Open problem in orthogonal polynomials

Transcription

1 Ope problem i orthogoal polyomials Abdlaziz D. Alhaidari Sadi Ceter for Theoretical Physics, P.O. Box 374, Jeddah 438, Sadi Arabia haidari@sctp.org.sa URL: Abstract: Usig a algebraic method for solvig the wave eqatio i qatm mechaics, we ecotered a ew class of orthogoal polyomials o the real lie. It cosists of a for-parameter polyomial with cotios spectrm o the whole real lie ad two of its discrete versios; oe with a fiite spectrm ad aother with cotably ifiite spectrm. A secod class of these ew orthogoal polyomials appeared recetly while solvig a e-type eqatio. Based o these reslts ad o or recet stdy of the soltio space of a ordiary differetial eqatio of the secod kid with for siglar poits, we itrodce a modificatio of the Askey scheme of hypergeometric orthogoal polyomials. Up to ow, these polyomials are defied by their three-term recrsio relatios ad iitial vales. owever, their other properties like the weight fctios, geeratig fctios, orthogoality, Rodriges-type formlas, etc. are yet to be derived aalytically. De to the prime sigificace of these polyomials i physics ad mathematics, we call po experts i the field of orthogoal polyomials to stdy them, derive their properties ad write them i closed form (e.g., i terms of hypergeometric fctios). Keywords: tridiagoal represetatio, orthogoal polyomials, potetial fctios, asymptotics, recrsio relatio, spectrm formla. MSC: 4C5, 33C47, 33C45, 33D45. Itrodctio The wave fctio i qatm mechaics cold be viewed as a vector field i a ifiite dimesioal space with local it vectors. Therefore, i oe of the formlatios of qatm mechaics, the wave fctio at a eergy E, E ( x ), is writte as a boded sm over a complete set of sqare itegrable basis fctios i cofigratio space with coordiate x: ( x) f ( E) ( x), () E where ( x ) are the basis elemets (local it vectors) ad f ( ) E are proper expasio coefficiets i the eergy (the compoets of the wave fctio alog the it vectors). All physical iformatio abot the system, both strctral ad dyamical, are cotaied i these expasio coefficiets. The Tridiagoal Represetatio Approach (TRA) is a algebraic method for solvig the wave eqatio (e.g., the Schrödiger or Dirac eqatio) [-4]. I the TRA, the basis elemets are chose sch that the matrix represetatio of the wave operator is tridiagoal. Coseqetly, the resltig matrix wave eqatio becomes a three-term recrsio relatio for the expasio coefficiets f ( ) E, which is solved i terms of orthogoal polyomials i some physical parameter(s) ad/or the eergy. If we write f( E) f( E) P( ), where is a appropriate fctio of the eergy ad physical parameters, the we have show that P ( ) is a complete set of orthogoal polyomials satisfyig the said recrsio relatio

