Orthogonal polynomials derived from the tridiagonal representation approach

Transcription

1 Orthogoal polyomials derived from the tridiagoal represetatio approach A. D. Alhaidari Saudi Ceter for Theoretical Physics, P.O. Box 374, Jeddah 438, Saudi Arabia Abstract: The tridiagoal represetatio approach is a algebraic method for solvig secod order differetial wave equatios. Usig this approach i the solutio of quatum mechaical problems, we ecouter two ew classes of orthogoal polyomials whose properties give the structure ad dyamics of the correspodig physical system. For a certai rage of parameters, oe of these polyomials has a mix of cotiuous ad discrete spectra makig it suitable for describig physical systems with both scatterig ad boud states. I this work, we defie these polyomials by their recursio relatios ad highlight some of their properties usig umerical meas. Due to the prime sigificace of these polyomials i physics, we hope that our short expose will ecourage experts i the field of orthogoal polyomials to study them ad derive their properties (weight fuctios, geeratig fuctios, asymptotics, orthogoality relatios, zeros, etc.) aalytically. PACS: 3.65.Pm, 3.65.Ge, 3.65.Fd, 3.65.Ca Keywords: tridiagoal represetatio, orthogoal polyomials, recursio relatio, asymptotics, eergy spectrum, phase shift. I. Itroductio The Tridiagoal Represetatio Approach (TRA) is a algebraic method for solvig the quatum mechaical wave equatio [-5] d V( x) E ( E, x) Mdx, () for a class of potetials V(x) that is larger tha the well-kow covetioal class (M is the effective mass of the model). I the TRA, the wavefuctio ( E, x) i cofiguratio space for a give eergy E is writte as a bouded sum over a complete set of square itegrable fuctios, ( x ). That is, we write ( E, x) f( E) ( x), where f ( ) E is a appropriate set of expasio coefficiets. The basis fuctios are chose such that the matrix represetatio of the wave operator is tridiagoal. This is possible oly for a fiite umber of special potetial fuctios. Noetheless, the class of exactly solvable potetials i this approach is larger tha the well-kow covetioal class. Recetly, we made a short review of this approach ad gave a partial list of these ew solvable potetials (see, for example, Table below) [6]. The tridiagoal requiremet maps the wave equatio () ito a three-term recursio relatio for the expasio coefficiets. Subsequetly, the recursio relatio is solved exactly i terms of orthogoal polyomials P ( ), where is some proper fuctio of the eergy ad f( E) Preset Address: 43 Ahmed Bi Bilal Street, Al-Muhamadiyah, Jeddah , Saudi Arabia

2 f ( ) ( ) E P. Completeess of the basis ad ormalizatio of the wavefuctio shows that f ( ) E is the positive weight fuctio associated with these eergy polyomials [7-9]. The asymptotics ( ) of these polyomials all have the same geeral form P ( ) A( )cos ( ) ( )log ( ), () where ad are real positive costats that deped o the particular polyomial. The studies i [7-] show that A( ) is the scatterig amplitude ad ( ) is the phase shift. Both deped o the eergy ad physical parameters of the correspodig system. Boud states, if they exist, occur at eergies E( m) that make the scatterig amplitude vaish, A( m). The size of this eergy spectrum is either fiite or ifiite. Therefore, all physical iformatio, structural ad dyamical, about the system are give by the properties of these polyomials. Therefore, derivig these properties aalytically is of prime importace. Such properties iclude the weight fuctio, geeratig fuctio, asymptotics, orthogoality relatio, zeros, differetial or differece relatios, etc. Moreover, with each polyomial that has a cotiuous spectrum there is a associated discrete versio that has a ifiite or fiite spectrum. These discrete polyomials costitute the expasio coefficiets for the pure boud states. I Ref. [6], we cosidered two classes of problems. Oe class is properly described i the Laguerre basis while the other is treated i the Jacobi basis. I the Laguerre class, the basis fuctios are writte as follows y ( x) Aye L( y), (3) where y y( x), L ( y) is the Laguerre polyomial of degree i y ad A ( ). The dimesioless real parameters,, are obtaied by the requiremets of square itegrability ad tridiagoalizatio. The dimesioless variable y is a fuctio of the dy a by physical cofiguratio space coordiate x such that yx (), ye dx ad b. The real dimesioless costats a ad b are fixed for each choice of coordiate trasformatio associated with a give problem. Moreover, the legth scale parameter has a iverse legth dimesio (see Appedix A i Ref. [6] for details). The two remaiig basis parameters ad are fixed by the physical parameters of the potetial. The secod order liear differetial equatio associated with all solvable problems i this class has the followig form i the atomic uits M d d A y aby A y ( y) ( y) dy dy y. (4) The correspodig potetial fuctios i this class all have the followig geeral structure [6] a by V( x) y e V Vy Vy, (5) where V, V, V is a set of potetial parameters that are related to the parameters A ad of the differetial equatio (4). The solvable problems i this class iclude the Coulomb problem, the 3D isotropic (or D harmoic) oscillator, ad the D Morse potetial. The eergy polyomials, P ( ), i this class are the two-parameter Meixer-Pollaczek polyomial or the three-parameter cotiuous dual Hah polyomial ad their discrete versios: the Meixer, Krawtchouk or dual Hah polyomials. However, these polyomials are well kow i the appropriate physics ad mathematics literature [3]. ( )

