Prepotential approach to solvable rational potentials and exceptional orthogonal polynomials
aa r X i v : . [ m a t h - ph ] A ug Prepotential approach to solvable rational potentialsand exceptional orthogonal polynomials
Choon-Lin Ho Department of Physics, Tamkang University, Tamsui 25137, Taiwan, R.O.C.
We show how all the quantal systems related to the exceptional Laguerre and Jacobipolynomials can be constructed in a direct and systematic way, without the need of shapeinvariance and Darboux-Crum transformation. Furthermore, the prepotential need not beassumed a priori. The prepotential, the deforming function, the potential, the eigenfunctionsand eigenvalues are all derived within the same framework. The exceptional polynomials areexpressible as a bilinear combination of a deformation function and its derivative. §
1. Introduction
In the last three years or so one has witnessed some interesting developmentsin the area of exactly solvable models in quantum mechanics: the number of ex-actly solvable shape-invariant models has been greatly increased owing to the dis-covery of new types of orthogonal polynomials, called the exceptional X ℓ polynomi-als. – Unlike the classical orthogonal polynomials, these new polynomials havethe remarkable properties that they still form complete sets with respect to somepositive-definite measure, although they start with degree ℓ polynomials instead ofa constant.Two families of such polynomials, namely, the Laguerre- and Jacobi-type X polynomials, corresponding to ℓ = 1, were first proposed by G´omez-Ullate et al.in Ref. 1), within the Sturm-Lioville theory, as solutions of second-order eigenvalueequations with rational coefficients. The results in Ref. 1) were reformulated inthe framework of quantum mechanics and shape-invariant potentials by Quesne etal. These quantal systems turn out to be rationally extended systems of thetraditional ones which are related to the classical orthogonal polynomials. The mostgeneral X ℓ exceptional polynomials, valid for all integral ℓ = 1 , , . . . , were discoveredby Odake and Sasaki (the case of ℓ = 2 was also discussed in Ref. 3)). Later, inRef. 5) equivalent but much simpler looking forms of the Laguerre- and Jacobi-type X ℓ polynomials were presented. Such forms facilitate an in-depth study of someimportant properties of the X ℓ polynomials, such as the actions of the forward andbackward shift operators on the X ℓ polynomials, Gram-Schmidt orthonormalizationfor the algebraic construction of the X ℓ polynomials, Rodrigues formulas, and thegenerating functions of these new polynomials. Structure of the zeros of the exceptionpolynomials was studied in Ref. 6).More recently, these exceptional polynomials have been studied in many ways.For instance, possible applications of these new polynomials were considered inRef. 7) for position-dependent mass systems, and in Ref. 8) for the Dirac and Fokker-Planck equations. The new polynomials were also considered as solutions associatedwith some conditionally exactly solvable potentials. These polynomials were re- typeset using
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TEX.cls h Ver.0.9 i C.-L. Ho cently constructed by means of the Darboux-Crum transformation.
Rationalextensions of certain shape-invariant potentials related to the exceptional orthogo-nal polynomials were generated by means of Darboux-B¨acklund transformation inRef. 12). Generalizations of exceptional orthogonal polynomials to discrete quantummechanical systems were done in Ref. 13). Structure of the X ℓ Laguerre polynomialswas considered within the quantum Hamilton-Jacobi formalism in Ref. 14). Gener-alizations of these new orthogonal polynomials to multi-indexed cases were discussedin Ref. 16),17). Recently radial oscillator systems related to the exceptional Laguerrepolynomials have been considered based on higher order supersymmetric quantummechanics.
