Exactly Solvable Discrete Quantum Mechanical Systems and Multi-indexed Orthogonal Polynomials of the Continuous Hahn and Meixner-Pollaczek Types

We present new exactly solvable systems of the discrete quantum mechanics with pure imaginary shifts, whose physical range of the coordinate is the whole real line. These systems are shape invariant and their eigenfunctions are described by the multi-indexed continuous Hahn and Meixner-Pollaczek orthogonal polynomials. The set of degrees of these multi-indexed polynomials are $\{\ell_{\mathcal{D}},\ell_{\mathcal{D}}+1,\ell_{\mathcal{D}}+2,\ldots\}$, where $\ell_{\mathcal{D}}$ is an even positive integer ($\mathcal{D}$ : a multi-index set), but they form a complete set of orthogonal basis in the weighted Hilbert space.


Introduction
Exactly solvable quantum mechanical systems in one dimension are closely related to the orthogonal polynomials. In ordinary quantum mechanics (oQM), whose Schrödinger equation is the second order differential equation, the Hermite, Laguerre and Jacobi polynomials appear in the harmonica oscillator, the radial oscillator and the Darboux-Pöschl-Teller potential, respectively. In discrete quantum mechanics (dQM) [1,2,3], whose Schrödinger equation is the second order difference equation, the Askey-Wilson, q-Racah polynomials etc. appear. Orthogonal polynomials satisfying second order differential or difference equations are severely restricted by the Bochner's theorem and its generalizations, and they are summarized as the Askey scheme of the (basic) hypergeometric orthogonal polynomials [4,5].
We have two types of dQM: dQM with pure imaginary shifts (idQM) and dQM with real shifts (rdQM). The coordinate of idQM is continuous and that of rdQM is discrete.
Recent developments in the theory of orthogonal polynomials and exactly solvable quantum mechanical systems are based on the discovery of new types of orthogonal polynomials: exceptional and multi-indexed polynomials {P D,n (η)|n ∈ Z ≥0 } [6]- [20]. These polynomials satisfy second order differential or difference equations and form a complete set of orthogonal basis in an appropriate Hilbert space in spite of missing degrees. We distinguish the following two cases; the set of missing degrees I = Z ≥0 \{deg P D,n (η)|n ∈ Z ≥0 } is case-(1):  1) is called stable in [11]. Our approach to orthogonal polynomials is based on the quantum mechanical formulation. We deform exactly solvable quantum mechanical systems by multi-step Darboux transformations and obtain multi-indexed polynomials as eigenfunctions of the deformed systems. In the quantum mechanical formulation, the multiindexed orthogonal polynomials appear as polynomials in the sinusoidal coordinate η(x) [21,22], P D,n (η(x)), where x is the coordinate of the quantum system.
The range of the coordinate x of oQM is a finite interval (0, 1 2 π) for the Darboux-Pöschl-Teller potential (Jacobi polynomial), the half real line (0, ∞) for the radial oscillator (Laguerre polynomial) and the whole real line (−∞, ∞) for the harmonic oscillator (Hermite polynomial). The counterparts of idQM to the Jacobi, Laguerre and Hermite polynomials of oQM are Askey-Wilson, Wilson and continuous Hahn polynomials, respectively. Their physical range of the coordinate x is a finite interval (0, π), the half real line (0, ∞) and the whole real line (−∞, ∞), respectively [5]. The situation of the multi-indexed polynomials for oQM and idQM constructed so far is given in Table 1. The case-(2) multi-indexed polynomials are obtained by taking the eigenfunctions as seed solutions of the Darboux transformations, and some eigenvalues are deleted from the original spectrum [23]- [27]. The Darboux transformations with the pseudo virtual state wavefunctions as seed solutions also give the case-(2) multi-indexed polynomials, and some eigenvalues are added to the original spectrum [28]- [31]. The case-(1) multi-indexed polynomials are obtained by taking virtual state wavefunctions as seed solutions of the Darboux transformations, and the deformed systems are isospectral to the original systems [13,15]. For rdQM systems, for example, see [32] for case- (2) and [17,20] for case- (1).
The purpose of the present paper is to study the case-(1) multi-indexed polynomials of idQM systems on the whole real line, namely, the case-(1) multi-indexed polynomials of the continuous Hahn and Meixner-Pollaczek types, which have not been studied except for continuous Hahn ?
: not yet studied

Original Continuous Hahn System
After recapitulating the discrete quantum mechanics with pure imaginary shifts, we present the data of the continuous Hahn system.