3 The same caot be said about the eergy polyomials i the Jacobi class to be preseted ext. I the Jacobi class, the basis fuctios are writte as follows (, ) ( x) A( y) ( y) P ( y), (6), where y y( x), P( ) ( y) is the Jacobi polyomial of degree i y ad the ormalizatio ( ) ( ) costat is chose as A. The dimesioless real ( ) ( ) parameters,,, are obtaied by the requiremets of square itegrability ad tridiagoalizatio. The dimesioless variable y is a fuctio of the physical dy a b cofiguratio space coordiate x such that yx ( ) ad ( y) ( y) dx. The real dimesioless costats a ad b are fixed for each choice of coordiate trasformatio associated with a give problem (see Appedix B i Ref. [6] for details). The basis parameters are fixed by the potetial parameters ad the tridiagoal requiremet o the matrix represetatio of the wave operator. The secod order liear differetial wave equatio associated with all solvable problems i this class has the followig form i the atomic uits M d d A A y a b y a b Ay ( y) ( y) dy. (7) dy y y The correspodig potetial fuctios i this class all have the followig geeral form [6] a b V V V( x) ( y) ( y) V Vy y y, (8) where V, V, V is the set of potetial parameters that are related to the parameters A, A, of the differetial equatio (7). The solvable problems i this class iclude the covetioal potetials (e.g., the Pöschl-Teller, Scarf, Eckart, Rose-Morse, etc.) as well as ew potetials or geeralizatio of the covetioal potetials. Table is a partial list of these potetials that are exactly solvable i the TRA but ot solvable usig the covetioal methods. The eergy polyomials associated with this class of problems are the target of this study. Noe of them is foud i the mathematics literature. We defie them by their three-term recursio relatios which eables us to obtai all of them aalytically (albeit ot i closed form) ad to ay desired degree startig with P ( ). The three-term recursio relatio is obtaied from the tridiagoal matrix represetatio of the wave equatio of the correspodig physical problem [6]. II. The first polyomial class (, ) The first polyomial, which we desigate as H ( z ;, ), is a four-parameter orthogoal polyomial associated with the potetial fuctio (8). It satisfies a three-term recursio relatio that could be writte i the followig form (, ) (, ) cos H ( z ;, ) z si H ( ;, ) ( )( ) z (9) ( )( ) (, ) ( )( ) (, ) H ( )( ) ( z ;, ) H ( )( ) ( z ;, ) 3

