Extending the class of solvable potentials II. Screened Coulomb potential with a barrier

Transcription

1 Extedig the class of solvable potetials II. Screeed Coulomb potetial with a barrier A. D. Alhaidari a,b,c,* a Saudi Ceter for Theoretical Physics, Dhahra, Saudi Arabia b KTeCS, P.O. Box 374, Jeddah 438, Saudi Arabia c Physics Departmet, Kig Fahd Uiversity of Petroleum & Mierals, Dhahra 36, Saudi Arabia This is the secod article i a series where we succeed i elargig the class of solvable problems i oe ad three dimesios. We do that by workig i a complete square itegrable basis that carries a tridiagoal matrix represetatio of the wave operator. Cosequetly, the wave equatio becomes equivalet to a three-term recursio relatio for the expasio coefficiets of the wavefuctio i the basis. Fidig solutios of the recursio relatio is equivalet to solvig the origial problem. This method gives a larger class of solvable potetials. The usual diagoal represetatio costrait results i a reductio to the covetioal class of solvable potetials. However, the tridiagoal requiremet allows oly very few ad special potetials to be added to the solvability class. I the preset work, we obtai S-wave solutios for a three-parameter /r sigular but short-rage potetial with a o-orbital barrier ad study its eergy spectrum. We argue that it could be used as a more appropriate model for the screeed Coulomb iteractio of a electro with exteded molecules. We give also its resoace structure for o-zero agular mometum. Additioally, we plot the phase shift for a electro scatterig off a molecule modeled by a set of values of the potetial parameters. PACS umbers: 3.65.Ge, 3.65.Fd, 34.8.Bm, 3.65.Ca Keywords: screeed Coulomb, parameter spectrum, electro-molecule iteractio, tridiagoal represetatios, /r sigular potetial, resoaces, scatterig, recursio relatio. I. INTRODUCTION Very few problems i quatum mechaics (relativistic ad orelativistic) are exactly solvable. Despite the limited umber of these problems, there are may advatages to obtaiig exact solutios (eergy spectrum ad eigefuctios) of their associated wave equatios usig as may methods as possible. Oe such advatage is that the aalysis of such solutios makes the coceptual uderstadig of quatum 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 models of realistic problems or approximatios thereof. Additioally, potetials associated with these solvable problems could be used i the uperturbed part of more realistic Hamiltoias. Cosequetly, all attempts at elargig the class of potetials for which a exact solutio is obtaiable are importat ad very fruitful. Aside from the three well-kow classes of solvability (exact, coditioally-exact [], ad quasi-exact []), we defie here the otio of exact solvability to be the ability to write the solutio of the wave equatio i a closed form as a coverget series i terms of quatities that are well-defied to all orders. Moreover, all physical quatities i the * haidari@mailaps.org, Fax:

2 problem (e.g., the eergy spectrum, phase shift, wavefuctio, resoaces, etc.) are obtaiable to ay desired accuracy limited oly by the computig machie precisio; o physical approximatios are ivolved. A subclass of this otio of exact solvability is the aalytic solvability i which all objects i the resultig expressio for ay physical quatity are mathematically well-defied i terms of the idepedet variables (e.g., positio, time, agular mometum, eergy, potetial parameters, etc.). We also reitroduce the cocept of a parameter spectrum where a exact solutio of the problem is obtaied at a sigle eergy but for a set (fiite or ifiite) of values of the potetial parameters (the parameter spectrum) [3]. I a recet publicatio (heceforth referred to as paper I) [4], we used the tools of our tridiagoal physics program [5], which was ispired by the J-matrix method [6], to obtai a exact L series solutio for the ifiite potetial well with a siusoidal bottom. No aalytic expressio was obtaied for the eergy spectrum formula. However, the accuracy of the calculated values of the eergy spectrum is limited oly by the computig machie precisio; o physical approximatios were ever ivolved. The wavefuctio was writte as a ifiite coverget series whose terms are the product of the trigoometric fuctio i cofiguratio space ad the dipole polyomial i the eergy [7]. I this work, we employ the same techique i paper I to solve a highly sigificat problem i three dimesios with the followig three-parameter cetral potetial λr e γ V() r = V e λr, (.) where V is the potetial stregth ad the rage parameter λ is positive with a iverse legth uits. The dimesioless parameter γ is i the ope rage < γ <. This is a short rage potetial with /r sigularity at the origi. Figure is a graphical represetatio of this potetial fuctio for V > (Fig. a) ad V < (Fig. b). The potetial trace crosses the radial axis at r = l( γ ) λ the reaches a local extremum value of V( r ) = V ( γ ) at r l ( γ ) =. It is iterestig to ote that at short distaces, λ there is a clear resemblace betwee this potetial (with V > ) ad the attractive Coulomb potetial for o-zero agular mometum. However, the potetial valley i Fig. a is ot due to the cetrifuge kiematics attributed to o-zero agular mometum. Moreover, the log-rage behavior is ot the same. Thus, i cotrast to the log-rage Coulomb potetial, we expect that the umber of boud states for this potetial to be fiite. I Fig. c, we show the effective potetial, which is the sum of V(r) for V < ad the orbital term with o-zero agular mometum. For certai rage of values of the potetial parameters ad agular mometum, we obtai the cofiguratio show i the figure with local maximum ad miimum. Additioally, with V < (see, Fig. b) this potetial exhibits a rather differet type of charge screeig. Near the origi, the electro experieces a strog attractio to a effective ucleus with effective charge Z = 4 πε V( γ) λe, which is ot ecessarily a iteger but is less tha Z (the sum of all proto charges i the uclei of the molecule). As the electro gets farther away from the origi, the screeig due to the electro cloud aroud the effective ucleus icreases ad becomes sigificat util a balace is reached at r, which is less tha the charge radius of the molecule. Beyod that, a local excess/deficiecy of egative/positive charges cotributed by electros/uclei from the outer/ier atoms i the exteded molecule repels the scattered electro util it gets far eough from the ceter of the molecule where the eff

