On an Energy-Dependent Quantum System with Solutions in Terms of a Class of Hypergeometric Para-Orthogonal Polynomials on the Unit Circle

: We study an energy-dependent potential related to the Rosen–Morse potential. We give in closed-form the expression of a system of eigenfunctions of the Schrödinger operator in terms of a class of functions associated to a family of hypergeometric para-orthogonal polynomials on the unit circle. We also present modiﬁed relations of orthogonality and an asymptotic formula. Consequently, bound state solutions can be obtained for some values of the parameters that deﬁne the model. As a particular case, we obtain the symmetric trigonometric Rosen–Morse potential for which there exists an orthogonal basis of eigenstates in a Hilbert space. By comparing the existent solutions for the symmetric trigonometric Rosen–Morse potential, an identity involving Gegenbauer polynomials is obtained.


Introduction
An energy-dependent Schrödinger equation appears for the first time in relativistic quantum mechanics with the Pauli-Schrödinger equation, given by Pauli [1] in the description of the spectrum of an electron in the presence of a magnetic field. Further developments in relativistic and non relativistic quantum mechanics was made by many authors [2][3][4][5][6][7][8][9][10]. The list is by no means exhaustive.
These classes of quantum potentials appear frequently in many areas of quantum mechanics. A relativistic scalar particle in presence of an electromagnetic field can be studied by means of a Klein-Gordon equation with an energy dependent potential [4,8]. In [11], the authors have applied energy dependent potentials with emphasis on confining potentials to the description of heavy quark systems. Furthermore, the description of systems of N bosons bound is considered in [12] and the Hamiltonian formulation of the relativistic many-body problem in [4,5,13] also lead to energy dependent potential models. For physical applications in hydrodynamics, see [14].
Mathematical aspects of wave equations with energy-dependent potentials have been studied by several authors. The presence of an energy-dependence in the potential in the nonrelativistic context requires a modification of the underlying quantum theory, principally affecting orthogonality relation and norm [15]. An analogous modification is required in the relativistic framework, [16]. An extension of the quantum mechanical formalism of systems with energy-dependent potentials to systems defined by generalized Schrödinger equations including a position-dependent mass was studied in [7]. Energy-dependence in the framework of noncommutative quantum mechanics has been recently considered in [17].
The search of solutions for energy dependent potentials in wave equations has attracted considerable attention since the appearance of Pauli's work. In the present manuscript we study a quantum system with energy dependent potential related to the Rosen-Morse trigonometric potential, used in describing the interatomic interaction of linear molecules and for describing polyatomic vibration states and energies of the NH 3 molecule [18]. It has long been known that some mathematical features of quantum systems with an energy-dependent potential have several non-trivial implications; for instance, it is necessary to modify the scalar product to guarantee the conservation of the norm [15,19]. In the present manuscript we give closed-form of solutions, modified relations of orthogonality given by a indefinite (in general) bilinear form and an asymptotic formula of the solutions. Consequently, bound state solutions can be obtained for some values of the parameters that define the model. The solutions are given in terms of a class of functions derived from a sequence of hypergeometric para-orthogonal polynomials on the unit circle. We obtain, as a particular case, the symmetric trigonometric Rosen-Morse potential. In such case, the solutions reduce to an orthogonal basis of eigenfunctions defined in a Hilbert space and are expressed in terms of the Gegenbauer or ultraspherical polynomials. By comparing this solution with other solutions given in the literature we obtain, as a consequence, an identity involving Gegenbauer polynomials. Our procedure to obtain the energy dependent potential is based in a classical technique developed by Bose in [20] to construct solvable one-variable Schrödinger potentials.
The manuscript is organized as follows. In Section 2 we give some background, notations and statement of the results, in Section 3 we give the proofs and in Section 4 we present a discussion and concluding remarks.