4 (, ) where,,..., ad H ( z ;, ). It is a polyomial of order i z ad i. I the limit z, this recursio relatio becomes that of the Jacobi, polyomial P( ) (cos ). The polyomial of the first kid satisfies this recursio relatio (, ) with H ( z ;, ) ad (, ) H ( z ;, ) cos z si 4, () (, ) which is obtaied from (9) with ad by defiig H ( z ;, ). The polyomial of the secod kid satisfies the same recursio relatio (9) with (, ) (, ) H ( z ;, ) but H ( z ;, ) c cz where the liearity coefficiets c ad/or c are differet from those i Eq. (). The orthoormal versio of (, ) (, ) (, ) H ( z ;, ) is defied as H ( z ;, ) H ( z ;, ) where!( ) ( ) ( ) ( a) ad ( a) ( )( )...( ) a a a a ( a) (see Appedix E i Ref. [6]). This polyomial has oly a cotiuous spectrum over the whole real z lie. This could be verified umerically by lookig at the distributio of its zeros for a very large order. Figure shows such distributio for a give set of values of the polyomial (, ) parameters. Whereas, Figure shows that the asymptotic behavior of H ( z ;, ) is siusoidal ad cosistet with the limit give by formula () for ad. Table gives the expressio for z ad the polyomial parameters i terms of the eergy ad potetial parameters of the correspodig geeralized potetials of Table. Cosequetly, if the properties of this polyomial were to be kow aalytically the we would have easily obtaied the physical features of the correspodig quatum mechaical system. For example, if all elemets i the asymptotics () were give explicitly, the we would have obtaied the phase shift ad eergy spectrum i a simple ad straightforward maer just by substitutig the parameter map of Table ito Eq. (). Additioally, the physics of the problems associated with this polyomial suggests that it should have two discrete versios. Oe with a ifiite discrete spectrum ad aother that has a fiite discrete spectrum [6]. These two polyomials are defied by their recursio relatios, which are obtaied by the replacemets i ad z iz k i Eq. (9), where k is a iteger of either fiite or ifiite rage. Doig so, results i the followig recursio relatio (, ) ( ) (, ) H ( k;, ) zk H ( ;, ) ( )( ) k () 4( )( ) (, ) 4( )( ) (, ) H ( k;, ) H ( k;, ) ( )() ()() where e with. We should ote that umerical istabilities or divergeces could arise i the calculatio with this polyomial. The reaso is see from the recursio relatio (9) where the diagoal recursio coefficiet goes as for large whereas the off diagoal coefficiets go like. For this reaso (as a example), we had to choose a large value for z to obtai a stable ad coverget evaluatio of the asymptotics show i Fig.. III. The secod polyomial class 4

5 The secod polyomial is a three-parameter orthogoal polyomial associated with the potetial fuctio (8) whe either V or V. We desigated the polyomial as G ( z ; ) ad it satisfies the followig there-term recursio relatio for,,... (, ) (, ) ( ) (, ) z G ( z ; ) B ( ) G ( ; ) ( )( ) z ( )( ) (, ) ( )( ) (, ) B G ( )( ) ( z ; ) B G ( )( ) ( z ; ) (, ) B ad G ( z ; ). It is a polyomial of order i where 5 () z. The (, ) polyomial of the first kid satisfies this recursio relatio with G ( z ; ) ad z (, ) ( ) G ( z ; ). (3) B The polyomial of the secod kid satisfies the same recursio relatio () with (, ) (, ) G ( z ; ) but G ( z ; ) c cz where the liearity coefficiets c, c are (, ) (, ) differet from those i Eq. (3). G ( z ; ) is the orthoormal versio of G ( z ; ) (, ) (, ), which is defied by G ( z ; ) G ( z ; ) (see Appedix E i Ref. [6]). If is positive the this polyomial has oly a cotiuous spectrum o the positive z lie. However, if is egative the the spectrum is a mix of a cotiuous part o the positive z lie ad a discrete part o the egative z lie. This could also be verified umerically by lookig at the distributio of the zeros of this polyomial for a very large order. Figure 3 is a plot of such a distributio for a give set of values of polyomial parameters with egative showig the cotiuous as well as the discrete spectrum. O the other had, (, ) Figure 4 is a plot of the asymptotics of G ( z ; ) showig clearly the siusoidal behavior portrayed i formula () where umerical aalysis gives ad shows that l( z). That is, i the asymptotic limit as G ( z ; ) A( z) cos l( ) ( z) ( z), (4) where, i geeral, the three fuctios A( z ), ( z) ad ( z) do also deped o the polyomial parameters,,. Table 3 gives expressios for z ad the polyomial parameters i terms of the eergy ad potetial parameters of the correspodig physical system. Here too, if the properties of this polyomial were to be kow aalytically the we would have obtaied the physical features of the correspodig quatum mechaical system. For example, if the asymptotics were give explicitly as i Eq. (4), we would have obtaied the phase shift ad eergy spectrum i a simple ad straightforward maer just by substitutig the parameter map of Table 3 ito Eq. (4). Noetheless, the wellkow eergy spectrum of the potetials associated with this polyomial gives its spectrum formula as [4] (, ) z, (5) where,,.., N ad N is the largest iteger less tha or equal to. Therefore, the zeros of the scatterig amplitude A( z ) i the asymptotics (4) are at i z. Moreover, the kow scatterig states of the quatum systems associated with