3 Coulomb iteractio dimiishes rapidly. As such, this potetial could be used as a more appropriate model for the iteractio of a electro with exteded molecules whose electro cloud is cogregated ear the ceter of the molecule (for example, due to a large cetered atom, see Fig. ). A measure of the molecule extesio could be give by the parameter σ = ( Zeff Z ). The usual Coulomb screeig iteractio potetials (e.g., the Hulthé [8] ad Yukawa [9] potetials) do ot have the barrier structure show i Fig. b ad they behave close to the origi like Z r e αr. I fact, the potetial (.) exhibits such behavior i the limit γ =, i which case Z = 4π ε V λ e ad α = λ. Therefore, the parameter γ has the physical iterpretatio as beig the measure of molecular extesio whe the potetial (.) is take as a model for the iteractio of a electro with Z Z = γ. exteded molecules. This is because ( eff ) Choosig γ < or γ > i the potetial (.) takes it outside the class of problems uder curret study. However, our approach ca still easily hadle the boud states solutio for these cases provided that γ V >. I fact, these two cases are simpler ad carry less complicated structure as opposed to the case where < γ <. For all real γ the solutio space splits ito three discoected subspaces with differet physical iterpretatios. We limit our ivestigatio here to the middle ope rage < γ < that has a richer structure. For the boudary case where γ =, the potetial (.) becomes the simple expoetial potetial, Ve λr. However, γ = results i a strogly decayig λr λr Hulthé potetial, Ve ( e ) (i.e., heavily screeed Coulomb) as oted above. The solutio obtaied i this work is valid oly for S-wave ( = ). Nevertheless, by usig the complex rotatio (scalig) method [], we obtai i Sec. 5 a highly accurate evaluatio of the resoace structure associated with this potetial for a give physical cofiguratio ad for o-zero agular mometum. I the same sectio, we utilize the J- matrix method to calculate the phase shift for the scatterig of a electro from a molecule modeled by a give set of values of the potetial parameters. But first, we start i the followig sectio by implemetig our approach i paper I to obtai a exact S-wave solutio for the 3D problem with the potetial (.). The reader is advised to cosult Sec. I ad Sec. II i paper I [4] for details o the theoretical formulatio, backgroud, ad motivatio of the approach. Sectio III i paper I gives a illustratio o how to implemet this approach to obtai a exact solutio for the ifiite potetial well with siusoidal bottom. A brief summary of the mai fidigs i this paper was published recetly i a letter []. II. SOLUTION IN THE TRIDIAGONAL REPRESENTATION The time-idepedet S-wave ( = ) Schrödiger equatio for a dyamical system modeled by a poit particle of mass m i the field of a spherically symmetric potetial V(r) is d V() r E ψ (, r E) =, (.) mdr 3

