An extended class of L 2 -series solutions of the wave equation

Transcription

1 A exteded class of L -series solutios of the wave equatio A D Alhaidari Physics Departmet, Kig Fahd Uiversity of Petroleum & Mierals, Dhahra 36, Saudi Arabia haidari@mailapsorg We lift the costrait of a diagoal represetatio of the Hamiltoia by searchig for square itegrable bases that support a ifiite tridiagoal matrix represetatio of the wave operator The class of solutios obtaied as such icludes the discrete (for boud states) as well as the cotiuous (for scatterig states) spectrum of the Hamiltoia The problem traslates ito fidig solutios of the resultig three-term recursio relatio for the expasio coefficiets of the wavefuctio These are writte i terms of orthogoal polyomials, some of which are modified versios of kow polyomials The examples give, which are ot exhaustive, iclude problems i oe ad three dimesios PACS umbers: 0365Ge, 030Gp, 0365Nk, 0365Ca I INTRODUCTION Oe of the advatages of obtaiig exact solutios of the wave equatio is that the aalysis of such solutios makes the coceptual uderstadig of physics straightforward ad sometimes ituitive Moreover, these solutios are valuable meas for checkig ad improvig models ad umerical methods itroduced for solvig complicated physical systems I fact, i some limitig cases or for some special circumstaces they may costitute aalytic solutios of realistic problems or approximatios thereof Most of the kow exactly solvable problems fall withi distict classes of, what is referred to as, shape ivariat potetials [] Each class carries a represetatio of a give symmetry group Supersymmetric quatum mechaics, potetial algebras, poit caoical trasformatios, ad path itegratio are four methods amog may which are used i the search for exact solutios of the wave equatio I orelativistic quatum mechaics, these developmets were carried out over the years by may authors where several classes of these solutios are accouted for ad tabulated (see, for example, the refereces cited i []) These formulatios were exteded to other classes of coditioally exactly [,3] ad quasi exactly [4,5] solvable problems where all or, respectively, part of the eergy spectrum is kow Recetly, the relativistic extesio of some of these fidigs was carried out where several relativistic problems are formulated ad solved exactly These iclude, but ot limited to, the Dirac-Morse, Dirac-Scarf, Dirac-Pöschl-Teller, Dirac-Hulthéetc [6] I these developmets, the mai objective is to fid solutios of the eigevalue wave equatio H ψ E ψ, 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 whe searchig for algebraic or umerical solutios, the wave fuctio ψ spas the space of square itegrable fuctios with discrete basis elemets { } 0 That is, the wavefuctio is expadable as ψ ( re, ) f ( E ) ( r ), where r is the set of coordiates of real space The basis fuctios must be compatible with the domai of the Hamiltoia They should also satisfy the boudary coditios Typically (ad especially whe calculatig the discrete spectrum) the choice of basis is limited to those that carry a

2 diagoal represetatio of the Hamiltoia That is, oe looks for a L basis set { } 0 such that H E givig the discrete spectrum of H The cotiuous spectrum 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 matrix represetatio of the Hamiltoia We oly require that the hermitia matrix represetatio of the wave operator be tridiagoal That is, the actio of the wave operator o the elemets of the basis is allowed to take the geeral form ( H E) ad such that H E ( a z) δ bδ b δ, () m, m, m, m where z ad the coefficiets { a, b} 0 are real ad, i geeral, fuctios of the eergy, agular mometum, ad potetial parameters Therefore, the matrix wave equatio, which is obtaied by expadig ψ as f m m m i ( H E) ψ 0 ad projectig o the left by, results i the followig three-term recursio relatio zf a f b f bf () Cosequetly, the problem traslates ito fidig solutios of the recursio relatio for the expasio coefficiets of the wavefuctio ψ I most cases this recursio is solved easily ad directly by correspodece with those for well kow orthogoal polyomials It is obvious that the solutio of () is obtaied modulo a overall factor which is a fuctio of the physical parameters of the problem but, otherwise, idepedet of The uiqueess of the solutio is achieved by the requiremet (for example) of ormalizability of the wavefuctio, that is ψψ It should also be oted that the solutio of the problem as depicted by Eq () above is obtaied for all E, the discrete as well as the cotiuous Moreover, the represetatio equatio () clearly shows that the discrete spectrum is easily obtaied by diagoalizatio which requires that: b 0, ad a z 0 (3) Oe could obtai two solutios to the recursio () by startig with two differet iitial relatios ( 0) Oe solutio is associated with the homogeeous recursio relatio that starts with: ( a0 z) f0 b0f 0 (4) The other is associated with a o-homogeeous recursio whose iitial relatio is ( a0 z) f0 b0f ξ, (5) where ξ is a real ad o-zero seed parameter Oe of these two solutios behaves asymptotically ( ) as sie-like while the other behaves as cosie-like These two solutios have the same asymptotic limits as the regular ad irregular solutios of the secod order differetial wave equatio Scatterig states ad the phase shift could be obtaied algebraically by studyig these asymptotic limits These are issues of cocer i algebraic scatterig theories such as the J-matrix method [7] I the preset work, however, we will oly be cocered with the regular solutios of the wave equatio I cofiguratio space, with coordiate x, the wavefuctio ψ E ( x ) is expaded as f 0 E x where the L basis fuctios could geerally be writte as ( x) Aw( x) P( x) (6)