So far most of the methods employed to generate the exceptional polynomialshave invoked in one way or another the idea of shape invariance and/or the relatedDarboux-Crum transformation. Furthermore, the so-called superpotentials (whichwe shall call the prepotential), which determine the potentials, have to be assumeda priori (often with good educated guesses).The aim of this paper is to demonstrate that it is possible to generate all thequantal systems related to the exceptional Laguerre and Jacobi polynomials by asimple constructive approach without the need of shape invariance and the Darboux-Crum transformation. The prepotential (hence the potential), eigenfunctions andeigenvalues are all derived within the same framework. We call this the prepotentialapproach, which is an extension of the approach we employed to construct all thewell-known one-dimensional exactly solvable quantum potentials in Ref. 19).The plan of this paper is as follows. Sect. 2 presents the ideas of prepotentialapproach to systems which are rational extensions of the traditional systems relatedto the classical orthogonal polynomials. In Sect. 3 the prepotential approach is em-ployed to generate the L2 Laguerre system. Construction of the J1 and J2 Jacobicases is then outlined in Sect. 4. Sect. 5 summarizes the paper. Appendix A col-lects some useful results on the classical Laguerre and Jacobi polynomials. The L1Laguerre system is then summarized in Appendix B. §
2. Prepotential approach
Main ideas
We shall adopt the unit system in which ~ and the mass m of the particle aresuch that ~ = 2 m = 1. Consider a wave function φ ( x ) which is written in terms ofa function W ( x ) as φ ( x ) ≡ e W ( x ) . (2.1)Operating on φ N by the operator − d /dx results in a Schr¨odinger equation H φ = 0,where H = − d dx + ¯ V , (2.2)¯ V ≡ ˙ W + ¨ W . (2.3) repotential approach to solvable rational potentials .... x . Since W ( x ) determines the potential¯ V , it is therefore called the prepotential. For clarity of presentation, we shall oftenleave out the independent variable of a function if no confusion arises.In this work we consider the following form of the prepotential W ( x, η ) = W ( x ) − ln ξ ( η ) + ln p ( η ) . (2.4)Here W ( x ) is the zero-th order prepotential, and η ( x ) is a function of x whichwe shall choose to be one of the sinusoidal coordinates, i.e., coordinates such that˙ η ( x ) is at most quadratic in η , since most exactly solvable one-dimensional quantalsystems involve such coordinates. The choice of η ( x ) and the final form of thepotential dictate the domain of the variable x . ξ ( η ) and p ( η ) are functions of η to bedetermined later. We shall assume ξ ( η ) to be a polynomial in η . The function p ( η )consists of the eigen-polynomial, but itself need not be a polynomial (see Sect. 3).With the prepotential (2.4), the wave function is φ ( x ) = e W ( x ) ξ ( η ) p ( η ) , (2.5)and the potential ¯ V = ˙ W + ¨ W takes the form¯ V = ˙ W + ¨ W + (cid:20) ˙ η (cid:18) ξ ′ ξ − ξ ′′ ξ (cid:19) − ξ ′ ξ (cid:16) W ˙ η + ¨ η (cid:17)(cid:21) + 1 p (cid:20) ˙ η p ′′ + (cid:18) W ˙ η + ¨ η − η ξ ′ ξ (cid:19) p ′ (cid:21) . (2.6)Here the prime denotes derivative with respective to η .For ξ ( η ) = 1, the prepotential approach can generate exactly and quasi-exactlysolvable systems associated with the classical orthogonal polynomials. The pres-ence of ξ in the denominators of φ ( x ) and V ( x ) thus gives a rational extension, ordeformation, of the traditional system. We therefore call ξ ( η ) the deforming function.To make ¯ V exactly solvable, we demand that: (1) W is a regular function of x ,(2) the deforming function ξ ( η ) has no zeros in the the ordinary (or physical) domainof η ( x ), and (3) the function p ( η ) does not appear in V .The requirement (3) can be easily met by setting the last term involving p ( η ) inEq. (2.6) to a constant, say “ −E ”, i.e.,˙ η p ′′ + (cid:18) W ˙ η + ¨ η − η ξ ′ ξ (cid:19) p ′ + E p = 0 . (2.7)If W , ξ can be determined, and Eq. (2.7) can be solved, then we would have con-structed an exactly solvable quantal system Hψ = E ψ defined by H = − d /dx + V ( x ), with the wave function (2.5) and the potential V ( x ) ≡ ˙ W + ¨ W + (cid:20) ˙ η (cid:18) ξ ′ ξ − ξ ′′ ξ (cid:19) − ξ ′ ξ (cid:16) W ˙ η + ¨ η (cid:17)(cid:21) . (2.8) C.-L. Ho