Discrete quantum mechanics with pure imaginary shifts
Let us recapitulate the discrete quantum mechanics with pure imaginary shifts (idQM) [2,3].
The dynamical variables of idQM are the real coordinate x (x 1 ≤ x ≤ x 2 ) and the conjugate momentum p = −i∂ x , which are governed by the following factorized positive semi-definite Hamiltonian: Here the potential function V (x) is an analytic function of x and γ is a real constant. The *operation on an analytic function f (x) = n a n x n (a n ∈ C) is defined by f * (x) = n a * n x n , in which a * n is the complex conjugation of a n . Since the momentum operator appears in exponentiated forms, the Schrödinger equation is an analytic difference equation with pure imaginary shifts instead of a differential equation.
Throughout this paper we consider those systems which have a square-integrable groundstate together with an infinite number of discrete energy levels: 0 = E 0 < E 1 < E 2 < · · · . The orthogonality relation reads The eigenfunctions φ n (x) can be chosen 'real', φ * n (x) = φ n (x), and the groundstate wavefunction φ 0 (x) is determined as the zero mode of the operator A, Aφ 0 (x) = 0. The norm of In the following, we assume that the eigenfunctions φ n (x) (2.3) have the following form: where η(x) is a sinusoidal coordinate [21,22] and P n (η) is a orthogonal polynomial of degree n in η. As a polynomial P n (η), we consider the Askey-Wilson, Wilson, continuous Hahn polynomials etc., which are members of the Askey-scheme of hypergeometric orthogonal polynomials [5]. We call the idQM system by the name of the orthogonal polynomial: Askey-Wilson system, Wilson system, continuous Hahn system etc. These idQM systems have the property of shape invariance, which is a sufficient condition for exact solvability. The shape invariant condition is the following [2,22,3]: where κ is a real positive constant and δ is the shift of the parameters. As a consequence of the shape invariance, we have where f n (λ) and b n−1 (λ) are some constants satisfying f n (λ)b n−1 (λ) = E n (λ). These relations can be rewritten as Here the forward and backward shift operators F (λ) and B(λ) are defined by where ϕ(x) is an auxiliary function (ϕ(x) ∝ η(x − i γ 2 ) − η(x + i γ 2 )). The difference operator H(λ) acting on the polynomial eigenfunctions is square root free: (2.12) H(λ)P n (x; λ) = E n (λ)P n (x; λ). (2.13)

Continuous Hahn system
Let us consider the continuous Hahn system. The lower bound x 1 , upper bound x 2 and the parameter γ are (2.14) Namely, the physical range of the coordinate x is a whole real line. A set of parameters λ is λ = (a 1 , a 2 ), a i ∈ C, Re a i > 0. (2.15) Here are the fundamental data [2]: (2.24) (Although the notation b 1 conflicts with b n−1 (λ), we think this does not cause any confusion.) Here p n (η; a 1 , a 2 , a 3 , a 4 ) in (2.20) is the continuous Hahn polynomial [5], and the symbol (a) n is the shifted factorial. Note that φ * 0 (x; λ) = φ 0 (x; λ) andP * n (x; λ) =P n (x; λ). It is not necessary to distinguishP n and P n since η(x) = x, but we will use both notations to compare with other cases in [15].

New Exactly Solvable idQM Systems and Multi-indexed Continuous Hahn Polynomials
In this section we deform the continuous Hahn system by applying the multi-step Darboux transformations with the virtual state wavefunctions as seed solutions. The eigenfunctions of the deformed systems are described by the case-(1) multi-indexed continuous Hahn polynomials.
) was used to construct the pseudo virtual state wavefunctions [31]. For each of t I and t II , the potential function V (x; λ) satisfies where α(λ) and α ′ (λ) are .