3 A is a ormalizatio costat, P ( x ) is a polyomial of degree i x, ad the weight fuctio satisfies w ( x ± ) 0, where x ( x ) is the left (right) boudary of cofiguratio space I the followig sectios we cosider examples i two spaces Oe is where x ± are fiite ad α β w ( x) ( x x ) ( x x ), (7) P ( x) F(, b; c; x) The other is semi-ifiite where x is fiite, x ifiite, ad where α β ( x x ) w ( x) ( x x ) e, (8) P ( x) F( ; c; x) F is the hypergeometric fuctio ad F is the cofluet hypergeometric fuctio The parameters α, β,b ad c are real with α ad β positive They are related to the physical parameters of the correspodig problem ad may also deped (for boud states) o the idex I the followig sectios we cosider examples of various problems i oe ad three dimesios ad fid their L series solutios The solutios of some of the classic problems such as the Coulomb ad Morse are reproduced addig, however, ew tridiagoal represetatios of the solutio space We also fid geeralizatios of others such as the Hulthé-type problems where we obtai a exteded class of solutios ad defie their associated orthogoal polyomials I additio, we ivestigate problems with hyperbolic potetials such as the Rose-Morse type ad preset its geeralized solutio These ivestigatios do ot exhaust the set of all solvable problems usig this approach Furthermore, this developmet embodies powerful tools i the aalysis of solutios of the wave equatio by exploitig the itimate coectio ad iterplay betwee tridiagoal matrices ad the theory of orthogoal polyomials I such aalysis, oe is at liberty to employ a wide rage of well established methods ad umerical techiques associated with these settigs such as quadrature approximatio ad cotiued fractios [8] II THE COULOMB PROBLEM We start by takig the cofiguratio space coordiate simply as x r, where is a legth scale parameter which is real ad positive ad r is the radial coordiate i three dimesios This problem belogs to the case described by Eq (8) with x 0 Sice F( ; c; z) is proportioal to the Laguerre polyomial c L ( z) [9], we could write the basis fuctios as α β () r A x e x L () x, () where A Γ ( ) Γ ( ), 0,,,, >, α ad β are real ad positive I the atomic uits, m, the radial time-idepedet Schrödiger wave equatio for a structureless particle i a spherically symmetric potetial V(r) reads d ( ) ( H E) ψ V( r) E ψ 0, () dr r 3