2 with P ( ). The correspodig positive defiite weight fctio is f ( E ). These polyomials are associated with the cotim scatterig states of the system where E is a cotios set. O the other had, the discrete bod states are associated with the discrete versio of these polyomials. We fod all sch polyomials that correspod to well-kow physical systems ad to ew oes as well. For example, the scatterig states of the Colomb problem are associated with Meixer-Pollaczek polyomial whereas the bod states are associated with oe of its discrete versios; the Meixer polyomial. Moreover, the scatterig states of the Morse oscillator are associated with the cotios dal ah polyomial whereas the fiite mber of bod states are associated with its discrete versio, the dal ah polyomial. Additioally, the cotim scatterig states of the hyperbolic Pöschl-Teller potetial correspod to the Wilso polyomial whereas the fiite mber of bod states are associated with the Racah polyomial, which is the discrete versio of the Wilso polyomial. Ad so o. Sice 5, however, we fod a ew class of exactly solvable problems that are associated with orthogoal polyomials, which were overlooked i the mathematics ad physics literatre [5-]. These polyomials are defied, p to ow, by their three-term recrsio relatios ad iitial vale P ( ). owever, their other importat properties are yet to be derived aalytically. These properties iclde the weight fctios, geeratig fctios, asymptotics, orthogoality relatios, Rodriges-type formlas, etc. Two classes of these ew polyomials ad their discrete versios are defied i the followig two sectios. I sectio 4, we itrodce a modificatio to the hypergeometric polyomials i the Askey scheme [3,4] that maifest itself i a particlar deformatio of their correspodig three-term recrsio relatios.. The first polyomial class The for-parameter orthogoal polyomial, which we desigate as followig three-term recrsio relatio (, ) (; z, ), satisfies the (, ) (, ) cos ( z;, ) zsi ( ;, ) ( )( ) z ( )( ) (, ) ( )( ) (, ) ( )( ) (; z, ) ( )( ) (; z, ) () where z, ad,,.... It is a polyomial of degree i z. Settig z trs, () ito the recrsio relatio of the Jacobi polyomial P( ) (cos ). Physical reqiremets dictate that ad are greater tha. The polyomial of the first kid satisfies this recrsio (, ) relatio together with (; z, ) ad (, ) (; z, ) cos zsi 4, (3) (, ) which is obtaied from () by settig ad (; z, ). This polyomial has oly a cotios spectrm over the whole real z lie. This cold be verified merically by lookig at the distribtio of its zeros for a very large degree. The asymptotics ( ) of

3 (, ) (; z, ) cold also be obtaied merically ad fod to be sisoidal as a fctio of, which is cosistet with the expected physical behavior. Additioally, the physics of the problems associated with this polyomial sggests that it shold have two discrete versios, oe with a ifiite spectrm (if is pre imagiary) ad aother with a fiite spectrm (for real ). This is similar to the Meixer-Pollaczek polyomial ad its discrete versios of the Meixer ad Krawtchok polyomials with ifiite ad fiite spectra, respectively. The discrete spectrm is obtaied as the set of vales of z, which are pre imagiary (e.g., iz k ), that make the amplitde of the sisoidal asymptotics vaish. The size of this set is either fiite or ifiite. The two discrete versios of the polyomial are defied by their three-term recrsio relatios, which are obtaied from Eq. () by the replacemets i ad z iz k givig ( ) ( )() 4( )( ) (, ) 4( )( ) (, ) ( )( ) ( k;, ) ( )( ) ( k;, ) (, ) (, ) ( k;, ) zk ( k;, ) (4) where k is a o-egative iteger of either fiite or ifiite rage ad e with. If we desigate the discrete polyomials that solve (4) for the ifiite or fiite sets, z or k k zk k (,, as ) (, h ( k;, ) ad g ) ( k; N, ), respectively, the Table shows some of the physical potetial fctios associated with the polyomials i this class. The size of the fiite spectrm, N, is the largest iteger less tha or eqal to. N 3. The secod polyomial class While solvig a e-type differetial eqatio, we ecotered recetly aother class of these ew orthogoal polyomials [5]. It is also a for-parameter polyomial, which we desigate (, as Q ) (; z, ). It satisfies the followig three-term recrsio relatio (, ) (, ) cos Q ( z;, ) zsi Q ( ;, ) ( )( ) z ( )( ) (, ) ( )( ) (, ) Q ( )( ) (; z, ) Q ( )( ) (; z, ) (5) where z, ad,,.... Note the iverse power o the sqare bracket, which (, ) costittes a major differece from the recrrece relatio (). ere too, Q (; z, ) ad (, ) (, ) Q (; z, ) is obtaied from (5) by settig ad Q (; z, ). This polyomial has a prely cotios spectrm over the etire real lie. Moreover, it has aother versio, (, G ) (; z, ), whose recrsio relatio is obtaied from (5) by the replacemet i ad z iz givig 3