New exactly solvable systems
By applying multi-step Darboux transformations to the continuous Hahn system in § 2.2, we can deform it and obtain new exactly solvable idQM systems. The virtual state wavefunctions in § 3.1 are used as seed solutions, and new systems are isospectral to the original one.
The deformed systems are labeled by , which are the degrees and types of the virtual state wavefunctions used in M-step Darboux transformations. Exactly speaking, D is an ordered set. Various quantities of the deformed systems are denoted as H D , φ D n , A D , etc. The general formula is as follows [15]: (3.10) To obtain the concrete forms of V D (x) and φ D n (x), we have to evaluate the Casoratians.

11)
Let us define the following functions: whose explicit forms are Furthermore, let us defineΞ D (x; λ) andP D,n (x; λ) as follows: where A and B are and X ; λ). (3.23) We call Ξ D (η; λ) the denominator polynomial and P D,n (η; λ) the multi-indexed polynomial.
If the deformed systems is well-defined, the general formula gives the orthogonality of the eigenfunctions [15]: Namely, the orthogonality relations of the multi-indexed polynomialsP D,n (x; λ) are The multi-indexed orthogonal polynomial P D,n (η; λ) has n zeros in the physical region η ∈ R (⇔ η(x 1 ) < η < η(x 2 )), which interlace the n + 1 zeros of P D,n+1 (η; λ) in the physical region, and (3.19) are slightly simplified, where r I j (x) = r I j (x; λ, M + 1). The cases of type II only (M I = 0, M II = M) are similar.

Shape invariance
The shape invariance of the original system is inherited by the deformed systems. By the same argument of [15], the Hamiltonian H D (λ) is shape invariant: As a consequence of the shape invariance, the actions of A D (λ) and A D (λ) † on the eigen- The forward and backward shift operators are defined by The similarity transformed Hamiltonian is square root free: 40) and the multi-indexed orthogonal polynomialsP D,n (x; λ) are its eigenpolynomials: H D (λ)P D,n (x; λ) = E n (λ)P D,n (x; λ).

Limit from the Wilson system
The continuous Hahn polynomial can be obtained from the Wilson polynomial [5]. The potential function V (x; λ), the energy eigenvalue E n (λ), the sinusoidal coordinate η(x) and the eigenfunctions φ n (x; λ) of the Wilson system are [15] λ = (a 1 , a 2 , a 3 , a 4 ), where W n (η; a 1 , a 2 , a 3 , a 4 ) is the Wilson polynomial [5].
Let us consider the following limit: Here the superscript W indicates the quantities of the Wilson system, and x, a 1 and a 2 are quantities of the continuous Hahn system. The physical range of x W (0 ≤ x W < ∞) gives the physical range of x (−∞ < x < ∞). The continuous Hahn polynomial is obtained from the Wilson polynomial [5], Other quantities are also obtained: lim t→∞ e π 2 (Im (a 1 +a 2 )+2t) The continuous Hahn system is obtained from the Wilson system by the limit (3.45).
Next let us consider the deformed case. We can show that the denominator polynomiaľ Ξ D (x; λ) and the multi-indexed polynomialsP D,n (x; λ) of the continuous Hahn type are obtained from those of the Wilson type by the same limit (3.45): where explicit forms ofΞ W D andP W D,n are found in [15]. We remark that the denominator polynomialΞ D (x; λ) is obtained as (3.48) algebraically, but the condition (3.29) is not inherited from that of the Wilson type in general. Therefore the limit (3.45) of the deformed Wilson systems do not give the deformed continuous Hahn systems in general.

New Exactly Solvable idQM Systems and Multi-indexed Meixner-Pollaczek Polynomials
In this section we deform the Meixner-Pollaczek system. Since the method is the same as in § 3, we present results briefly. The eigenfunctions of the deformed systems are described by the case-(1) multi-indexed Meixner-Pollaczek polynomials.