4 where is the agular mometum quatum umber The actio of the first term (the secod order derivative) o the basis fuctio results i the followig d α βx d d Ax e α α ( x α β) ( x β) L ( x) (3) dr dx dx x Usig the differetial equatio ad differetial formula of the Laguerre polyomial [Eqs (A3) ad (A4) i the Appedix] we obtai the followig actio of the wave operator () o the basis H E ( α αα αβ β x ) β x x x (4) A V E ( β α x ) x A The orthogoality relatio for the Laguerre polyomials [show i the Appedix as Eq (A5)] requires that β if we were to obtai a tridiagoal represetatio for H E m Additioally, we ed up with oly two possibilities to achieve the tridiagoal structure of the wave operator (4): () α, α, ad V Z r (5a) () α, 8E, ad V Z B / r r (5b) where Z ad B are real potetial parameters Z is the particle s charge ad B is a cetripetal potetial barrier parameter We start by cosiderig the first possibility described by (5a) where we have: () x r Ax e L () x (6) Substitutig i Eq (4) with β ad projectig o m we obtai m H E ( )( E 4 ) Z δ, m (7) E ( 4 ) ( ) δm, ( )( ) δ m, Therefore, the resultig recursio relatio () for the expasio coefficiets of the wavefuctio becomes σ Z ( ) σ σ f ( ) f ( )( ) f 0, (8) where σ E ± ± 4 Rewritig this recursio i terms of the polyomials P ( E ) Γ ( ) Γ ( ) f( E), we obtai the more familiar recursio relatio σ Z ( ) σ σ P ( ) P ( ) P 0, (9) which is that of the Pollaczek polyomials [0] provided that E > 0 [see Eq (A) i the Appedix] Thus, we ca write E ( E 8 ) Γ ( ) 8 Γ ( ) E f ( E) P Z,cos, (0) givig the followig L -series solutio of the Coulomb problem for positive eergies E ( E 8 ) (, re) N Γ ( ) 8 ( ),cos r P E r e L r Γ 0 ψ Z, () where N is a ormalizatio costat This solutio of the Coulomb problem was obtaied by Yamai ad Reihardt [] Restrictig the represetatio (7) to the 4

5 diagoal form gives the discrete spectrum via the requiremet (3) which, i this case, reads as follows: E 0, 4 () ( E )( 4 ) Z 0 This gives the followig well kow eergy spectrum for the boud states of the Coulomb problem E Z, ( ) Z, (3) where 0,,, Therefore, the correspodig boud states wavefuctios are ψ () r r A r e L ( r) (4) For the secod possibility described by (5b) the basis fuctios are () r A x e x L () x, x Er, (5) where E < 0 Thus, the basis parameter i this case is the Laguerre polyomial idex Substitutig i Eq (4) ad projectig o we get H E 4 ( )( τ ) ( ) ( ½ E ) B δ m m, m τ ( ) δ τ ( )( ) δ m, m, (6) where τ Z E The resultig recursio relatio for the expasio coefficiets of the wavefuctio i terms of the polyomials P( E) Γ ( ) Γ ( ) f ( E) reads B P ( )( τ ) ( ½) ( τ) ( )( τ) P P 0 (7) This is a special case of the recursio relatio for the cotiuous dual Hah orthogoal polyomial [], which is show as Eq (A4) i the Appedix As a result we obtai S ( z;, τ ), B ( ½) Γ < f( E) Γ ( ) (8) S ( z;, τ ), B> ( ½) where z ( ½) B ad S µ (; z a, b) is a modified versio of the cotiuous dual Hah polyomial defied as, x, x S µ µ µ µ ( x; a, b) S ( ix; a, b) F (9) 3 ( µ a, µ b ) The discrete eergy spectrum is evidetly obtaied by diagoalizatio of (6) which traslates ito the requiremets τ 0, (0) ½ B 0, givig, E Z, ( ½) B () This is the same eergy spectrum as that i Eq (3) above whe B 0 5