Determining W ( x ) , ξ ( η ) and p ( η )As mentioned before, we require that ξ ( η ) has no zeros in the ordinary domainof the variable η , but ξ ( η ) may have zeros in the other region in the complex η -plane.Suppose ξ ( η ) satisfies the equation c ( η ) ξ ′′ + c ( η ) ξ ′ + e E ξ = 0 , e E = real constant . (2.9)Here c ( η ) and c ( η ) are functions of η to be determined. We want Eq. (2.9) to beexactly solvable. This is most easily achieved by matching (2.9) with the (confluent)hypergeometric equation, and this we shall adopt in this paper. Thus c ( η ) and c ( η )are at most quadratic and linear in η , respectively.If the factor exp( W ( x )) /ξ ( η ) in Eq. (2.5) is normalizable, then p ( η ) = constant(in this case we shall take p ( η ) = 1 for simplicity), which solves Eq. (2.7) with E = 0,is admissible. This gives the ground state φ ( x ) = e W ( x ) ξ ( η ) . (2.10)However, if exp( W ( x )) /ξ ( η ) is non-normalizable, then φ ( x ) cannot be the groundstate. In this case, the ground state, like all the excited states, must involve non-trivial p ( η ) = 1. Typically it is in such situation that the exceptional orthogonalpolynomials arise. To determine states involving non-trivial p ( η ), we proceed asfollows. Since both ξ and ξ ′ appear in Eq. (2.7), we shall take the ansatz that p ( η )be a linear combination of ξ and ξ ′ : p ( η ) = ξ ′ ( η ) F ( η ) + ξ ( η ) G ( η ) , (2.11)where F ( η ) and G ( η ) are some functions of η . Then using Eq. (2.9) we have p ′ ( η ) = ξ ′ (cid:20) − c c F + F ′ + G (cid:21) + ξ " − e E c F + G ′ . (2.12)We demand that Eq. (2.7) be regular at the zeros of ξ . This is achieved if p ′ ∝ ξ ,which requires that the coefficient of ξ ′ in Eq. (2.12) be zero, thus giving a relationthat connects F and G , G = c c F − F ′ . (2.13)Putting Eq. (2.13) into (2.7), we get ξ ′ " − ˙ η − e E c F + G ′ ! + E F + ξ " ˙ η ddη − e E c F + G ′ ! + (cid:16) W ˙ η + ¨ η (cid:17) − e E c F + G ′ ! + E G = 0 . (2.14)Since ξ and ξ ′ are independent for any polynomial ξ , Eq. (2.14) implies the coefficientsof ξ and ξ ′ are zero. Setting the terms in the two square-brackets to zero, and using repotential approach to solvable rational potentials .... G , we arrive at the following equations satisfied by F ( η ) and c ( η ), respectively: − ˙ η F ′′ + ˙ η c c F ′ + ˙ η c (cid:20) c ddη (cid:18) c c (cid:19) − e E (cid:21) F = E F, (2.15)and c ( η ) = c ˙ η (cid:20) ddη (cid:0) ˙ η (cid:1) − (cid:16) W ˙ η + ¨ η (cid:17)(cid:21) = c ˙ η (cid:20) ddη (cid:0) ˙ η (cid:1) − Q ( η ) (cid:21) , (2.16)where Q ( η ) ≡ ˙ W ˙ η , and we have used the identity ¨ η = ( d ˙ η /dη ) / F ( η ) = c ( η ) V ( η ) , (2.17)with some function V ( η ) in order to avoid any possible singularity from c . Eqs. (2.15)and (2.13) then reduce to c V ′′ + (cid:0) c ′ − c (cid:1) V ′ + (cid:20) c ′′ − c ′ + e E + c ˙ η E (cid:21) V = 0 , (2.18)and G ( η ) = (cid:0) c − c ′ (cid:1) V − c V ′ . (2.19)As mentioned before, in this paper we shall take c ( η ) and c ( η ) to be at mostquadratic and linear in η , respectively. This means the coefficients of the first andsecond terms in Eq. (2.18) are also at most quadratic and linear in η , respectively. SoEq. (2.18) can be matched with the (confluent) hypergeometric equation, providedthat the coefficient of the last term in (2.18) is a constant. This then requires c ( η ) = ± ˙ η . (2.20)(Note: in general one has c ( η ) = ± constant × ˙ η . But it is evident that the constantcan be factored out of Eq. (2.18), together with a rescaling of e E . Thus without lossof generality we set the constant to unity). From Eq. (2.16) this leads to c ( η ) = ± (cid:20) ddη (cid:0) ˙ η (cid:1) − Q ( η ) (cid:21) . (2.21)Now we summarize the procedure or algorithm for constructing an exactly solv-able quantal system, whose potential as well as its eigenfunctions and eigenvaluesare all determined within the some approach: • choose ˙ η from a sinusoidal coordinate; this then fixes the form of c ; C.-L. Ho • by matching Eq. (2.9) with the (confluent) hypergeometric equation, one deter-mines ˜ E , Q ( η ), c and ξ ( η ). Integrating Q ( x ) = ˙ W ˙ η then gives the prepotential W ( x ): W ( x ) = Z x dx Q ( η ( x ))˙ η ( x )= Z η ( x ) dη Q ( η )˙ η ( η ) ; (2.22) • by matching Eq. (2.18) with the (confluent) hypergeometric equation, one de-termines V , and thus F ( η ) , G ( η ) , p ( η ) and E ; • the exactly solvable system is defined by the wave function (2.5) and the po-tential (2.8), which, by Eqs. (2.9) and (2.20), can be recast in the form V ( x ) ≡ ˙ W + ¨ W + ξ ′ ξ (cid:20) η (cid:18) ξ ′ ξ (cid:19) − (cid:16) W ˙ η + ¨ η (cid:17) ± c (cid:21) ± ˜ E . (2.23)2.3. Orthogonality of p ( η )Using the relations dξdη = ˙ ξ ˙ η , dpdx = ˙ ηp ′ , d pdx = ˙ η p ′′ + ¨ ηp ′ , (2.24)one can recast Eq. (2.7) into a differential equation in variable