taylor, Phys. Rev. A, 9 (97) [8] T. S. Murtaugh ad W. P. Reihardt, Chem. Phys. Lett., 56 (97); W. P. Reihardt, D. W. Oxtoby, ad T. N. Rescigo, Phy. Rev. Lett. 8, 4 (97); E. J. Heller, T. N. Rescigo, ad W. P. Reihardt, Phys. Rev. A 8, 946 (973) [9] J. T. Broad, Phys. Rev. A 6, 378 (98) [] P. C. Ojha, J. Phys. A, 875 (988) [] H. A. Yamai ad W. P. Reihardt, Phys. Rev. A, 44 (975) [] A. D. Alhaidari, A. Phys. (N.Y.), i productio.

13 [3] R. Koekoek ad R. F. Swarttouw, The Askey-scheme of hypergeometric orthogoal polyomials ad its q-aalogue, Report o (Delft Uiversity of Techology, Delft, 998) pp [4] See, for example, E. Witte, Nucl. Phys. B 85, 53 (98); F. Cooper ad B. Freedma, A. Phys. (NY) 46, 6 (983); C. V. Sukumar, J. Phys. A 8, 97 (985); A. Arai, J. Math. Phys. 3, 64 (989); F. Cooper, A. Khare, ad U. Sukhatme, Phys. Rep. 5, 67 (995) [5] P. M. Morse, Phys. Rev. 34, 57 (99); S. Flügge, Practical Quatum Mechaics (Spriger-Verlag, Berli, 974) p. 8 [6] These fuctios are orthogoal (with respect to the measure dx = ω dz z ). The more α z α commoly used fuctios, Φ ~ z e L ( z), are ot. However, both are square itegrable. [7] Examples of textbooks ad moographs o special fuctios ad orthogoal polyomials are: W. Magus, F. Oberhettiger, ad R. P. Soi, Formulas ad Theorems for the Special Fuctios of Mathematical Physics (Spriger-Verlag, New York, 966); T. S. Chihara, A Itroductio to Orthogoal Polyomials (Gordo ad Breach, New York, 978); G. Szegö, Orthogoal polyomials, 4 th ed. (Am. Math. Soc., Providece, RI, 997); R. Askey ad M. Ismail, Recurrece relatios, cotiued fractios ad orthogoal polyomials, Memoirs of the Am. Math. Soc., Vol. 49 Nr. 3 (Am. Math. Soc., Providece, RI, 984) 3