6 III THE SPHERICAL OSCILLATOR I this case, we write the cofiguratio space coordiate as x ( r) Thus, the basis elemets of the L space is writte as α β () r A x e x L () x, (3) where agai α ad β are real ad positive, >, ad A Γ ( ) Γ ( ) d dr d dx Usig x, we obtai the followig α ½ α α ½ α ( x ) ( x )( x ) d α βx d d 4 Ax e β β β L x (3) dr dx dx x Employig the differetial formulas of the Laguerre polyomials, which are show i the Appedix, gives the followig d ½ ( ½) ( ½) 4 x α α α β α ( x ) dr β β x x x (33) 4 ½ A α ( x β) A The tridiagoal structure ca oly be achieved if ad oly if β ½ resultig i the followig actio of the wave operator o the basis (α ½) ( α ¼) ( ½) 4 x ( H E) α x 4 4 (34) ( )(α ½) A ( V E) x A The orthogoality relatio (A5) ad the tridiagoal requiremet limit the possibilities to either oe of the followig two: 4 () α, α, ad V ω r (35a) 4 () α 3, ad V r B r (35b) where ω is the oscillator frequecy ad B is a cetripetal potetial barrier parameter The first possibility gives the followig tridiagoal matrix represetatio of the wave operator 4 m H E ( ω )( 4 ) E δ, m (36) 4 ω 4 ( δ ) m, ( )( ) δ m, Writig the resultig recursio relatio i terms of the polyomials P ( E ) Γ ( ) Γ ( ) f( E) gives the followig E ( σ ) σ σ P ( ) P ( ) P 0, (37) where σ 4 ± ω ± We compare this with the recursio relatio of the Pollaczek polyomial [Eq (A) i the Appedix] while carefully cosiderig the rage of values of the parameters Usig the well kow relatios, coshθ cos iθ ad sihθ isi iθ, we could defie a Hyperbolic Pollaczek polyomial as follows 6

7 Γ ( µ ) Γ ( ) Γ( µ ) P x P ix i e F x e µ (, ) µ θ (, θ ) θ (, µ ; µ ; θ ), (38) where θ > 0 These polyomials satisfy the followig modified three-term recursio relatio: µ µ µ ( [ µ )coshθ xsih θ ] P ( µ ) P ( ) P 0 (39) Therefore, the expasio coefficiets of the oscillator wavefuctio could be writte i terms of these polyomials ad i oe of two alterative expressios, depedig o the values of the parameters, as follows: 3 ( P E ρ 4 Γ ( ) ω ρ f ) E Γ ( 3) 3 E ρ P ( ω ρ4),sih, < ω (30) ( ),sih, > ω where ρ ω The discrete eergy spectrum is easily obtaied by diagoalizatio of the Hamiltoia i (36) resultig i the followig requiremets ω, ad E, (3) which gives E ω ( ), (3) where ad 0,,, Thus, the boud states wavefuctios are ω ψ () r A ( ωr) e r L ( ω r ) (33) Now, we cosider the secod possibility (35b) i which the parameter, aside from beig >, is arbitrary It is to be oted that i this case the first term i the potetial, x, is essetial to cacel the cotributio of the term 4 x i Eq (34) which destroys the tridiagoal structure Therefore, the basis scale parameter should be idetified with the oscillator frequecy, whereas, the arbitrary basis parameter is the Laguerre polyomial idex Followig the same procedure as outlied above we obtai the followig matrix represetatio of the wave operator ½ m H E ( )( τ B ) ( ) ( ) δ 4 m, (34) τ ( ) δ τ ( )( ) δ, m, m, where τ E Similarly, oe could easily show that the solutio of the recursio relatio () resultig from the matrix wave equatio is writte i terms of the cotiuous dual Hah orthogoal polyomial ad its modified versio as follows: S ( z;, τ ), B ( ½) Γ < f( E) Γ ( ) (35) S ( z;, τ ), B> ( ½) where z ( ½) B ad S µ ( ;, ) x a b is defied by Eq (9) Additioally, the discrete eergy spectrum is obtaied from (34) by the requiremet that τ 0, (36) ½ B 0 4 Givig:, ( ½) B, (37) E 7