x , d dx p ( η ( x )) + 2 ˙ W − ˙ ξξ ! ddx p ( η ( x )) + E p ( η ( x )) = 0 . (2.25)This can further be put in the Sturm-Liouville form ddx (cid:20) W ddx p ( η ( x )) (cid:21) + EW p ( η ( x )) = 0 , (2.26)where W ( x ) ≡ exp Z x dx ˙ W − ˙ ξξ !! = e W ( x ) ξ ( η ( x )) . (2.27)According to the standard Sturm-Liouville theory, the functions p E (here we add asubscript to distinguish p corresponding to a particular eigenvalue E ) are orthogonal,i.e., Z dx p E ( η ( x )) p E ′ ( η ( x )) W ( x ) ∝ δ E , E ′ (2.28)in the x -space, or Z dη p E ( η ) p E ′ ( η ) W ( x ( η ))˙ η ∝ δ E , E ′ (2.29)in the η -space. repotential approach to solvable rational potentials .... §
3. L2 Laguerre case
We now employ the above algorithm to generate the deformed radial oscillatoras given in Ref. 2)–4).Let us choose η ( x ) = x ∈ [0 , ∞ ). Then ˙ η = 4 η . For c and c , we takethe positive signs in Eqs. (2.20) and (2.21) (the opposite situation is considered inAppendix B) . Thus c ( η ) = 4 η and c = 2(1 − Q ( η )).3.1. W , ξ and ˜ E Eq. (2.9) becomes ηξ ′′ + 12 (1 − Q ( η )) ξ ′ + ˜ E ξ = 0 . (3.1)Comparing Eq. (3.1) with the Laguerre equation ηL ′′ ( α ) ℓ + ( α + 1 − η ) L ′ ( α ) ℓ + ℓL ( α ) ℓ = 0 , ℓ = 0 , , , . . . , (3.2)where L ( α ) ℓ ( η ) is the Laguerre polynomial, we have ξ ( η ) ≡ ξ ℓ ( η ; α ) = L ( α ) ℓ ( η ) , ˜ E = 4 ℓ, Q ( η ) = 2 (cid:18) η − α − (cid:19) . (3.3)For ξ ℓ ( η ; α ) not to have zeros in the ordinary domain [0 , ∞ ), we must have α < − ℓ (see Appendix A). By Eq. (2.22), the form of Q ( η ) gives W ( x ) = x − (cid:18) α + 12 (cid:19) ln x. (3.4)We shall ignore the constant of integration as it can be absorbed into the normal-ization constant.3.2. p ( η ) , φ ( η ) and E The above results implies that exp( W ) ∝ exp( x / x − ( α + ) ( α < − ℓ ). Theterm exp( x /
2) will make φ ( x ) non-normalizable if p ( η ) = 1, or if V ( η ) is a poly-nomial in η . To remedy this, we try V = exp( − η ) U ( η ) with some function U ( η ).Eq. (2.18) becomes ηU ′′ + ( − α + 1 − η ) U ′ + E + ˜ E α ! U = 0 . (3.5)Comparing this equation with the Laguerre equation (3.2) (replacing ℓ by anotherinteger n = 0 , , . . . . In the rest of this paper, the index n will always take on thesevalues), one has U ( η ) = L ( − α ) n ( η ) , E ≡ E n = 4( n − α ) − ˜ E = 4( n − α − ℓ ) . (3.6) C.-L. Ho
From Eqs. (2.17) and (2.19), one eventually obtains p ( η ) ≡ p ℓ,n ( η ) = ξ ′ F + ξG = 4 e − η P ℓ,n ( η ; α ) , (3.7) P ℓ,n ( η ; α ) ≡ ηL ( − α ) n ξ ′ ℓ + (cid:16) αL ( − α ) n − ηL ′ ( − α ) n (cid:17) ξ ℓ = ηL ( − α ) n ξ ′ ℓ + ( α − n ) L ( − α − n ξ ℓ . (3.8)Use has been made of Eqs.(A.1)-(A.3) in obtaining the last line in Eq. (3.8). Wenote that P ℓ,n ( η ; α ) is a polynomial of degree ℓ + n . It is just the L2 type exceptionalLaguerre polynomial. We will show in the next subsection that it is equivalent tothe form presented in Ref. 5) (to be called HOS form for simplicity). By Eq. (2.29),one finds that P ℓ,n ( η ; α )’s are orthogonal in the sense Z ∞ dη e − η η − ( α +1) ξ ℓ P ℓ,n ( η ; α ) P ℓ,m ( η ; α ) ∝ δ nm . (3.9)The exactly solvable potential is given by Eq. (2.23) with W ( x ) and ξ ℓ ( η ; α )given by Eqs. (3.4) and (3.3), respectively. The eigenvalues E n are given in Eq. (3.6),i.e. E n = 4( n − α − ℓ ) . Explicitly, the potential is V ( x ) = x + (cid:0) α + (cid:1) (cid:0) α + (cid:1) x + 8 ξ ′ ℓ ξ ℓ (cid:20) η (cid:18) ξ ′ ℓ ξ ℓ − (cid:19) + α + 12 (cid:21) + 2(2 ℓ − α ) . (3.10)It is easily shown that V ( x ) is equivalent to the potential for L2 Laguerre case inRef. 4), 5), 11) with α = − g − ℓ − ( g > φ ℓ,n ( x ; α ) ∝ e − x x − ( α + ) ξ ℓ P ℓ,n ( η ; α ) , α < − ℓ. (3.11)For ℓ = 0, we have ξ = 1 and ξ ′ ℓ = 0, and the system reduces to the radialoscillator. From Eq. (3.8) one has P ℓ,n → L ( − α − n and α < − / Reducing P ℓ,n ( η ; α ) to HOS form The polynomial P ℓ,n ( η ; α ) is expressed as a bilinear combination of ξ ℓ ( η ; α ) andits derivative ξ ′ ℓ ( η ; α ). The HOS form instead expresses the exceptional polynomialas a bilinear combination of ξ ℓ ( η ; α ) and its shifted form, i.e., ξ ℓ ( η ; α − P ℓ,n ( η ; α ) and the HOS form, we make use ofthe identities Eqs. (A.1) and (A.3) to express ηξ ′ ℓ ( η ; α ) in the first term of P ℓ,n ( η ; α )as ηξ ′ ℓ ( η ; α ) = − ηL ( α +1) ℓ − ( η )= − αL ( α ) ℓ − ( η ) + ℓL ( α − ℓ ( η ) . (3.12)Then we have P ℓ,n ( η ; α ) = (cid:16) − αL ( α ) ℓ − + ℓL ( α − ℓ (cid:17) L ( − α ) n + (cid:16) αL ( − α ) n − ηL ′ ( − α ) n (cid:17) L ( α ) ℓ = (cid:16) α (cid:16) L ( α ) ℓ − L ( α ) ℓ − (cid:17) + ℓL ( α − ℓ (cid:17) L ( − α ) n − ηL ′ ( − α ) n L ( α ) ℓ . (3.13) repotential approach to solvable rational potentials .... L ( α ) ℓ ( η ) − L ( α ) ℓ − ( η ) = L ( α − ℓ ( η ). Finally, we arrive at P ℓ,n ( η ; α ) = ( α + ℓ ) L ( − α ) n ( η ) ξ ℓ ( η ; α − − ηL ′ ( − α ) n ( η ) ξ ℓ ( η ; α ) , (3.14) ξ ℓ ( η ; α − ≡ L ( α − ℓ ( η ) . (3.15)Setting α = − g − ℓ − and ξ ℓ ( η ; g ) ≡ L ( − g − ℓ − ) ℓ ( η ) into (3.14), we have P ℓ,n ( η ; α ) = − (cid:20)(cid:18) g + 12 (cid:19) L ( g + ℓ + ) n ( η ) ξ ℓ ( η ; g + 1)+ ηL ′ ( g + ℓ + ) n ( η ) ξ ℓ ( η ; g ) (cid:21) . (3.16)This is, up to a multiplicative constant, the HOS form of the L2 Laguerre polynomial.The example in this section demonstrates that the prepotential approach de-scribed in Sect. 2 can indeed generate the exactly solvable quantal system which hasthe L2 Laguerre polynomials as the main part of its eigenfunctions. The prepoten-tial W ( x ), the potential V ( x ), the deforming function ξ ℓ ( η ; α ), the eigenfunction φ ℓ,n ( x ; α ) and eigenvalues E n are all determined from first principle.In the next section and in the Appendix, we shall generate systems associatedwith the exceptional Jacobi and L1 Laguerre polynomials. Our description for thesecases will be concise, since the main steps are similar to those described in thissection. §
4. Exceptional Jacobi cases
Let us choose η ( x ) = cos(2 x ) ∈ [ − , c and c give the same equations that determine ξ and V , i.e. Eqs. (2.9) and (2.18).So for definiteness, we shall take the upper signs, which give c ( η ) = 4(1 − η ) and c = − η + Q ( η )).4.1. W , ξ and ˜ E Equation determining ξ is(1 − η ) ξ ′′ ( η ) + (cid:18) − η − Q ( η )2 (cid:19) ξ ′ ( η ) + ˜ E ξ ( η ) = 0 . (4.1)Comparing this with the differential equation satisfied by the Jacobi polynomial P ( α,β ) ℓ ( η ), namely,(1 − η ) P ′′ ( α,β ) ℓ ( η )+ (cid:0) β − α − ( α + β +2) η (cid:1) P ′ ( α,β ) ℓ ( η )+ ℓ ( ℓ + α + β +1) P ( α,β ) ℓ ( η ) = 0 , (4.2)we have ξ ( η ) ≡ ξ ℓ ( η ; α, β ) = P ( α,β ) ℓ ( η ) , ˜ E = 4 ℓ ( ℓ + α + β + 1) ,Q ( η ) = 2 [ α − β + ( α + β + 1) η ] (4.3)0 C.-L. Ho for some parameters α and β . The form of Q ( η ) gives, from Eq. (2.22), W ( x ) = − (cid:18) α + 12 (cid:19) ln sin x − (cid:18) β + 12 (cid:19) ln cos x. (4.4)The equation of V is(1 − η ) V ′′ + [ − β + α − ( − β − α + 2) η ] V ′ + E + ˜ E α + β ! V = 0 . (4.5)From Eq. (4.4) we have e W ∝ (1 − η ) − ( α + ) (1 + η ) − ( β + ) . (4.6)The exponents in Eq. (4.6) naturally divide the parameters α and β into four groups:(i) α > − / , β > − /
2, (ii) α > − / , β < − /
2, (iii) α < − / , β > − / α < − / , β < − /
2. Group (i) should be excluded, or ξ ℓ ( η ; α ) will have zerosin the ordinary domain [ − ,
1] (see Appendix A). So we shall study the other threecases. It turns out that these three cases correspond, respectively, to quantal systemsrelated to the type J1, J2 exceptional Jacobi polynomials, and a rationally extendedJacobi system obtained from the Darboux-P¨oschl-Teller system by deleting the lowest ℓ excited states according to the Crum-Adler method discussed in Ref. 21). We stresshere that the actual admissible parameters α and β in each case are dictated by thefinal form of V , as will be shown below.4.2. J1 Jacobi case
Consider the case with parameters α > − / , β < − /
2. The deforming func-tion ξ ℓ ( η ; α, β ) is given in Eq. (4.3), ξ ℓ ( η ; α, β ) = P ( α,β ) ℓ ( η ) . (4.7)We demand that ξ ℓ ( η ; α, β ) has no zeros in the ordinary domain [ − , β < − ℓ for α > − /
2. Forthis choice of the parameters the first term (1 − η ) − ( α + ) of Eq. (4.6) will make theeigenfunction φ ( x ) non-normalizable, if V is a polynomial.This prompted us to try V = (1 − η ) γ U ( η ) where γ is a real parameter and U ( η )a function of η . From Eq. (4.5) we find that U ( η ) satisfies(1 − η ) U ′′ + ( − γ − β + α − (2 γ − β − α + 2) η ) U ′ + E + ˜ E α + β + γ ( γ + β − α −