4 (, ) ( ) (, ) G ( z;, ) z G ( ;, ) ( )( ) z 4( )( ) (, ) 4( )( ) (, ) G ( )( ) (; z, ) G ( )( ) (; z, ) (6) where z ad e with. If is pre imagiary the the spectrm is prely cotios. owever, if is real the the spectrm is a mix of a cotios positive spectrm ad a discrete egative spectrm of fiite size N, where N is the largest iteger less tha or eqal to. I this case, the polyomial satisfies a geeralized orthogoality relatio of the form N (, ) (, ) (, ) (, ) ( z) G ( z;, ) G m ( z;, ) dz ( k) G ( zk;, ) G m ( zk;, ), m k. (7) where, ( z) ad ( k) are the positive defiite cotios ad discrete weight fctios, respectively. The fiite discrete spectrm z N k cold be determied from the k (, coditio that forces the asymptotics ( ) of G ) (; z, ) to vaish. 4. Deformatio of the Askey scheme of orthogoal polyomials The reslts of or recet stdies i [5] ad [6], seem to sggest that the type of deformatio i the recrsio relatio like that of the Jacobi polyomial i Eq. () is, i fact, commo to a larger class of orthogoal polyomials: the Askey scheme of hypergeometric polyomials [3,4]. This scheme cosists of two chais of hypergeometric orthogoal polyomials. Oe of them is a cotios set with the Wilso polyomial at the top of the chai that cotais the cotios dal ah, cotios ah, Meixer-Pollaczek, Jacobi, Lagerre, etc. The other is a discrete set with the Racah polyomial at the top of the chai that icldes, the dal ah, ah, Meixer, Krawtchok, Charlier, etc. The polyomials i each chai are obtaied from that at the top by certai limits of the hypergeometric fctios (i.e., 4 F 3 3 F F F ). We write the three-term recrsio relatio of the origial polyomials i the scheme geerically as follows xp ( x) a P ( x) b P ( x) c P ( x), (8) where stads for a fiite set of parameters ad x is a cotios or discrete set (fiite or cotably ifiite) or both. As a example, for the Lagerre polyomial L ( x), x is cotios with x, ad a, b ( ), c ( ). Now, the deformatio of this recrsio relatio is itrodced by modifyig it sch that it reads xp ( x) a ( ) P ( x) b P ( x) cp ( x), (9) where is the deformatio parameter ad is a fctio of the parameter set. As a example, the recrsio relatio () above is obtaied by deformig that of the Jacobi polyomial, P( ) (cos ) where x cos, z si ad. Moreover, i Ref. [5] ad [6], 4

5 we also ecotered modified versios of orthogoal polyomials i the Askey scheme while searchig for series soltios of the followig secod order liear differetial eqatio d y( x) a b c dy( x) A B C x( x)( rx) d xd E y( x) dx x, () x r x dx x x r x where abcdr,,,,, ABCDE,,,, are real parameters with r,. For d, the eqatio has for reglar siglar poits at x,, r, ad oe of its soltios, which we referred to as geeralized soltio [5], is writte as a series of sqare itegrable basis fctios like () with the expasio coefficiets beig modified versio of the Wilso polyomial W ( ;,,, ) z that satisfies the deformed recrsio relatio (9) where x, z, r, 4, () ad the polyomial parameters,,, are related to the differetial eqatio parameters i a particlar way. I Ref. [7], the Athors refer to W ( ;,,, ) z as the Racah-e polyomial bt oe of its aalytic properties was give. O the other had, for d Eq. () x,, r ad oe irreglar at ifiity. I Ref. [6], has for siglarities with three reglar at we obtaied a series soltio of this differetial eqatio where the expasio coefficiets are modified versio of the cotios ah polyomial p (; z,,, ) that satisfies the deformed recrsio relatio (9) with x iz, d, abc D. (), 4 The polyomial parameters,,, are related to the differetial eqatio parameters abcdr,,,,, ABCDE,,,, via oe of two alterative ways depedig o the sig of. 5. Coclsio De to the prime sigificace of these ew (or modified) polyomials alog with their discrete versios to the soltio of varios problems i physics ad mathematics, we rge experts i the field of orthogoal polyomials to stdy them, derive their aalytic properties ad write them i closed form (e.g., i terms of hypergeometric fctios). The soght-after properties of these polyomials iclde the weight fctios, geeratig fctios, asymptotics, orthogoality relatios, Rodriges-type formlas, Forward/Backward shift operator relatios, zeros, etc. 5