8 which is the same eergy spectrum as that i Eq (3) whe B 0 IV POWER-LAW POTENTIALS AT ZERO ENERGY The cofiguratio space coordiate for this problem is x ( r), where > 0 ad the real parameter 0,, The three dismissed values of correspod to the Morse, Coulomb, ad Oscillator problems, respectively [3] A elemet of the basis fuctios could be take as follows α () x r Ax e L() x (4) (0) ( ) 0 for positive ad egative values of, which is fixed oce ad for all The itegratio measure i terms of x is x dx 0 x dx sice for ± > 0 we get dr 0 ± Therefore, the ormalizatio costat is A Γ ( ) Γ ( ) Usig the derivative chai rule, which gives x, we ca write dr dx ( H E) ½ x ( ) ( α ) ( α ) x x( α ) x ( V E ) 4 A x α d A d (4) The costat E term must idepedetly vaish (for 0,, ) if the represetatio of the wave operator is to be tridiagoal Cosequetly, this case is aalytically solvable oly for zero eergy It is to be oted that this coditio does ot dimiish the sigificace of these solutios Zero eergy solutios have valuable applicatios i scatterig calculatios (eg, effective rag ad scatterig legth parameters [4]) ad i the ivestigatio of low eergy limit cases For this problem we also ed up with two possibilities for obtaiig the tridiagoal structure of the Hamiltoia: () α, α ( ), > 0 V Ar Br (43a), < 0 r V A Br ( r) r, ad () α (43b) where A ad B are real potetial parameters The last term of the potetial i the secod case (43b) is ecessary to elimiate the o-tridiagoal compoet comig from the cotributio of the term 4 x i Eq (4), ad The resultig represetatio of the Hamiltoia for the first case (43a) is B m H ( 4 )( A ) δ, m (44) B ( 4 ) ( δ ) m, ( )( ) δ m, The recursio relatio obtaied from this represetatio has two solutios depedig o whether B is positive or egative For B < 0 ad ± > 0, we get: 8

9 Γ ( ) Γ ( ± ) A ρ ( B ρ ) ½ ± P f (),cos, (45) where ρ B O the other had, for B > 0 ad ± > 0 the solutio is writte i terms of the Hyperbolic Pollaczek polyomial as follows: ½ ± P A ρ (,sih ), ρ > Γ ( ) B ρ f() Γ ( ± ) (46) ½ ± ( ) P A ρ (,sih ), ρ < B ρ The diagoal represetatio of the Hamiltoia is obtaied from Eq (44) by the requiremet that the potetial parameters assume the followig values: B ( ), ad A ( ) ( ± ) for ± > 0 (47) Thus, the discrete spectrum of the Hamiltoia occurs for positive values of the potetial parameter B ad for egative, -depedet, ad discrete values of A with A The eigefuctios of the Hamiltoia that correspod to this discrete represetatio ad for ± > 0 are as follows: ½ ± ( ½) r ± ( ) ψ () r A( r) e L ( r ), (48) This diagoal represetatio has already bee obtaied by this author [3] ad others [5] The secod possibility defied i (43b) produces the followig tridiagoal matrix represetatio for H ½ m H ( )( τ) A m, δ (49) τ ( ) δ τ ( )( ) δ m, m, where τ B ad > but, otherwise, arbitrary The associated recursio relatio is solved for the expasio coefficiets of the wavefuctio i terms of the cotiuous dual Hah polyomial ad its modified versio as follows: S ( z;, τ ), A ( ½) Γ < f() Γ ( ) (40) S ( z;, τ ), A> ( ½) where z ( ½) A potetial ad basis parameters to satisfy B, ( ) ( ) A ½, The diagoal represetatio is obtaied by restrictig the (4) which is the same results as that i (47) above with A 0 ad B A as it is also evidet by comparig the potetial i (43a) with that i (43b) V THE ONE-DIMENSIONAL MORSE OSCILLATOR I this example, we oly list the results without givig details of the calculatio: x µ e y, where µ, > 0 ad y R (the real lie) (5) 9

10 α () r Ax e L() x, A Γ ( ) Γ ( ) (5) Case () α : r r V ( Ae Be ), E ( α) (53) m H E ( B )( 4 ) A µ δ, m µ B ( 4 ) ( δ ) m, ( )( ) δ µ m, (54) where E ad E < 0 Γ ( ) Γ ( ) The solutio for B < 0 is A ( B ρ ) f ( E ) P,cos ρ, (55) where ρ B µ However, if B is positive the we obtai the followig A ρ ( B ρ ) A P ρ ( B ρ ) P,sih, µ < B Γ ( ) f( E) Γ ( ) ( ),sih, µ > B The discrete eergy spectrum requiremet puts A ( ) µ B µ ad gives (56) E (57) Therefore, the correspodig boud states wavefuctios are as follows A y A y µ µ y ψ y Ae exp Be L Be (58) Case () α ( ) : µ r r V Ae e (59) m H E ( )( A ) E µ ( ) δm, (50) A ( ) m, A µ δ µ ( )( ) δm, ( ) ( ) Γ ( ) ( ;, A µ ) f E S z, (5) Γ ( ) where z E The solutio of this case coicides with the fidigs i Refs [6] Moreover, we also obtai by diagoalizatio of (50) the followig discrete eergy spectrum A ( ) µ E, (5) which is idetical to (57) above VI THE S-WAVE HULTHÉN PROBLEM The Hulthé potetial [7], which is writte as () r r V r Z e ( e ) with > 0, is used as a model for a screeed Coulomb potetial, where is the screeig r parameter This is so, because for small we ca write the potetial as V() r Z r e The cofiguratio space coordiate which is compatible with these kid of problems is 0