1) + 2 γ ( γ − α ) η − η ! U = 0 . (4.8)If γ = 0 , α , the coefficient of U in the last term of the above equation can be reducedto a constant, so that Eq. (4.8) can be compared with the Jacobi differential equation(4.2). As γ = 0 does not solve our original problem with normalizability of the wavefunction, so we shall take γ = α . This leads to(1 − η ) U ′′ + ( − β − α − ( − β + α + 2) η ) U ′ + E + ˜ E β ( α + 1) ! U = 0 . (4.9) repotential approach to solvable rational potentials .... U ( η ; α, β ) = P ( α, − β ) n ( η ) , E≡ E n = 4 [ n ( n + α − β + 1) − ℓ ( ℓ + α + β + 1) − β ( α + 1)] . (4.10)Putting all these results into F ( η ) and G ( η ) gives p ( η ) ≡ p ℓ,n ( η ; α, β ) = 4(1 − η ) α +1 P ℓ,n ( η ; α, β ) ,P ℓ,n ( η ; α, β ) ≡ n (1 + η ) P ( α, − β ) n ( η ) ξ ′ ℓ + h βP ( α, − β ) n ( η ) − (1 + η ) P ′ ( α, − β ) n ( η ) i ξ ℓ o = (1 + η ) P ( α, − β ) n ( η ) ξ ′ ℓ − ( n − β )) P ( α +1 , − β − n ( η ) ξ ℓ . (4.11)We have made use of Eq. (A.9) to arrive at the last line of (4.11). By Eq. (2.29), theorthogonality relations of P ℓ,n ( η ; α )’s are Z − dη (1 − η ) ( α +1) (1 + η ) − ( β +1) ξ ℓ P ℓ,n ( η ; α, β ) P ℓ,m ( η ; α, β ) ∝ δ nm . (4.12)The exactly solvable potential is given by Eq. (2.23) with W ( x ) and ξ ℓ ( η ; α, β )given by Eqs. (4.4) and (4.3), respectively. The eigenvalues E n are given in Eq. (4.10).The complete eigenfunctions are φ ℓ,n ( x ; α, β ) ∝ (1 − η ) ( α + ) (1 + η ) − ( β + ) ξ ℓ P ℓ,n ( η ; α, β ) , (4.13) α > − / , β < − ℓ. Using the identity (A.9) one can show easily that P ℓ,n ( η ; α, β ) = ( ℓ + β ) P ( α, − β ) n ( η ) ξ ℓ ( η ; α + 1 , β − − (1 + η ) P ′ ( α, − β ) n ( η ) ξ ℓ ( η ; α, β ) , (4.14)Up to a multiplicative constant, this is just the HOS form of the J1 Jacobi polynomialpresented in Ref. 5), with the substitution α = g + ℓ − / β = − h − ℓ − . Itis easy to show that V ( x ) and E n are equivalent to those for J1 Jacobi case given inRef. 4), 5), 11) with these values of α and β .As ℓ →
0, the system reduces to the trigonometric Darboux-P¨oschl-Teller po-tential, where α and β can now take the values α > − / , β < − / J2 Jacobi case
One can proceed in a similar manner to construct the exactly solvable systemswith α < − / , β > − /
2. This turns out to lead to the system involving the J2Jacobi polynomials.We shall not bore the readers with similar details here. Instead, we point out thatit is easier to obtain the system by symmetry consideration. One notes that underthe parity transformation η → − η , together with interchange α ↔ β , Eqs. (4.1) (with Q ( η ) given by (4.3)) and (4.5) are invariant in form. This is in complete accordancewith the parity property of the Jacobi polynomials, namely P ( α,β ) n ( − η ) = ( − n P ( β,α ) n ( η ) . (4.15)2 C.-L. Ho
This implies that the J2 Jacobi system is simply the mirror image of the J1 Jacobisystem, and thus it can be obtained from the J1 case by taking the above transfor-mations.4.4.
Rationally extended Jacobi case
Let α, β < − /
2. In this case, the factors in Eq. (4.6) cause no problem withnormalizability of the wave function even if V ( η ) is a polynomial. For ξ ℓ ( η ; α, β )to be nodeless in the ordinary domain [ − , α and β such thatthe conditions in (A.13) are satisfied. For example, if ℓ = 1, one can have α < − , − < β < − /
2, or β < − , − < α < − /
2. For ℓ = 2, we have α, β < − − < α, β < −
1. For ℓ odd, we must have α = β , or ξ ( η ) will have a zero at η = 0 in view of Eq. (4.15).Comparing Eqs. (4.5) and (4.2), one obtains V ( η ) = P ( − α, − β ) n ( η ) , E ≡ E n = 4 [ n ( n − α − β + 1) − ℓ ( ℓ + α + β + 1) − α − β ] . (4.16)From F ( η ) and G ( η ) we get p ( η ) ≡ P ℓ,n ( η ; α, β ) ≡ n (1 − η ) P ( − α, − β ) n ( η ) ξ ′ ℓ + h ( β − α − ( β + α ) η ) P ( − α, − β ) n ( η ) − (1 − η ) P ′ ( − α, − β ) n ( η ) i ξ ℓ o . (4.17)Again, by applying the identity (A.7) and (A.9), one can reduce P ℓ,n ( η ; α, β ) to P ℓ,n ( η ; α, β )= 4 n ( ℓ + β )(1 − η ) P ( − α, − β ) n ( η ) ξ ℓ ( η ; α + 1 , β − n − α )(1 + η ) P ( − α − , − β +1) n ( η ) ξ ℓ ( η ; α, β ) o . (4.18)One notes that P ℓ,n ( η ; α, β ) is a polynomial of degree ℓ + n + 1, and has n + 1 nodes.Thus the wave function with P ℓ, ( η ; α, β ) has one node, and does not correspond tothe ground state. In fact, in this case the ground state wave function is given byEq. (2.10) with p ( η ) = 1 and E = 0, since φ ( x ) is normalizable. To ensure thatthe energies of the excited states are positive, i.e., E n > n = 0 , , , . . . , onemust have, besides the constraints stated at the beginning of this subsection, thecondition α + β < − ℓ , which can be easily checked from