6 this polyomial (see, for example, Refs. [5-7]) give the phase shift ( z) i the asymptotics (4) as ( z ) arg i i i z arg z arg z. (6) Cosiderig this result together with the spectrum formula (5) suggest that the scatterig amplitude Az ( ) i the asymptotics (4) is proportioal to i i A( z) i z z z. (7) These fidigs also imply that the cotiuous weight fuctio for this polyomial takes the followig form ( z ) i A ( z) z z i. (8) Fially, the orthogoality relatio for egative reads as follows N (, ) (, ) (, ) (, ) ( z) G ( z ; ) G ( z ; ) dz G ( z ; ) G ( z ; ), (9) m k k m k, m k where k is the ormalized discrete weight fuctio. For positive, however, oly the itegratio part of this orthogoality survives. Moreover, the physics of the problems associated with this polyomial suggest that it has oe discrete versio with a fiite spectrum [6]. This is defied by its recursio relatios, which is obtaied by the substitutio z iz k i Eq. (), where k,,,.., Nad is some proper fuctio of N. IV. Coclusio With this short umerical expose, we hope that we did demostrate the sigificace of (, ) (, ) the two polyomials H ( z ;, ) ad G ( z ; ) alog with their discrete versios to physics. Thus, we call upo experts i the field of orthogoal polyomials to study them ad derive their properties aalytically. Refereces: [] A. D. Alhaidari, A exteded class of L -series solutios of the wave equatio, A. Phys. 37 (5) 5 [] A. D. Alhaidari, Aalytic solutio of the wave equatio for a electro i the field of a molecule with a electric dipole momet, A. Phys. 33 (8) 79 [3] A. D. Alhaidari ad H. Bahlouli, Extedig the class of solvable potetials: I. The ifiite potetial well with a siusoidal bottom, J. Math. Phys. 49 (8) 8 [4] A. D. Alhaidari, Extedig the class of solvable potetials: II. Screeed Coulomb potetial with a barrier, Phys. Scr. 8 () 53 [5] H. Bahlouli ad A. D. Alhaidari, Extedig the class of solvable potetials: III. The hyperbolic sigle wave, Phys. Scr. 8 () 58 [6] A. D. Alhaidari, Solutio of the orelativistic wave equatio usig the tridiagoal represetatio approach, J. Math. Phys. 58 (7) 74 [7] A. D. Alhaidari ad M. E. H. Ismail, Quatum mechaics without potetial fuctio, J. Math. Phys. 56 (5) 77 6