11 x e r, r 0, This problem belogs to the situatio described by Eq (7) with x ± ± Sice F(, b; c; z) is proportioal to the Jacobi polyomial ( c, b c ) P ( z) [9], the the L basis fuctios that satisfy the boudary coditios for this case could be writte as α β ( µ, ) () r A( x)( x) P () x, (6) where α, β > 0, µ, > ad the ormalizatio costat is It maps real space ito a bouded oe That is, x [ ] for [ ] ( µ ) Γ ( ) Γ ( µ ) A µ Γ ( µ ) Γ ( ) (6) Usig the differetial formulas of the Jacobi polyomials [Eqs (A8) ad (A9) i the Appedix], ad d ( x) d, we ca write dr dx µ µ β α µ { ( x ) x µ µ β α A µ A } d x ( µ ) α(β ) dr x x x (63) β x α α x x x x Notig that dr 0 dx ad usig the orthogoality relatio for the Jacobi x polyomials [Eq (A0) i the Appedix], we arrive at the followig coclusios First, this problem admits oly S-wave ( 0) exact solutios sice the orbital term creates itractable o-tridiagoal represetatios Secod, the tridiagoal requiremet o the actio of the wave operator limits the possibilities to the followig three: () β µ, ad α ( ) (64a) () β µ, ad α (64b) (3) β ( µ ), ad α ( ) (64c) The first possibility elimiates the term from Eq (63), whereas the last two allow this term to cotribute to the matrix elemets above ad below the diagoal The calculatio i the first possibility (64a) gives the followig actio of the S-wave Schrödiger operator o the basis: x ( H E ) ( µ ) ( µ )( ) x (65) µ x x x ( V E ) 4 x 4 x x Therefore, to obtai the tridiagoal represetatio, our choice of potetial fuctios is limited to those that satisfy the followig costrait: x µ x x A B ( V E) ( ± x), (66) x 4 x 4 x where A ad B are real potetial parameters The first two terms o the right had side of the equatio are ecessary to cacel the cotributio of the correspodig terms i Eq (65) that destroy the tridiagoal structure Equatio (66) results i the followig E µ, (67) C A V Be r ( e ) r r e r e (68)

12 C where ad E < 0 The two alteratives i the last term of the potetial 4 correspod to the ± sig i Eq (66) From ow o, we will adopt the sig makig the last potetial term i (68) pure expoetial, Be r However, the other choice could easily be obtaied from this oe by the parameter map: B B, A A B The first two terms i V(r) are the Hulthé potetial ad its square After some maipulatios, we obtai the followig matrix represetatio of the wave operator ( B µ µ m H E µ ) ( µ )( µ ) B ( )( ) A µ µ µ δm, δ µ ( µ )( µ ) m, (69) B ( )( µ )( )( µ ) δ µ µ m,, µ ( )( 3) where µ µ ( E) as give by Eq (67) The resultig three-term recursio relatio () could be writte i terms of polyomials defied by µ Γ ( µ ) Γ ( ) Γ ( ) Γ ( µ ) P( E) f ( E), (60) i which case it reads µ ( µ ) ( µ )( ) zp P ( )( ) µ µ ( µ )( ) ( )( µ ) P µ µ µ µ ( )( ) ( )( ) P (6) where B ad ze ( C A E) B We are ot aware of ay kow threeparameter orthogoal polyomials that satisfy the above recursio relatio However, comparig it to the recursio (A6) i the Appedix suggests that these polyomials could be cosidered as deformatios of the Jacobi polyomials with beig the deformatio parameter ( 0 correspods to the Jacobi polyomial) Pursuig the aalysis of these polyomials would be too mathematical ad iappropriate for the preset settig Noetheless, we fid it pressig to make the followig remark For large values of the idex the term i the recursio (6) goes like whereas the rest of 0 the terms go like Therefore, to obtai reasoable ad meaigful umerical results, should be take very small (ie, B ) Now back to the matrix represetatio (69) of the wave operator The discrete eergy spectrum is easily obtaied by diagoalizig this represetatio which requires that B 0 ad ( µ ) ( µ )( ) A 0, (6) givig the followig discrete spectrum for 0,,, ( A C) E µ ( ) 8 (63) The correspodig boud states could be writte as follows µ r r ( µ, ) r ψ () r Ae ( e ) P ( e ), (64) where µ E ad 8C These fidigs, for the special case where B 0 i the potetial fuctio (68), agree with the results obtaied i Refs [8] Repeatig the same aalysis for the secod possibility (64b) ad ivestigatig the tridiagoal structure of the resultig actio of the wave operator o the basis we coclude