the form of E in Eq. (4.16).The functions P ℓ,n ( η ; α, β ) ( n = 0 , , , . . . ), together with p ( η ) = 1, form acomplete set and are orthogonal with respect to the weight function(1 − η ) − ( α +1) (1 + η ) − ( β +1) ξ ℓ . (4.19)The complete eigenfunctions are given by φ ( x ; α, β ) ∝ (1 − η ) − ( α + ) (1 + η ) − ( β + ) ξ ℓ ,φ ℓ,n ( x ; α, β ) ∝ (1 − η ) − ( α + ) (1 + η ) − ( β + ) ξ ℓ P ℓ,n ( η ; α, β ) . (4.20) repotential approach to solvable rational potentials .... W ( x ) and ξ ℓ ( η ; α, β )given by Eqs. (4.4) and (4.3), respectively. Since the polynomials in the eigenfunc-tions start with degree zero, the polynomials P ℓ,n ( η ; α, β ) cannot be considered asexceptional. In fact, this system corresponds to the system discussed in Ref. 21),which is obtained from the Darboux-P¨oschl-Teller system by deleting the lowest ℓ excited states according to the Crum-Adler method. It belongs to the same class ofrationally extended exactly solvable systems discussed in Ref. 22), 23). It is easy toshow, using the identities in Appendix A.2, that the polynomials in Eq. (4.18) areproportional to those given by Eq. (A.33) of Ref. 21) for the Jacobi case. §
5. Summary
We have demonstrated how all the quantal systems related to the exceptionalLaguerre and Jacobi polynomials can be constructed in a direct and systematic way.In this approach one does not need to rely on the requirement of shape invarianceand the Darboux-Crum transformation. Even the prepotential need not be assumeda priori. The prepotential, the deforming function, the potential, the eigenfunctionsand eigenvalues are all derived within the same framework. It is worth to note thatthe main part of the eigenfunctions, which are the exceptional orthogonal polyno-mials, can be expressed as bilinear combination of the deformation function ξ ( η )and its derivative ξ ′ ( η ). However, they are equivalent of the forms given in Ref. 5).We have also derived easily a rationally extended Jacobi model obtained from theDarboux-P¨oschl-Teller system by deleting the lowest ℓ excited states according tothe Crum-Adler method discussed in Rer. 21).We have not discussed the related hyperbolic Darboux-P¨oschl-Teller systems(which are of J2 type). They can be generated in the same way by choosing theappropriate sinusoidal coordinates. They can also be obtained from the trigonometriccase by suitable analytic continuation. Acknowledgments
This work is supported in part by the National Science Council (NSC) of theRepublic of China under Grant NSC NSC-99-2112-M-032-002-MY3.
Appendix A
Useful identities
In this Appendix, we collect some useful identities satisfied by the Laguerre andJacobi polynomials which are used in the main text.A.1.
Laguerre Polynomials
Some useful relations among Laguerre polynomials are: ddη L ( α ) ℓ ( η ) = − L ( α +1) ℓ − ( η ) , (A.1)4 C.-L. Ho L ( α ) ℓ ( η ) + L ( α +1) ℓ − ( η ) = L ( α +1) ℓ ( η ) , (A.2) ηL ( α +2) ℓ − ( η ) − ( α + 1) L ( α +1) ℓ − ( η ) = − ℓL ( α ) ℓ ( η ) , (A.3)According to the Theorem 6.73 of Ref. 24), for an arbitrary real number α = − , − , . . . , − ℓ , the number of the positive zeros of L ( α ) ℓ ( η ) is ℓ if α > −
1; it is ℓ + [ α ] + 1 if − ℓ < α < −
1; it is 0 if α < − ℓ . Here [ a ] denotes the integral part of a .Furthermore, η = 0 is a zero when and only when α = − , − , . . . , − ℓ .A.2. Jacobi polynomials
Some useful relations among the Jacobi polynomial are: ddη P ( α,β ) ℓ ( η ) = ℓ + α + β + 12 P ( α +1 ,β +1) ℓ − ( η ) , (A.4)2( β + 1) P ( α − ,β +1) ℓ ( η ) + ( ℓ + α + β + 1)( η + 1) P ( α,β +2) ℓ − ( η )= 2( ℓ + β + 1) P ( α,β ) ℓ ( η ) , (A.5)( ℓ + α ) P ( α − ,β +1) ℓ ( η ) − αP ( α,β ) ℓ ( η )= 12 ( ℓ + α + β + 1)( η − P ( α +1 ,β +1) ℓ − ( η ) , (A.6)( n + α )(1 + η ) P ( α,β +1) ℓ − ( η ) − β (1 − η ) P ( α +1 ,β ) ℓ − ( η ) = 2 ℓP ( α,β − ℓ ( η ) . (A.7)Using Eqs. (A.4) and (A.6) to eliminate the P ( α +1 ,β +1) ℓ − ( η ) term gives(1 − η ) ddη P ( α,β ) ℓ ( η ) = αP ( α,β ) ℓ ( η ) − ( ℓ + α ) P ( α − ,β +1) ℓ ( η ) . (A.8)Combining Eqs. (A.4) and (A.5) to eliminate the P ( α,β +2) ℓ − ( η ) term gives(1 + η ) ddη P ( α − ,β +1) ℓ ( η ) = − ( β + 1) P ( α − ,β +1) ℓ ( η ) + ( ℓ + β + 1) P ( α,β ) ℓ ( η ) . (A.9)Setting α → α + 1 , β → β −
1, we get(1 + η ) ddη P ( α,β ) ℓ ( η ) = ( ℓ + β ) P ( α +1 ,β − ℓ ( η ) − βP ( α,β ) ℓ ( η ) . (A.10)According to the Theorem 6.72 of Ref. 24), for arbitrary real values of α and β ,the number of zeros of P ( α,β ) ℓ ( η ) in ( − ,