6 Refereces: [] A. D. Alhaidari ad M. E.. Ismail, Qatm mechaics withot potetial fctio, J. Math. Phys. 56 (5) 77 [] A. D. Alhaidari ad T. J. Taiwo, Wilso-Racah Qatm System, J. Math. Phys. 58 (7) [3] A. D. Alhaidari,. Bahloli, ad M. E.. Ismail, The Dirac-Colomb Problem: a mathematical revisit, J. Phys. A 45 () 3654 [4] A. D. Alhaidari, Soltio of the orelativistic wave eqatio sig the tridiagoal represetatio approach, J. Math. Phys. 58 (7) 74 [5] A. D. Alhaidari, A exteded class of L -series soltios of the wave eqatio, A. Phys. 37 (5) 5 [6] A. D. Alhaidari, Aalytic soltio of the wave eqatio for a electro i the field of a molecle with a electric dipole momet, A. Phys. 33 (8) 79 [7] A. D. Alhaidari ad. Bahloli, Extedig the class of solvable potetials: I. The ifiite potetial well with a sisoidal bottom, J. Math. Phys. 49 (8) 8 [8] A. D. Alhaidari, Extedig the class of solvable potetials: II. Screeed Colomb potetial with a barrier, Phys. Scr. 8 () 53 [9]. Bahloli ad A. D. Alhaidari, Extedig the class of solvable potetials: III. The hyperbolic sigle wave, Phys. Scr. 8 () 58 [] A. D. Alhaidari, For-parameter $/r^$ siglar short-rage potetial with a rich bod states ad resoace spectrm, Theor. Math. Phys. 95 (8) 86 [] A. D. Alhaidari ad T. J. Taiwo, For-parameter potetial box with iverse sqare siglar bodaries, Er. Phys. J. Pls 33 (8) 5 [] I. A. Assi, A. D. Alhaidari ad. Bahloli, Soltio of spi ad psedo-spi symmetric Dirac eqatio i + space-time sig tridiagoal represetatio approach, Comm. Theor. Phys. 69 (8) 4 [3] R. Koekoek ad R. Swarttow, The Askey-scheme of hypergeometric orthogoal polyomials ad its q-aaloges, Reports of the Faclty of Techical Mathematics ad Iformatics, Nmber 98-7 (Delft Uiversity of Techology, Delft, 998) [4] R. Koekoek, P. A. Lesky ad R. F. Swarttow, ypergeometric Orthogoal Polyomials ad Their q-aaloges (Spriger, eidelberg, ) [5] A. D. Alhaidari, Series soltios of e-type eqatio i terms of orthogoal polyomials, J. Math. Phys. 59 (8) 357 [6] A. D. Alhaidari, Series soltio of a te-parameter secod order differetial eqatio with three reglar ad oe irreglar siglarities, arxiv:8.66 [math-ph] [7] F. A. Grübam, L. Viet, ad A. Zhedaov, Tridiagoalizatio ad the e eqatio, J. Math. Phys. 58 (7) 373 Table Captio: Table : The physical potetial fctios associated with the ew polyomial class of sectio. The polyomial parameters,,, are related to the potetial parameters V, V, V as show, where V ad E. i i 6

7 Table V( x ) Polyomials cos cosh z V Vsi( x L) V V si x L cos ( xl) L x L, V V 4L 4 V V ( xl) ( xl) ( xl) x L, V L V L x V V Ve x x e e x, V V 4 V ( x L) V V V x sih ( x) cosh ( x) [tah ( ) ] x, V 8 V Vtah( x) cosh ( x) x (, h ) ( k;, ) (, h ) ( k;, ) (, ) (; z, ) (, g ) ( k; N, ) (, ) (; z, ) (, g ) ( k; N, ) (, ) (; z, ) (, g ) ( k; N, ) k k k k k 4 4 L 6 4 L