13 the followig First, the parameter µ is related to the eergy by Eq (67) the same way as i the first case Secod, the potetial is required to take the followig fuctioal form r x x A Be A B B V A B (65) r r r r x ( x) e ( e ) e ( e ) The matrix represetatio of the S-wave Schrödiger operator is obtaied as m H E ( µ ) ( µ ) ( µ )( ) µ { z ( ) } µ µ µ µ µ µ ( ) δ m, µ ( µ )( µ ) ( )( µ )( )( µ ) µ ( µ )( µ 3), where z B µ δ m, δm (66) 4 ad ( E A) The resultig recursio relatio i terms of the polyomials defied by Eq (60) above reads as follows ( µ ) ( µ ) ( µ )( ) µ zp { ( ) } P µ µ µ µ ( ) ( µ )( ) P ( µ )( µ ) µ ( ) ( )( µ ) P ( µ )( µ ) (67) Fig : A graph of the desity (weight) fuctio ρ ( z) associated with the orthogoal polyomials satisfyig the three-term recursio relatio (67) The Dispersio Correctio method developed i Ref [9] was used to geerate this plot usig the recursio coefficiets { a, } 50 b i Eq (66) with µ 0 0, 5 ad for several values of the parameter as show o the traces The orthogoal polyomials defied by this recursio relatio are, to the best of our kowledge, ot see before The mathematical aalysis of this recursio relatio ad the correspodig polyomials will ot be carried out here However, we fid it appropriate 3

14 ad physically sufficiet to give a fairly accurate graphical represetatio of the desity (weight) fuctio ρ ( z) associated with the orthogoality of these polyomials We use oe of three umerical methods developed i Ref [9] to obtai a good approximatio of the desity fuctio associated with a fiite tridiagoal Hamiltoia matrix Figure () shows ρ ( z) for a give value of µ ad ad for several choices of the parameter The discrete eergy spectrum is obtaied (by diagoalizatio) from Eq (66) as A E 8 where D ad D is defied by D B 4 where, 0,,, (68) B > 8 We leave it to the iterested reader to fid the recursio relatio for the third possibility (64c) ad to verify the followig results: C A V (69) r r ( e ) e where C ad A is a real potetial parameter Moreover, m 4 H E ( ) ( µ ) ( µ )( µ ) µ { z ( ) } µ µ µ µ µ µ ( ) δ m, µ ( µ )( µ ) ( )( µ )( )( µ ) µ µ ( µ ( ) δ )( µ 3) m, Aside from some sig chages ad the exchage µ, this is the same as Eq (66) above However, here we have E µ z ( ) eergy spectrum is obtaied as E µ ( ) ad ( A C) 8 δ m, (60) ( E A C) The discrete (6) VII ROSEN-MORSE TYPE POTENTIALS The basis fuctios for the oe-dimesioal problem associated with this potetial could be writte as follows α β ( µ, ) ( x) A( x) ( x) P ( x), (7) where x tah( y) ad y R The parameters α, β, µ,, are real with α, β ad positive The ormalizatio costat A is give by Eq (6) Aalytic solutios of this problem are obtaiable for three cases where the parameters are related as: ( α, β ) (, µ ), (, µ ), or (, µ ) As a example, we cosider oly the first case where the potetial fuctio assumes the followig form which is compatible with the tridiagoal represetatio 4