Virtual state wavefunctions
Let us introduce a twist operation t, which is an involution t 2 = id and satisfies t(λ + βδ) = t(λ) + βδ (β ∈ R). The potential In the following, we assume a > 1 2 , which gives α ′ (λ) < 0. The virtual state wavefunctions which satisfy the Schrödinger equation The virtual state polynomial ξ v (η; λ) is a polynomial of degree v in η, and the virtual energyẼ v (λ) is Let us define the following functions: whose explicit form is The denominator polynomial and the multi-indexed polynomial are defined by (3.23) and , .
To check the regularity and hermiticity of H D (λ), let us consider the function g(x), . The necessary and sufficient condition for g(x) to have no poles in the rectangular domain D γ is a − 1 2 M > 0. This condition is automatically satisfied because of (4.11). By the same argument as § 3.2, the deformed Hamiltonian H D (λ) is well-defined and hermitian, if the condition (3.29) is satisfied. To satisfy the condition (3.29), the degree of Ξ D (η; λ), ℓ D , should be even. Although we have no analytical proof that there exists a range of parameters λ satisfying the condition (3.29), numerical calculation (for small M and d j ) suggests the following conjecture.
In the following we assume the condition (3.29) is satisfied.

Limit from the continuous Hahn system
The Meixner-Pollaczek polynomial can be obtained from the continuous Hahn polynomial [5]. Let us consider the following limit:  Other quantities are also obtained:

Limit to the harmonic oscillator
The harmonic oscillator is an ordinary quantum mechanical system and its eigenfunctions are the following: where H n (η) is the Hermite polynomial [5].
Let us consider the following limit of the Meixner-Pollaczek system: We remark that the Meixner-Pollaczek polynomial P (a) n (x; φ) with any φ reduces to the Hermite polynomial as [5] lim t→∞ n! t n 2 but this limit does not lead to a good limit of the quantum system.
Next let us consider the deformed case. There is no virtual state in the harmonic oscillator [29]. Hence the limit (4.28) of the virtual state wavefunction of the Meixner-Pollaczek system can not be a virtual state wavefunction. In fact, the limit ofφ MP v is This is the pseudo virtual state wavefunction of the harmonic oscillatorφ v (x) = i −v φ v (ix) [29]. The deformed harmonic oscillator system, which is obtained by the Darboux transformations withφ v (v ∈ D) as seed solutions, has energy eigenvalues E n (n = 0, 1, . . .) andẼ d j (j = 1, . . . , M). The eigenfunctions with E n are obtained as the limit of φ MP D n , but those with E d j can not obtained from the eigenfunctions of the deformed Meixner-Pollaczek system. In this sense the limit (4.28) of the deformed Meixner-Pollaczek system is not a good limit.

Summary and Comments
The continuous Hahn and Meixner-Pollaczek idQM systems are exactly solvable and their physical range of the coordinate is the whole real line. We deform them by the multistep Darboux transformations with the virtual state wavefunctions as seed solutions, and obtain new exactly solvable idQM systems and the case-(1) multi-indexed continuous Hahn and Meixner-Pollaczek polynomials. By this result, the construction of the multi-indexed polynomials in idQM is essentially completed. The remaining task is to study the properties of various multi-indexed polynomials and to use them to investigate quantum mechanical systems. The deformed quantum system labeled by an index set D may be equivalent to another labeled by a different index set D ′ with shifted parameters, which means that the corresponding two multi-indexed orthogonal polynomials labeled by D and D ′ with shifted parameters are proportional. Such equivalence is studied for the case-(2) multi-indexed polynomials of Hermite, Laguerre, Jacobi, Wilson and Askey-Wilson types [29,31] and for the case-(1) multi-indexed polynomials of Laguerre, Jacobi, Wilson and Askey-Wilson types [33] (see also [34]). The case-(1) multi-indexed continuous Hahn polynomials obtained in this paper have equivalence in the same form as the (Askey-)Wilson cases [33], which is derived from the properties (A.4)-(A.5). The case-(1) multi-indexed Meixner-Pollaczek polynomials also have similar equivalence derived from the property (B.4).
The multi-indexed orthogonal polynomials do not satisfy the three term recurrence relations, which are characterizations of the ordinary orthogonal polynomials [4], because they are not ordinary orthogonal polynomials. Instead, they satisfy the recurrence relations with more terms [35]- [43]. The case-(1) multi-indexed continuous Hahn and Meixner-Pollaczek polynomials satisfy such recurrence relations. The recurrence relations with constant coefficients are related to the generalized closure relations [42], which give the creation and annihilation operators of the deformed quantum systems. We will report these topics elsewhere [44].