4 where E is the particle s eergy ad ψ ( re, ) is the wavefuctio that describes the state. Now, we make a trasformatio, y = y( λr), to a referece cofiguratio space with coordiate y [, ], where λ is a positive scale parameter havig the dimesio of iverse legth. This trasformatio takes the wave equatio (.) ito d d ( y ) y U( y) ψ ( y, E) =, (.) dy dy m where U( y( λr) ) = [ E V( r) ] ad the prime stads for the derivative with respect to λ λ r. As show i paper I, our choice of a complete square itegrable basis that is compatible with this problem ad carries a faithful descriptio of the wavefuctio, ψ ( ye, ), has the followig elemets α β (, ν) φ ( y) = A( y) ( y) P ( y), (.3) ν, where P( ) ( y) is the Jacobi polyomial of degree =,,,.. []. The parameters ad ν are larger tha whereas the values of α ad β deped o the boudary coditios ad square itegrability. For boud states, they are real ad mostly positive; however, for scatterig states they may assume complex values. The ormalizatio costat is chose ν Γ ( ) Γ ( ν ) as A = ν. For a give physical system modeled by the potetial Γ ( ν ) Γ ( ) fuctio i U(y), a crucial costrait o the choice of coordiate trasformatio, y= y( λr), is that the basis (.3) should result i a tridiagoal matrix represetatio for the wave operator J(y) i (.), where d d dy J = ( y ) y U ( y ). That is, the matrix elemets Jm = φ J φm must vaish for all m. Stated differetly, for a give coordiate trasformatio y= y( λr), the tridiagoal requiremet limits the umber of solvable potetial fuctios i U(y) to a very special set. We expad the wavefuctio i the basis (.3) as ψ( ye, ) f( E) φ ( y) f are the (Fourier) expasio = dy =, where { } coefficiets. Therefore, a complete solutio of the problem is obtaied if all { f( E) } = are determied. As explaied i paper I, it might be sufficiet from the physical poit of view (without goig ito rigorous mathematical aalysis) to see that completeess ad square itegrability guaratee boudedess ad covergece of this series. This will also be demostrated umerically below. Now, if the tridiagoal costrait is satisfied the the wave equatio (.) becomes equivalet to the followig three-term recursio relatio for the expasio coefficiets J f J f J f =. (.4),,, With a proper choice of ormalizatio, the solutios of this recursio relatio are polyomials i some give physical parameter (e.g., the eergy or potetial parameter) [3]. For most cases i the class of solvable potetials obtaied by our approach, the orthogoal polyomials associated with the resultig recursio relatio do ot belog to ay of the kow classic polyomials (e.g., the Hermite, Chebyshev, Laguerre, etc.) [4]. That is, their associated weight fuctios, geeratig fuctios, zeros, etc. are ot foud elsewhere. Noetheless, they are completely defied for all degrees by their recursio relatios ad iitial seed values. As stated i the Itroductio, the theoretical formulatio, backgroud, ad motivatio for our tridiagoal physics approach could be foud i paper I ad i several other publicatios cited therei. 4

5 Next, we will show that the coordiate trasformatio y = e λr will meet the tridiagoal requiremet for a suitable choice of potetial fuctio U(y) ad basis parameters. The itegratio measure is dy... dr =... λ ad the wave equatio (.) y becomes d d y ( y ) ( y) U( y) ψ ( y, E) =, (.5) dy dy y where y has bee factored away. Sice this factor vaishes at y =, the it might y ivalidate the wave equatio as r. Cosequetly, we must pay extra attetio to the solutio of (.5) at the boudaries, which will be doe below. The actio of the wave operator o the basis elemets (.3) is calculated with the help of the differetial equatio, differetial formula, ad recursio relatios of the Jacobi polyomials []. The result is as follows y ν y J ( ) ( ) ( y α ν β φ = α β ν ν)( y y ) (.6) y y y ( )( ν) β A α ( α ) y β y y U φ α ν ν ( y y ) A φ The recursio relatio ad orthogoality formula of the Jacobi polyomials show that the matrix represetatio for the wave operator φ J φ m becomes tridiagoal oly i three cases: y y y ν = α, = β : U = A B Cy D (.7a) y y y ν = α, = β : y y y D U = A C (.7b) y y y y ν = α, = β : y y y D U = B C (.7c) y y y y where A, B, C, ad D are real dimesioless parameters ad the basis parameters α ad β must satisfy α( α ) = A ad β = B. The secod ad third cases above correspod to the geeralized Hulthé potetial, which is the sum of the Hulthé potetial ad its square. This 3D S-wave problem is already kow ad its exact solutio has bee obtaied ad classified by may researchers [5]. Thus, we will ot cosider these two cases. O the other had, the first case results i the followig four-parameter potetial fuctio (i the uits = m = ) λr D A e V() r = C, (.8) λr λr λr λ e ( e ) e Ad gives the basis parameter assigmets: = 4B= 8E λ ad ν = 4A. Thus, possible real solutios of this problem are cofied to egative eergy (boud states) ad for A 4. Moreover, the basis (.3) becomes eergy depedet through the parameter ( E) ad the wave equatio (.5) becomes d d y y ( y ) ( y) A ε Cy D ψ( y, ε) =, (.5) dy dy y y where ε = E λ is the eergy parameter. Defiig the dimesioless ratio γ = D C for C, we ca rewrite the potetial fuctio (.8) as C 5