15 A tah( y) V Ctah( y) B cosh( y) ± cosh( y), (7) where C ( µ ) ( ) We cosider, i what follows, the potetial with the top sig i (7) The tridiagoal represetatio of the wave operator becomes A B µ 4B µ µ m H E 4 δ ( )( ) m, µ µ 4B ( µ )( )( µ ) δ µ ( µ )( µ ) m, (73) 4B ( )( µ )( )( µ ) δ µ ( µ )( µ 3) m, The resultig recursio relatio is similar to (6) The discrete eergy spectrum is obtaiable oly for B 0 which correspods to the hyperbolic Rose-Morse potetial [0] I this case we obtai: E ( µ ) ( D C ) ( ) ( D ) (74) where D is defied by D A 4 ad A < 8 The correspodig boud states wavefuctios are (, ) ( tah ) µ ( tah ) µ y A y y P ψ (tah y), (75) where µ ( E C) ad E C Fially, we ote that the examples preseted i this work do ot exhaust all possible potetials i this larger class of aalytically solvable systems Moreover, it might be possible that this approach could be exteded to the study of quasi exactly ad coditioally exactly solvable problems I additio, the relativistic extesio of this developmet is also possible I fact, the Dirac-Coulomb ad Dirac-Morse problems have already bee worked out [] APPENDIX The followig are useful formulas ad relatios satisfied by the orthogoal polyomials that are relevat to the developmet carried out i this work They are foud i most books o orthogoal polyomials [9] We list them here for ease of referece () The Laguerre polyomials L ( x), where > : xl ( ) L ( ) L ( ) L (A) Γ ( ) L x ( ) F x Γ Γ (A) d d x ( x) L ( x) 0 dx dx (A3) d x L L ( ) L dx (A4) 0 x Γ ( ) Γ ( ) xe L xl xdx (A5) m δ m 5

16 µ, () The Jacobi polyomials P ( x), where µ >, > : (, ) (, ) µ µ ± x µ µ µ ( ± ) µ P P µ µ ( µ )( ) ( µ, ) ( )( µ ) ( µ, ) ± P ( ) P µ µ ± ( µ )( µ ) (A6) ( µ, ) Γ ( µ ) x (, µ ) P ( x) F(, µ ; µ ; ) ( ) P x Γ Γ µ (A7) d d ( µ, ) ( x ) ( µ ) x µ ( µ ) P ( x) 0 dx dx (A8) d ( µ, ) µ ( µ, ) ( µ )( ) ( µ, ) ( x ) P ( x ) P P dx µ µ (A9) µ ( µ, ) ( µ, ) µ Γ ( µ ) Γ ( ) ( x) ( x) P ( x) Pm ( x) dx δ µ Γ ( ) Γ ( µ ) m (A0) µ (3) The Pollaczek polyomials P ( x, θ ), where µ > 0 ad 0 < θ < π : µ µ µ ( [ µ )cosθ xsi θ ] P ( µ ) P ( ) P 0 (A) ( ) i i P µ ( x, θ ) Γ e F(, µ ix; µ ; e ) µ (A) Γ ( ) Γ Γ ( ) Γ ( ) ρ ( x, θ) P ( x, θ) Pm ( x, θ) dx δm (A3) µ µ µ µ µ µ ( θ π) x where ρ ( x, θ) (si θ ) e Γ ( µ ix) π µ (4) The cotiuous dual Hah polyomials S ( x; a, b), where x > 0 ad µ, a, b are positive except for a pair of complex cojugates with positive real parts: µ µ x S ( µ a)( µ b) ( a b ) µ S (A4) µ µ ( a b ) S ( µ a)( µ bs ) µ, µ ix, µ ix ( ;, ) 3 µ a, µ b S x a b F (A5) 0 Γ ( ) Γ ( a b) Γ ( µ a) Γ ( µ b) µ µ µ ρ ( xs ) ( xabs ;, ) m( xabdx ;, ) δm (A6) Γ ( ix) Γ ( a ix) Γ ( b ix) where ρ ( x) π Γ ( µ a) Γ ( µ b) Γ( ix) µ µ 6

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