1) is N ( α, β ) = (cid:2) X +12 (cid:3) , if ( − ℓ (cid:18) ℓ + αℓ (cid:19) (cid:18) ℓ + βℓ (cid:19) > (cid:2) X (cid:3) + 1 , if ( − ℓ (cid:18) ℓ + αℓ (cid:19) (cid:18) ℓ + βℓ (cid:19) < . (A.11)Here X ≡ E (cid:20)
12 ( | ℓ + α + β + 1 | − | α | − | β | + 1) (cid:21) , (A.12) repotential approach to solvable rational potentials .... E ( u ) is the Klein’s symbol defined by E ( u ) = , u ≤ u ] u > , u non-integral; u − u = 1 , , , . . . From this theorem, we conclude that the conditions for P ( α,β ) ℓ ( η ) to have nozeros in the ordinary domain ( − ,
1) are | ℓ + α + β + 1 | − | α | − | β | + 1 ≤ , and( − ℓ (cid:18) ℓ + αℓ (cid:19) (cid:18) ℓ + βℓ (cid:19) > . (A.13)It is noted that η = +1( −
1) is a zero of P ( α,β ) ℓ ( η ) if and only if α ( β ) = − , − , . . . , − ℓ with multiplicity | α | ( | β | ). Appendix B
L1 Laguerre case
As with the L2 Laguerre case discussed in Sect. 3, let us take η ( x ) = x . Butnow the negative signs in Eqs. (2.20) and (2.21) will be taken leading to c ( η ) = − η and c = − − Q ( η )).B.1. W , ξ and ˜ E Equation determining ξ is − ηξ ′′ ( η ) −
12 (1 − Q ( η )) ξ ′ ( η ) + ˜ E ξ ( η ) = 0 . (B.1)We shall take E >
0, otherwise the problem reduces to the L2 Laguerre case discussedin Sect. 3. The first term of Eq. (B.1) differs in sign from that of the the Laguerreequation (3.2). Suppose we make a parity change η → − η in Eq. (B.1), then we willhave ηξ ′′ ( − η ) + 12 (1 − Q ( − η )) ξ ′ ( − η ) + ˜ E ξ ( − η ) = 0 . (B.2)This equation has the form of Eq. (3.2), provided that ξ ( − η ) ≡ ξ ℓ ( − η ; α ) = L ( α ) ℓ ( η ) , ˜ E = 4 ℓ, Q ( − η ) = 2 (cid:18) η − α − (cid:19) (B.3)for some parameter α . This means ξ ℓ ( η ; α ) = L ( α ) ℓ ( − η ) , (B.4)and Q ( η ) = − (cid:18) η + α + 12 (cid:19) . (B.5)6 C.-L. Ho
For ξ ℓ ( η ; α ) not to have zeros in the ordinary domain [0 , ∞ ), we must have α > − Q ( η ) then leadsto W ( x ) = − x − (cid:18) α + 12 (cid:19) ln x. (B.6)As before we ignore the constant of integration.B.2. p ( η ) , φ ( η ) and E Consider exp( W ) ∝ exp( − x / x − ( α + ) ( α > − x − ( α + ) that could cause φ ( x ) non-normalizable(when α > − /
2) if V ( η ) is a polynomial in η . So we try V = η β U ( η ) where β is areal parameter and U ( η ) a function of η . From Eq. (2.18) we get ηU ′′ + (2 β − α + 1 − η ) U ′ + β ( β − α ) η + E − ˜ E − β − ! U = 0 . (B.7)If β = 0 , α , the η -dependent term in the last term of the above equation can beeliminated, and Eq. (B.7) can be reduced to the Laguerre equation (3.2). As β = 0does not solve our original problem with normalizability of the wave function, weshall take β = α . This leads to U ( η ) = L ( α ) n ( η ) , E ≡ E n = 4( n + α + ℓ + 1) . (B.8)Putting all these result into F ( η ) and G ( η ) gives p ( η ) ≡ p ℓ,n ( η ) = − η α +1 P ℓ,n ( η ; α ) P ℓ,n ( η ; α ) ≡ L ( α ) n ξ ′ ℓ + (cid:16) L ( α ) n − L ′ ( α ) n (cid:17) ξ ℓ = L ( α ) n ξ ′ ℓ + L ( α +1) n ξ ℓ , (B.9)where use has been made of Eqs. (A.1) and (A.2) to get the last line. P ℓ,n ( η ; α )is a polynomial of degree ℓ + n . It will be shown below that it is just the L1type exceptional Laguerre polynomial. It is also easy to check that P ℓ,n ( η ; α )’s areorthogonal with respect to the weight function e − η η ( α +1) ξ ℓ . (B.10)The exactly solvable potential is given by Eq. (2.23) with W ( x ) and ξ ℓ ( η ; α )given by Eqs. (B.6) and (B.4), respectively. The eigenvalues are E n = 4( n + α + ℓ + 1). It is easy to show that V ( x ) is equivalent to the potential for L1 Laguerre case inRef. 4), 5), 11) with α = g + ℓ − / g > φ ℓ,n ( x ; α ) ∝ e − x x ( α + ) ξ ℓ P ℓ,n ( η ; α ) , α > − . (B.11)As in the L2 case, this system reduces to the radial oscillator system in the limit ℓ → repotential approach to solvable rational potentials .... Reducing P ℓ,n ( η ; α ) to HOS form Using Eqs. (A.1) and (A.2), we have ξ ′ ℓ ( η ; α ) = L ( α +1) ℓ ( − η ) − L ( α ) ℓ ( − η ) . (B.12)Then it is easy to check that P ℓ,n ( η ; α ) = L ( α ) n ( η ) ξ ℓ ( η ; α + 1) − L ′ ( α ) n ( η ) ξ ℓ ( η ; α ) (B.13) ξ ℓ ( η ; α + 1) ≡ L ( α +1) ℓ ( − η ) . (B.14)This is, up to a multiplicative constant, the HOS form of the L1 Laguerre polynomial,with the substitution α = g + ℓ − / References
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