6 λr λ A γ e V() r = λ C λr λr ( e ) e. (.9) Usig the recursio relatio ad orthogoality formula of the Jacobi polyomials, we obtai the followig matrix elemets of the wave operator, J m = D A ( ν )( ν ) δm C y m, (.) where y m is the followig tridiagoal matrix (see, the Appedix i paper I) ν y m = δ m, ( ν)( ν ) ( )( ν)( ν) δ m, (.) ν ( ν )( ν ) ( )( )( ν )( ν ) δ m, ν ( ν )( ν 3) Therefore, the off-diagoal etries i the wave operator matrix (.) are due oly to the last term, which is proportioal to the potetial parameter C. Hece, the diagoal represetatio costrait requires that C = givig ( D A) 4 ε α =, where α( α ) = A. This is the well-kow eergy spectrum formula for the geeralized Hulthé potetial [5]. However, we are iterested i the ew compoet of the potetial (.9) that correspods to the special case where A = ad C : λr e γ V() r = λ C e λr. (.) Obviously, diagoalizig the Hamiltoia with this potetial i the basis (.3) will ot lead to a exact solutio because it requires C =. This is the reaso why we relax the diagoal costrait by workig i a more geeral tridiagoal represetatio that makes it possible to search for such a solutio, if it existed. Now, for all fiite values of the real parameter ratio γ, this potetial has /r sigularity at the origi ad a expoetially decayig tail (i.e., it is short-rage). Moreover, simple aalysis of this potetial fuctio shows that the most iterestig physical situatio occurs whe there is a local extremum (maximum or miimum) of the potetial. That is, dv dr = at some fiite radius. This situatio is possible oly if the value of γ falls withi the ope rage < γ <, which is equivalet to D < C. Thus, this potetial becomes idetical to the oe give i the Itroductio sectio by Eq. (.) with V = λ C ad, thus, has all of its iterestig features ad physical iterpretatio. Additioally ad as stated above, our approach ca still hadle the boud states solutio for the case γ or γ provided that γ V >. Now, takig A = gives ν = ad results i the followig basis elemet φ ( ) ( ) λ r λr (,) λr () r e e P ( ) e =, (.3a) where = ε. Alteratively, we ca write it i terms of the hypergeometric fuctio F ab, ( c ) z as follows ( ) ( ) ( ) λ r λr λr λ r λ, λ ( ) ( ) Γ ( ), Γ ( ) Γ ( ) φ () r = e e F e, or (.3b) r r φ () r = () ( )( )( ) e e F e. (.3c) α 6

7 It is obvious that this fuctio vaishes at the boudary (r = ad r ) for all eergies, which settles our cocer regardig the boudary coditios that was raised below the wave equatio (.5). Oe must remember, though, that a eigefuctio of the problem that correspods to a give (egative) eergy i the spectrum is a ifiite sum of these elemets that also depeds o the potetial parameters C ad γ via the expasio coefficiets { } f. III. POTENTIAL PARAMETER SPECTRUM VERSUS ENERGY SPECTRUM Because the basis (.3) is eergy depedet, our solutio strategy will differ from that i paper I. Here, we adopt the followig scheme: For a arbitrarily chose (egative) value of the eergy, we fid the set of values of the potetial parameters that leads to a exact solutio. Depedig o the eergy ad physical costraits, this set could be fiite or ifiite. We call this set, the potetial parameter spectrum or simply the parameter spectrum. The cocept of a parameter spectrum was itroduced for the first time i the solutio of the wave equatio i [3]. If the map that associates the parameter spectrum with the eergy is ivertible, the we could easily obtai the eergy spectrum for a give choice of potetial parameters. Now, for a fixed value of the eergy ε (equivaletly, a costat o-egative parameter = ε ), the tridiagoal matrix represetatio of the wave operator (.) i the basis (.3) becomes ( ) C Jm = (γ ) C ( )( ) δ ( )( 3) m, (3.) C ( )( )( ) C ( )( )( )( ) δ ( )( ) m, 3 δ ( )( 4) m, Hece, the resultig three-term recursio relatio for the expasio coefficiets of the wavefuctio becomes: ( γ ) f = ( C a d ) f b f b f, where (3.) a( ε) = ( )( ), d( ε) =, ( )( 3). (3.3) ( )( )( )( ) b ( ε ) = 3 ( )( 4) We write (3.) as the eigevalue equatio TC f = ( γ ) f, where T C is the tridiagoal C T = a d δ b δ b δ. Thus, for a give eergy m (equivaletly, a give ) ad potetial stregth C, this eigevalue equatio produces the γ- parameter spectrum. It is evidet that the eigevalue ( γ ), which for our class of problems is supposed to be limited to the ope rage [, ], grows rapidly with the size symmetric matrix ( C) ( ) m, m, m N of the matrix T C as ( γ ) at which D C N C due to the etry a i the diagoal term. The two limits ± correspod to the two critical values C ± of the potetial stregth parameter =. I fact, there is a ifiite umber of these critical values at each ± ± eergy. We write this set as { C ( ε )} where ± C ( ε ) are positive ad each represets the miimum value of the potetial stregth for which a th level boud state with eergy greater tha or equal to ε appears i the spectrum. To calculate these critical values for a give eergy, we proceed as follows. We start by rewritig (3.) as 7