7 [8] A. D. Alhaidari, Formulatio of quatum mechaics without potetial fuctio, Quatum Physics Letters 4 (5) 5 [9] A. D. Alhaidari ad T. J. Taiwo, Wilso-Racah Quatum System, J. Math. Phys. 58 (7) [] K. M. Case, Orthogoal polyomials from the viewpoit of scatterig theory, J. Math. Phys. 5 (974) 66 [] J. S. Geroimo ad K. M. Case, Scatterig theory ad polyomials orthogoal o the real lie, Tras. Amer. Math. Soc. 58 (98) [] J. S. Geroimo, A relatio betwee the coefficiets i the recurrece formula ad the spectral fuctio for orthogoal polyomials, Tras. Amer. Math. Soc. 6 (98) 658 [3] R. Koekoek ad R. Swarttouw, The Askey-scheme of hypergeometric orthogoal polyomials ad its q-aalogues, Reports of the Faculty of Techical Mathematics ad Iformatics, Number 98-7 (Delft Uiversity of Techology, Delft, 998) [4] See Sectio III.B i Ref. [6] [5] A. Frak ad K. B. Wolf, Lie algebras for systems with mixed spectra. I. The scatterig Pöschl-Teller potetial, J. Math. Phys. 6 (985) 973 [6] A. Khare ad U. P. Sukhatme, Scatterig amplitudes for supersymmetric shapeivariat potetials by operator methods, J. Phys. A (988) L5 [7] R. K. Yadav, A. Khare ad B. P. Madal, The scatterig amplitude for ratioally exteded shape ivariat Eckart potetials, Phys. Lett. A 379 (5) 67 7

8 Table Captios: Table : Partial list of ew ad/or geeralized potetials that are exactly solvable usig the TRA as outlied i Ref. [6]. (, ) Table : The argumet ad parameters of the polyomial H ( z ;, ) i terms of the eergy ad potetial parameters of the correspodig geeralized potetial of Table with u V ad E. i i (, ) Table 3: The argumet ad parameters of the polyomial G ( z ; ) i terms of the eergy ad potetial parameters of the correspodig potetial with u V ad E. The depedece of o the physical parameters (expected to be a simple liear relatio) is yet to be derived with N beig the umber of boud states. i i Figure Captios: (, ) Fig. : Distributio of the zeros of the polyomial HN ( z ;, ) o the z-axis for a large order N showig that the spectrum of this polyomial is cotiuous over the whole real lie. We took, 3,,.7 ad N 5. (, ) Fig. : Plot of the asymptotics of H ( z ;, ) showig clearly the siusoidal behavior portrayed i Eq. () with ad. We took, 3, 5,. radias, 6 z ad rages from 4 to 47. (, ) Fig. 3: Distributio of the zeros of the polyomial GN ( z ; ) o the z -axis for a large order N ad for egative values of. It is evidet that the spectrum i this case is a mix of a cotiuous part o the positive z -axis ad a discrete fiite part o the egative z - axis. The bottom graph is obtaied by zoomig i the top ear the origi to show the discrete 4-poit spectrum. These four poits do ot chage if we chage the polyomial order N. Moreover, their values ad umber are exactly those give by formula (5). We took, 3, 35 ad N. (, ) Fig. 4: Plot of the asymptotics of G ( z ; ) showig clearly the siusoidal behavior depicted by Eq. (4). We took,, 3, z ad rages from 5 to. 8

9 Fig. Fig. Fig. 3 Fig. 4 9

10 Geeralized Potetial Table cos z Trigoometric k Scarf u New u u Hyperbolic Eckart Hyperbolic Pöschl-Teller Hyperbolic Rose- Morse u k u u u u u u u u u u u u u u 4 u 4 u u u 6 u 4 u 4 u u Table 3 Potetial V( x ) z V V sih ( x) cosh ( x) 6 u f ( Nu,, u ) 4 u ( V V) ( V V)si( x L) V cos ( xl) u u f ( u, u ) 8 4 u V V x x e e u f ( Nu,, u ) u

11 Table yx ( ) V( x) Geeralized Potetial Boud Scatterig y( x) si x L W Wsi( x L) V V si x L cos ( xl) L x L, L xl yx ( ) x L, L yx yx ( ) e x x x ( ) tah ( ) x y( x) tah( x) x W V V, Trigoometric Scarf Ifiite No V 4L 4 V V ( xl) V V ( xl) ( xl) ( xl) ( xl) V L V L New Ifiite No x V V Ve x x e e Hyperbolic Eckart Fiite Yes V V V V[tah ( x) ] sih ( x) cosh ( x) V 8 Hyperbolic Pöschl-Teller Fiite Yes V Vtah( x) cosh ( x) Hyperbolic Rose-Morse Fiite Yes V, E