Basic Notations and Statement of the Results
Let µ be a measure on the unit circle T = ∂D, D = {z ∈ C : |z| < 1} with support consisting of an infinite number of points. We remind that (φ n ) ∞ n=0 is the sequence of orthonormal polynomials on the unit circle associated to µ (also termed as Szegő polynomials, after their introduction by G. Szegő where φ n (z) = κ n z n + a n−1 z n−1 + lower order terms and κ n > 0.
An exposition of the theory of orthogonal polynomials systems on the unit circle is presented in the monographs [21][22][23]. More recent surveys in [24,25].
If P n is a polynomial of degree n, the reciprocal polynomial P * n is defined as z n P n (1/z), or equivalently a k z k and a n = 0.
Let 2 F 1 (a, b; c; z) denote the Gauss hypergeometric function of the variable z with parameters a, b, c ∈ C, c / ∈ Z ≤0 ; cf. [26] (p. 56), given by for z ∈ D and for other values of z ∈ C by analytic continuation appropriately; the Pochhammer symbol is defined by (a) n = a(a + 1) · · · (a + n − 1), (a) 0 = 1.
For α ∈ C, the function z α will be defined on the branch for which arguments are restricted between −π and π. We also denote by x the floor function, defined as the greatest integer less than or equal to x.
A fundamental role in this manuscript is played by the sequence of functions where λ, η and x are such that b = λ + ηı and 2x = √ z + 1 √ z , z = e ıθ , θ ∈ [0, 2π]. The functions G n were introduced in [27] and are defined from the sequence (R n ) ∞ n=0 of para-orthogonal polynomials, cf. [28] associated to the Szegő hypergeometric polynomials, cf. [29,30] These polynomials satisfy the orthogonality relations in the unit disk through the parametrization here the constant is such that the moment µ In the sequel, we denote by S n and s n the monic and orthonormal polynomials of degree n respectively associated to S n . From (4) it follows that is the main coefficient of s n . It should be noted that the R n polynomials are of hypergeometric type. According to [30] (Th. 5.1), one has This last relation can also be extended to for n ≥ 1.
In the present manuscript we prove the following results.
Theorem 1. Let λ > − 1 2 , η ∈ R. Then, the stationary one-dimensional Schrödinger equation with energy dependent potential where V(θ, E; λ, η) = −V 0 (θ; λ, η) − (−λ + E(n; λ, η) + η 2 )V 1 (θ; λ, η), has by solution the system of wave functions The usual continuity equation where P denotes the probability density and J the probability current, governs the conservation of mass, charge, and probability of any closed system. If the potential is energy-dependent, it is necessary to modify the definition of the usual orthogonality relations in order to satisfy the continuity equation, [5,15]. More precisely, let (Ψ n ) ∞ n=0 be a system of normalizable wave functions solutions of an energy dependent potential Schrödinger equation defined through the boundary value problem where V is of class C 1 (I) with respect to the variable E, being I an open interval of the real line and a, b ∈ R.
The continuity equation read as and is satisfied by the probability density P and probability current J The orthogonality relation between two states n and m, n = m reads as now, by using the smooth dependence of V in relation to E one obtains In that regard, for the present quantum model we have the following relations of orthogonality. Notice that the presence of the function cot θ in the definition implies that the associated bilinear form is not in general of a definite sign. When |η| < λ, we have bound states solutions.
This potential has also been studied in [36,37] and solved in terms of the real Romanovski (also known as Romanovsky-Routh or Pseudo-Jacobi polynomial as in [40]). These polynomials are defined as the polynomial solution of degree n of the differential equation and can be expressed in terms of the Jacobi polynomials P For convenience, we will adopt the parametrization given in [41] for the solution to (10) in terms of the real Romanovski polynomials (expressed also in terms of author's parametrization), which reads for n ≥ 0, and the corresponding energies In particular, the symmetric trigonometric Rosen-Morse is obtained by taking η = 0 in (10). In such case we have that is a solution for the Schrödinger equation associated to the symmetric Rosen-Morse potential. By identifying the parameters λ = a we obtain is also a solution of (9), which coincides, up to a multiplicative factor with Ψ n , as shows the following identity Theorem 3. Let λ ∈ C and n ∈ N ∪ {0}. Then, For those values of λ, say λ = λ 0 , for which we have a zero or pole in the expressions (λ) n the formula may be interpreted as a limit when λ → λ 0 . In such cases, the limit exists and is finite.
The function G n (x) satisfy the differential equation, [27] (Th. 2.2) When η = 0, we obtain the differential equation that defines the ultraspherical polynomials cf. [23] (p. 80). Notice that from (1) and (2), G n reduces to a polynomial. Therefore, G n coincides, up to a constant factor, with the n degree Gegenbauer polynomial.

Remark 1.
We remark that in [27] (Th. 2.2), the term m should be corrected in the last summand of the left hand side of the differential equation.