8 ( ) ( γ ) C f = a d f b f bf, (3.4) If we defie the ew coefficiets g = a a f, the we ca write this recursio relatio i terms of them as C g = A g B g B g, =,,..., (3.5) where A = (γ d) a ad B = b aa. The iitial relatio ( = ) for this recursio is C g = A g B g. (3.6) The ew recursio coefficiets A ad B approach the limit of large as. Thus, usig (3.5) to calculate the C-parameter spectrum gives a more rapidly coverget result tha usig (3.) to calculate the γ-parameter spectrum. Figure 3 shows the lowest critical ± potetial stregth { ( ) } = C 4 ε for all boud states i a give eergy rage. We were able to = obtai these values, as solutios of Eq. (3.5) whe writte as the eigevalue equatio, Tγ g = C g with γ = or, to machie precisio with as low matrix size as. Out of these critical values, the most importat for the system are those at zero eergy (i.e., at the boudary of the eergy spectrum). We refer to this subset by the symbol { C ˆ }. I Table, we list some of these values displayed coveietly to a accuracy of decimal places. I fact, for each γ oe ca calculate these critical values of the potetial stregth at which a boud state gets created or destroyed. At these critical values, the state experieces a trasitio from boud to resoace or vice versa. Recetly, this pheomeo was also demostrated for the Yukawa potetial ad the trasitio process was displayed as video aimatio showig the trajectories of the eergy eigevalues i the complex eergy plae [6]. Table, gives a list of the lowest positive Cˆ ( γ ) for several choices of γ. A obvious relatio ad egative values i the set { } betwee the two sets of critical potetial stregth is: C ˆ () = C () ad C ˆ () = C (). It will be evidet from the followig aalysis of the eergy spectrum that for a give parameter γ, a potetial stregth C that lies i the rage Cˆ ( ) ˆ γ < C < C ( γ ) will result i boud states. Therefore, these critical values are very importat for boud states umber coutig. Now, sice λ is the oly dimesioful parameter i the problem, the it might be obvious that the oly possible depedece of the eergy spectrum o λ is via a overall factor of λ. Aother way to see that, is by writig the radial Schrödiger equatio (.) with the potetial (.) ad the rescalig the radial coordiate ad eergy as r r λ ad E λ E causig the parameter λ to disappear from the equatio. For this reaso, we work with the eergy variable ε = E λ rather tha E so that λ disappears ad we deal oly with two parameters, C ad γ, istead of three. To obtai the potetial stregth parameter spectrum for fixed values of γ ad for all eergies i a coveietly chose rage, we solve the recursio relatio (3.5) as the eigevalue equatio T g = C g. Figure 4 shows the result of this calculatio for a give choice of γ ad for eergy rages that are chose appropriately depedig o the sig of the resultig 5 sigificat digits o our laptop usig Mathcad computatioal software. γ 8

9 potetial stregth V. The figure is show with C o the horizotal axis ad ε o the vertical axis to make it more coveiet to visualize the eergy spectrum. Thus, a vertical lie that crosses the C-axis at a value, say, C itersects the curves at the eergy spectrum correspodig to the problem with potetial parameters C ad γ. The figure shows that for a give γ, the spacig of the eergy spectrum is larger for positive C tha for egative C (i.e., the eergy spectrum is deser for egative C but is more stretched for positive C). I fact, this should have already bee obvious from the physics of the potetial (.), as portrayed i Fig. a ad Fig. b. Moreover, reproducig Fig. 4 for differet values of γ leads to the followig coclusio about aother iterestig property of the eergy spectrum: For a give C, the eergy spectrum is larger for positive C whe γ < but larger for egative C whe γ >. For a fixed γ, we ca make a fit o each trace i the eergy versus C-parameter spectrum of Fig. 4 with a M th order cotiued fractio usig the ratioal fractio approximatio of Haymaker ad Schlessiger similar to that i the Padé method [7]. Cosequetly, the boud state eergy could be writte as a fuctio of C ad for that fixed γ. I other words, give the potetial parameters C ad γ, we ca use this cotiued fractio to obtai the eergy spectrum { ε( γ, C) }. Table 3 illustrates the computatioal stability ad covergece of the eergy spectrum with the order of the cotiued fractio. The Mathcad program codes used for this ad other calculatios i this work are available upo request from the author. Fially, it is worth otig that the same process above could be repeated to obtai the γ-parameter spectrum usig the recursio relatio (3.) for a fixed potetial stregth C ad a chose eergy rage. Figure 5 shows such results for ucostraied rage of γ. IV. BOUND STATES WAVEFUNCTION The solutio of the three-term recursio relatio (3.5) for a give eergy is defied modulo a overall o-sigular fuctio of the potetial parameters C ad γ. If we call this fuctio ω ( γ, C), the we ca write g ( C) = ω ( γ, C) Q ( C). Substitutig this i the recursio (3.5) with its iitial relatio (3.6) ad choosig the stadard ormalizatio, Q =, determies Q ( C) as polyomials of degree i C for all. For example, the first few are Q ( C) = (4.a) Q ( C) = C A (4.b) B ( ) ( )( ) ( ) = BB Q C C A C A B (4.c).... ( ) Q ( C) = B C Q ( C) Q ( C) A B (4.d) We show below that completeess of the basis ad ormalizatio of the wavefuctio give ω ( γ, C) = K ( γ), where K ( γ ) is the kerel operator associated with these 9

10 polyomials at the ifiite order limit. Now, the polyomial Q ( C) does ot belog to ay of the kow classes of orthogoal polyomials. However, if we defie the polyomial P ( γ ) = a a Q ( C), the we ca write f( ε ) = ω ( γ, C) P ( γ) ad P ( γ ) satisfies the recursio relatio (3.) with the iitial seed value P ( γ ) =. It is a polyomial i ( γ ) ad satisfies the same recursio relatio typical of those that we have foud recetly while attemptig to fid a exteded class of solutios to the geeralized Hulthé ad Rose-Morse problems [4] (see, Eq. (6.9) ad Eq. (7.3)). The theory of orthogoal polyomials gives a alterative meas for obtaiig the parameter spectrum [3]. It goes as follows: The C-parameter (or γ-parameter) spectrum is obtaied form the zeros of the polyomial Q ( C) ( or P ( γ )) at a give i the limit as, respectively. I fact, the calculated values of the lowest part of the spectrum coverge quickly for relatively low polyomial degrees (as low as = with digits accuracy). Figure 6 is a plot of the lowest boud state eergy eigefuctios [ormalized by ω ( γ, C) ] for a give set of potetial parameters C ad γ. A wavefuctio that correspods to a boud state with eergy ε = E λ is computed as ψ ( r, ε ) N m= ω ( γ, C) Pm ( γ) φm( r), for some large eough iteger N ad where C ad γ belog to the parameter spectrum associated with ε. Numerically, we fid that the sum coverges quickly but becomes ustable if the umber of terms, N, becomes too large exceedig a iteger that depeds o the potetial parameters ad eergy level. For the choice of parameters i Fig. 6, our umerical routie produced the stable plots show for ψ (, r ε) ω (, γ C) ; but as N is icreases beyod N = 5 it becomes ustable. Better umerical routies with higher degree of precisio might be developed such that istability occurs at larger values of N. Moreover, tryig to evaluate the wavefuctio at a eergy that does ot belog to the eergy spectrum will ever achieve stable results. It will oly produce rapidly icreasig oscillatios with large amplitudes. I fact, the sum of these oscillatios for large N leads to destructive iterferece that should result i zero et value for the wavefuctio. Now, the orm of the wavefuctio at a eergy eigevalue is evaluated as follows k k k m k r dr C P Pm k m, = y φ φm m y = ψ = λ ψ(, ε ) = [ ω ( γ, )] Θ ( γ) ( γ), (4.) where Θ m is the basis overlap matrix = [8]. Aside from Θ m, which is symmetric ad geerally ot separable i the idices as θ θ m, this sum is the usual kerel for orthogoal polyomials [3]. That is, for the polyomials p( z ) associated with orthogoal represetatio, this is usually writte as K (, z z N N ) = p () z p ( z ) =. Therefore, for orthogoal represetatios, where Θm δ m, the sum i (4.) becomes the ifiite limit of the usual kerel, K( z) = lim K ( z, z). However, for o-orthogoal represetatios, the kerel is evaluate as N N N N = Θ m, = m m K (, zz ) p () zp ( z ). Thus, Eq. (4.) gives ω ( γ, C) = K ( γ) ad we write the boud state wavefuctio as

11 ( ) λr λ r (,) K ( γ ) m m m m= λr ψ (, r ε ) = ( e ) e m Q ( C) P ( e ), (4.3) where = ε. V. ENERGY RESONANCE AND SCATTERING PHASE SHIFT The solutio of the 3D problem with the radial potetial (.) that we have obtaied usig our tridiagoal represetatio approach is valid oly for S-wave ( = ) boud sates. I this sectio ad for o-zero agular mometum, we obtai a highly accurate evaluatio of the boud state ad resoace eergies for this /r sigular potetial usig the complex rotatio method i the tridiagoal J-matrix basis [9]. Recetly, we have applied this method successfully i obtaiig the boud state ad resoace structure for the Yukawa potetial [6]. The details of implemetatio of the method o /r sigular short-rage potetials are foud i [9]. Figure 7 shows the spectrum of the potetial i the complex eergy plae for several choices of agular mometum usig a fiite dimesioal Hamiltoia represetatio. The spectrum cosists of: (i) Discrete poits o the egative real axis that correspod to the boud states; (ii) A clockwise-rotated lie of dese poits approximatig the rotated brach cut (discotiuity) of the fiite Gree s fuctio; ad (iii) Exposed resoaces show as well-separated poits i the lower half of the complex eergy plae located i the sector boud by the rotated cut lie ad the positive eergy axis. The locatio of poits correspodig to boud states ad resoaces remai stable agaist variatios i all ophysical computatioal parameters []. Table 4 lists the boud states ad resoace eergies for several physical cofiguratios (differet values of C, γ, ad ). O the other had, Table 5 compares the boud states eergy spectrum for = obtaied idepedetly by the fiite complex rotatio method ad the parameter spectrum method give i Table 3. Next, we obtai the phase shift for electro scatterig off a exteded molecule modeled by the potetial (.) usig the J-matrix method [6]. This method is a algebraic method of quatum scatterig that gives exact scatterig iformatio for a model potetial represeted by its matrix elemets i a fiite subset of a complete square itegrable basis. The basis is chose such that it supports a ifiite tridiagoal matrix represetatio of the referece Hamiltoia H, which is the part of the total Hamiltoia that is exactly solvable. Therefore, H is accouted for aalytically exactly ad i full whereas the cotributio of the potetial is approximated by its fiite matrix represetatio. Now, the potetial fuctio (.) is /r sigular, where lim V( r) = Zeff r. The its matrix elemets, r which are obtaied as itegrals over the L basis, may have large errors. To deal with this problem, we absorb the /r sigularity of the potetial ito H, which ca still be hadled exactly i the J-matrix method. Therefore, we rewrite the total Hamiltoia as H = H V, where H = H Z r ad V = V Z r. The V becomes regular eff eff everywhere resultig i accurate evaluatio of its matrix elemets. The details of this scheme is give i [9]. Figure 8 shows the scatterig phase shift for a chose physical cofiguratio. The figure idicates strog resoace activity aroud ε = 4.. Detailed ivestigatio cofirms the presece of a sharp resoace at ε = i.465.