The Schrödinger Invariant of a Second Order Differential Equation
Quantum systems with energy-dependent potentials have been studied following several approaches such as supersymmetric quantum mechanics, Darboux transformations, exceptional orthogonal polynomials, among others, see [7] for a review. In this subsection we summarize the method we followed, introduced by Bose in [20] in order to construct one-variable Schrödinger solvable potentials.
Let us have a second order differential equation being I an open subset of the real line and where p and q are functions defined on I. A straightforward calculation shows that the middle term in (19) can be eliminated by taking the substitution Under the above substitution, the Equation (19) transforms to the canonical form where I is given by the term I is named by Milson in [43] as the Bose invariant.
By applying now the transformation v = √ x ψ, x = dx dθ , we obtain the normal form the Schrödinger invariant, named by Bose in [20] and {x, θ} is the Schwartzian derivative

Proof of Theorem 2.
On the one hand, by applying iterated integration to the left hand side of (7) and taking into account the definition of P we obtain On the other hand, for the right hand side of (7) we have Now, from the definition of J and the fact that Ψ n (0; λ, η) = Ψ n (π; λ, η) = 0 when λ > 0 one has Hence, from (27), (28) and (29) By substituting the value of V given in Theorem 1 we obtain (8).
To evaluate the numerical value of the constant c n , notice that if λ > 1 2 , η ∈ R, from the recurrence relations (15) and (16) we have This completes the proof of the theorem.
From the orthogonality relation for Gegenbauer polynomials one has that Ψ n (θ; λ) is an orthogonal system in the Hilbert space L 2 [0, π] with the scalar product Proof of Theorem 3. From Corollary 1 and the relation (14) we have that Φ n and Ψ n are solutions of the differential equation hence, the transformations (21) and (23) give that are solutions of the differential equation notice that y 1 reduces, up to a constant factor, to the Gegenbauer polynomial of degree n.
From (14) and (31) one has and from [44] [(6.4.12) p. 303], Notice that the relation (34) defines the left hand side as a polynomial of degree n whose coefficients are rational functions of the variable λ varying in C. Now, (33) is a solution to (32). Since this solution is a polynomial of degree n, it follows from [23] (Th. 4.2.2 p. 61) that Let us consider n fixed. Since the coefficients of y 2 and y 1 are rational functions of λ and are equal when [λ] ∈ [ 3 2 , +∞), it follows from [45] (Th. 17.1 p. 369) that the relation is valid for λ ∈ C, with exception of a finite number of special values of λ. Notice that from [44] [(6.4.12) p. 303], the zeros of the functions (−n − λ + 1 2 ) n and (λ) n are removable singularities of the left hand side and right hand side respectively of (36). Furthermore, when λ is a zero of (λ + 1 2 ) n 2 or (1 − n − λ) n 2 we have simple poles in the main coefficients of the left hand side or right hand side accordingly. For these values of λ 0 , the formula may be interpreted as a limit as λ → λ 0 . This completes the proof of the theorem.

Discussion
The present work is devoted to the study an energy-dependent potential related to the Rosen-Morse potential. The system is obtained by the addition of a potential term which depends on the function cot θ and an energy dependence through a square root.
In order to show some numerical comparisons we will identify accordingly the parameters λ, η of the quantum model given by Theorem 1 and the parameters a, b of the asymmetric trigonometric Rosen-Morse model, following the form given in [41]. In effect, by identifying the parameters one has b = −λη, a = λ. Consequently, the quantum system defined by Theorem 1, in terms of the parameters a and b reads as, which has, when n ∈ N ∪ {0}, the system of solutions and the corresponding energy levels, On the other hand, following Theorem 3, we will multiply by an adequate numerical constant γ n the expression that defines the Φ n functions (12), By (11) and (12), we have that the rescaled function Φ n , n ∈ N ∪ {0} can be expressed as Φ n (θ; a, b) = γ n (−ı) n P (−n−a− ıb a+n ,−n−a+ ıb a+n ) n (ı cot θ)e − bθ n+a sin (a+n) θ = γ n (−ı) n −a − ıb a+n n 2 F 1 −n, 1 − n − 2a; 1 − n − a − ıb a + n ; 1 − ı cot θ 2 e − bθ n+a sin (a+n) θ.
In Figures 1 and 2 we plotted the wave functions for several toy values of the parameters.
In Figures 3 and 4 we plotted the densities 1 + b cot θ a(n+a) |Ψ n (θ; a, b)| 2 (see Theorem 2) and |Φ n (θ; a, b)| 2 for the same values of the parameters a, b with n fixed (n = 5). As can be appreciated, by fixing b and making a variable, the abscissas of the local maxima of the densities tend to be localized at the same points in both models, as a increases. These points correspond to the regions where it is most likely the particle to be found. It should be also noticed from the expressions of the energies (13) and (42) that lim a→∞ E Ψ (n; a, b) − E Φ (n; a, b) = 0, for b and n fixed. It could be interesting the further study of these facts.