12 VI. CONCLUSION I this article, which is the secod i the series, we have succeeded i elargig the class of solvable problems i 3D by addig to it the radial potetial (.), which is /r sigular ad short-raged. We achieved that by workig i a complete square itegrable basis that supports a tridiagoal matrix represetatio for the wave operator. This makes the wave equatio equivalet to a three-term recursio relatio for the expasio coefficiets of the wavefuctio. Cosequetly, fidig a solutio of the recursio relatio is equivalet to solvig the origial problem. This method gives a larger class of solvable potetials. The usual diagoal represetatio costrait results i a reductio to the covetioal class of solvable potetials. We foud that the radial potetial (.) meets the tridiagoal requiremet for S-wave problems. Moreover, usig the complex rotatio method i a fiite basis, we were also able to obtai a highly accurate evaluatio of the resoace ad boud states structure associated with this potetial for several o-zero agular mometa. The physical properties of the potetial ad the structure of its spectrum are highly o-trivial. We argued that it could be very useful as a more appropriate model for the iteractio of a electro with exteded molecules whose electro cloud is cogregated ear the ceter of the molecule. As a illustratio, we used the J-matrix method to calculate the phase shift for the scatterig of a electro from a molecule modeled by a give set of values of the potetial parameters. I this work, we also reitroduced the cocept of a parameter spectrum where a exact solutio of the problem is obtaied at a sigle eergy but for a ifiite set of values of the potetial parameters (the parameter spectrum). We foud that the map that associates the parameter spectrum with the eergy is ivertible, thus we were able to obtai the eergy spectrum for a give choice of potetial parameters. We also defied the otio of exact solvability to be the ability to write the wavefuctio i a closed form as a coverget series i terms of orthogoal polyomials. These polyomials, which are fuctios of the cofiguratio space ad of the eergy, are well-defied to all orders. Additioally, all physical quatities i the problem (e.g., the eergy spectrum, phase shift, wavefuctio, resoaces, etc.) are obtaied to ay desired accuracy limited oly by the computig machie precisio; o physical approximatios are ivoked. I the ear future, we will also report o a exact solutio for the oe-dimesioal sigle-wave potetial V( x) = V [ tah( λx) γ] cosh ( λx) with < γ <, which has o previously kow exact solutio. Fially, it is worth otig that the formulatio of this approach could easily be exteded to ocetral [] as well as relativistic problems. I such relativistic extesio, oe searches for a tridiagoal matrix represetatio of the Dirac operator i a suitable spior basis similar to what has bee doe i []. ACKNOWLEDGEMENTS This work is sposored by the Saudi Ceter for Theoretical Physics. Partial support by Kig Fahd Uiversity of Petroleum ad Mierals uder project SB-9 is highly appreciated.

13 REFERENCES [] A. de SouzaDutra, Phys. Rev. A 47, R435 (993); 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. 64, 7 (999). [] A. V. Turbier, Commu. Math. Phys. 8, 467 (988); M. A. Shifma, It. J. Mod. Phys. A 4, 335 (989); A. G. Ushveridze, Quasi-exactly Solvable Models i Quatum Mechaics (Istitute of Physics, Bristol, 994); V. V. Ulyaov ad O. B. Zaslavskii, Phys. Rep. 6, 79 (99). [3] A. D. Alhaidari, J. Phys. A 4, 635 (7). [4] A. D. Alhaidari ad H. Bahlouli, J. Math. Phys. 49, 8 (8). [5] E. El Aaoud, H. Bahlouli, ad A. D. Alhaidari, It. Rev. Phys., 7 (8); A. D. Alhaidari ad H. Bahlouli, Phys. Rev. Lett., 4 (8); A. D. Alhaidari, J. Phys. A 4, 4843 (7); ad refereces to our previous work therei. [6] 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.), The J-matrix method: developmets ad applicatios (Spriger-Verlag, Dordrecht, 8) [7] A. D. Alhaidari, A. Phys. 33, 79 (8). [8] L. Hulthé, Ark. Mat. Astro. Fys., 8A, 5 (94); 9B, (94); L. Hulthé ad M. Sugawara, i Ecyclopedia of Physics, edited by S. Flügge, Vol. 39 (Spriger, Berli, 957) [9] H. Yukawa, Proc. Phys. Math. Soc. Jp. 7, 48 (935) [] J. Aguilar ad J. M. Combes, Commu. Math. Phys., 69 (97); Y. K. Ho, Phys. Rep. 99, (983). [] A. D. Alhaidari ad H. Bahlouli, J. Phys. A 4, 6 (9) [] See, for example, W. Magus, F. Oberhettiger, ad R. P. Soi, Formulas ad Theorems for the Special Fuctios of Mathematical Physics (Spriger, New York, 966); T. S. Chihara, A Itroductio to Orthogoal Polyomials (Gordo ad Breach, New York, 978); G. Szegö, Orthogoal polyomials (Am. Math. Soc., Providece, RI, 997. [3] M. E. H. Ismail, Classical ad Quatum Orthogoal Polyomials i Oe Variable (Cambridge Uiversity Press, Cambridge, 5) [4] A. D. Alhaidari, A. Phys. 37, 5 (5). [5] C. S. Lai ad W. C. Li, Phys. Lett. A 78, 335 (98); B. Roy ad R. Roychoudhury, J. Phys. A, 35 (987); P. Matthys ad H. de Meyer, Phys. Rev. A 38, 68 (988); M. Demiralp, Appl. Math. Comp. 68, 38 (5); S. M. Ikhdair ad R. Sever, J. Math. Chem. 4, 46 (6); O. Bayrak ad I. Boztosu, Phys. Scr. 76, 9 (7); S. H. Patil ad K. D. Se, It. J. Qua. Chem. 7, 864 (7); A. K. Roy, A. F. Jalbout, J. Math. Chem. 44, 6 (8). [6] A. D. Alhaidari, H. Bahlouli, ad S. Abdelmoem, J. Phys. A 4, 3 (8). [7] R. W. Haymaker ad L. Schlessiger, The Padé Approximatio i Theoretical Physics, edited G. A. Baker ad J. L. Gammel (Academic Press, New York, 97). [8] The matrix elemets F( y) m for ay itegrable fuctio F(y) defied over the iterval y [, ] is give i the Appedix of [4] by Eq. (A9). Gauss quadrature could be used to give a highly accurate approximatio of its itegral formula. [9] M. S. Abdelmoem, I. Nasser, H. Bahlouli, U. Al-Khawaja, A. D. Alhaidari, Phys. Lett. A 373, 48 (9). [] A. D. Alhaidari, It. J. Mod. Phys. A, 657 (5). 3

14 [] A. D. Alhaidari, J. Phys. A 38, 349 (5). [] A. D. Alhaidari, A. Phys. 3, 44 (4); Phys. Lett. A 36, 58 (4) Table captios: Table : Lowest set of the critical potetial stregth at zero eergy, C ± (), to a accuracy of decimal places. Table : The smallest set of values of the critical potetial stregth C ˆ ( γ ) for several choices of γ. Table 3: Covergece of the values of the lowest part of the spectrum (for a give C ad γ) with the cotiued fractio fit order M. Table 4: Boud states ad resoace eergies associated with the potetial (.) for several values of C, γ, ad agular mometum. These values were obtaied by the complex scalig (rotatio) method. Table 5: Compares the boud states eergy spectrum for = obtaied idepedetly by the fiite complex scalig method ad the parameter spectrum method of Table 3. 4

15 Figure captios: Fig. : The potetial fuctio (.) i uits of V versus the radial coordiate i uits of λ : (a) for V > ad for several values of γ withi the rage < γ <, (b) for V < ad γ = /, ad i (c) we show the effective potetial, which is the sum of V(r) for V < ad the orbital term with o-zero agular mometum ( = 3 ad γ = /). Fig. : Electro iteractig with a exteded molecule whose electro cloud is cogregated ear the ceter of the molecule (for example, due to a large cetered atom). ± Fig. 3: The lowest critical potetial stregth parameters { C } 4 ε eergies i the rage ε. ( ) = = for boud states Fig. 4: The eergy spectrum as a fuctio of the potetial stregth parameter C (positive ad egative) for γ =.. Fig. 5: The eergy spectrum as a fuctio of ucostraied values of γ for C =. Note that the solvability coditio for values of γ outside the rage < γ <, which is γ V >, is satisfied. Fig. 6: A plot of the lowest four eergy eigefuctios ψ ( r, ε ) [ormalized by ω ( γ, C) ] versus the radial coordiate (i uits of λ ) for γ =.7, C = 7. Fig. 7: The potetial spectrum (boud states ad resoace eergies) i the complex ε- plae for γ =.5, C = 8, ad for several values of the agular mometum. Boud states (Resoaces) are show as boxed (circled) dots, while the strig of bare dots represets the rotated cut lie (discotiuity of the Gree s fuctio). Oe (two) boud state eergy (eergies) is (are) outside the rage of the figure with = ( = ) o the egative real lie. Fig. 8: P-wave sigle-chael scatterig phase shift for a electro off a exteded molecule modeled by the parameters Z = V λ = 7 ad Z ( ) eff = V γ λ = 4 ad λ =. a.u. (i.e., γ =.4 ad C = 7). The presece of sharp resoace aroud ε = 4. is very clear. 5

16 Table C () C () Table C ˆ (.) C ˆ (.4) C ˆ (.6) C ˆ (.8) V < V > Table 3 M = M = M = 5 M =

17 Table 4 γ =.3, C = 5 γ =.5, C = 8 γ =.7, C = i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i Table 5 Parameter Spectrum Complex Scalig

18 Fig. a Fig. b Fig. c 8

19 Fig. Fig. 3 9

20 Fig. 4 Fig. 5

21 Fig. 6 Fig. 7

22 Fig. 8