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Prepotential approach to solvable rational extensions of Harmonic Oscillator and Morse potentials

We show how the recently discovered solvable rational extensions of Harmonic Oscillator and Morse potentials can be constructed in a direct and systematic way, without the need of supersymmetry, shape invariance, Darboux-Crum and Darboux-B\"acklund transformations


II. PREPOTENTIAL APPROACH
The main ideas of the prepotential approach are summarized here. We refer the reader to [18] for the details of the procedure. We adopt the unit system in which and the mass m of the particle are such that = 2m = 1.
Consider a wave function φ(x) which is written in terms of a function W (x) as φ(x) ≡ exp(W (x)). The function W (x) is assumed to have the form W (x, η) = W 0 (x) − ln ξ(η) + ln p(η). (1) Here η(x) is a function of x which we shall choose to be one of the sinusoidal coordinates, i.e., coordinates such thaṫ η(x) 2 , where the dot denotes derivative with respect to x, is at most quadratic in η.
The functions W 0 (x), ξ(η) and p(η) are functions to be determined later. We shall assume ξ(η) to be a polynomial in η. The wave function is Operating on φ(x) by the operator −d 2 /dx 2 results in a Schrödinger equation Hφ = 0, where H = −d 2 /dx 2 +V ,V ≡ W 2 +Ẅ . For simplicity of presentation, we shall often leave out the independent variable of a function if no confusion arises.
Since W (x) determines the potentialV , it is therefore called the prepotential. To makeV exactly solvable, we demand that: (1) W 0 is a regular function of x, (2) the function ξ(η) has no zeros in the the ordinary (or physical) domain of η(x), and (3) the function p(η) does not appear inV .
For ξ(η) = 1, the prepotential approach can generate exactly and quasi-exactly solvable systems associated with the classical orthogonal polynomials [25]. The presence of ξ in the denominators of φ(x) and V (x) thus gives a rational extension, or deformation, of the traditional system. We therefore call ξ(η) the deforming function.
If the factor exp(W 0 (x))/ξ(η) in Eq. (2) is normalizable, then p(η) = constant (in this case we shall take p(η) = 1 for simplicity) is admissible. This gives the ground state However, if exp(W 0 (x))/ξ(η) is non-normalizable, then φ 0 (x) cannot be the ground state. 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 orthogonal polynomials arise [18]. The cases considered in [23], which we shall rederived by means of the prepotential approach in this paper, are such that exp(W 0 (x))/ξ(η) is normalizable and thus φ 0 (x) is the ground state. Following the procedure in [18], we assume ξ(η) to satisfy the equation Here the prime denotes derivative with respective to η. We choose c 2 (η) = ±η 2 , and c 1 is determined by where Q(η) ≡Ẇ 0η . The function E(η) was taken to be a real constant in [18], but here we allow the possibility that E(η) may be a function of η. Nonetheless, it turns out that all formulae presented in [18] remain intact. By matching Eq. (4) with the (confluent) hypergeometric equation, one determinesẼ, Q(η) and ξ(η). Integrating Q(x) =Ẇ 0η then gives the prepotential W 0 (x): The function p(η) is then given by a linear combination of ξ and ξ ′ : Here the two functions F (η) and G(η) are determined by with the function V(η) satisfying By matching Eq. (10) with the (confluent) hypergeometric equation, one determines V, and thus F (η), G(η), p(η) and E. Once all the relevant functions and parameters are determined, we would have constructed an exactly solvable quantal system Hφ = Eφ defined by H = −d 2 /dx 2 + V (x), with the wave function (2) and the potential Lastly, we note that the functions p E (here we add a subscript to distinguish p corresponding to a particular eigenvalue E) are orthogonal , i.e., in the η-space with the weight function
Eq. (4) becomes Comparing Eq. (14) with the Hermite equation where H ℓ (η) is the Hermite polynomial, we would have But this choice is not viable, as ξ ℓ (η; α) = H ℓ (η) has zeros in the ordinary domain (−∞, ∞), which we want to avoid. A simple way to solve this is to make the zeros of Hermite polynomials lie on the imaginary axis. This is achieved if we set η → iη in Eq. (15), giving Matching Eq. (14) with (17) gives One notes that the Hermite polynomials are odd functions in η for odd ℓ. So in this case ξ ℓ has a zero at η = 0. Further study of this case reveals that the wave functions are not normalizable. So here we shall only consider the case with even ℓ = 2m(m = 1, 2, . . .). By Eq. (6), the form of Q(η) leads to We shall ignore the constant of integration as it can be absorbed into the normalization constant. As noted in Sect. II, in this case p(η) = 1 is admissible, as exp(W 0 (x))/ξ(η) is normalizable. So the energy and eigenfunction of the ground state of this system are E 0 = 0 and φ 0 (x) = exp(W 0 (x))/ξ(η). Below we determine the energies and eigenfunctions of the excited states. With the solutions in Eq. (18), Eq. (10) becomes Comparing Eqs. (20) and (15) (with ℓ replaced by n = 0, 1, 2, . . .), we get From Eqs. (8) and (9), one eventually obtains Use has been made of the identity H ′ n = 2ηH n − H n+1 in obtaining the last line in Eq. (22). We note that p ℓ,n (η) is a polynomial of degree ℓ + n + 1.

IV. MORSE POTENTIAL
Now we consider rational extension of the Morse potential. It turns out that in this caseẼ cannot be a constant. Let us choose η(x) = e −x ∈ (0, ∞), withη 2 = η 2 . For definiteness we shall take the upper signs for c 2 and c 1 , as the lower signs lead to the same results. So we have c 2 (η) = η 2 and c 1 = (η − 2Q(η)).
Putting all these results into F (η) and G(η) gives p(η) ≡ p ℓ,n (η; α) = η −n−ℓ−1 P ℓ,n (η; α) P ℓ,n (η; α) is a polynomial of degree ℓ + n + 1. It is also easy to check that p ℓ,n (η; α)'s are orthogonal with respect to the weight function The exactly solvable potential is given by The complete eigenfunctions are where p ℓ,n (η; α)'s are given in (40). The corresponding eigen-energies are E 0 = 0 and E ℓ,n in (39), respectively. For the wave functions to be regular at x = 0, one must have n < −α/2 − ℓ − 1. This means the system admits only finite number of bound states.

V. SUMMARY
We have shown how the recently discovered solvable rational extensions of Harmonic Oscillator and Morse potentials can be constructed in a direct and systematic way, without the need of supersymmetry, shape invariance, Darboux-Crum and Darboux-Bäcklund transformations. In our approach, the prepotential, the deforming function, the potential, the eigenfunctions and eigenvalues are all derived within the same framework.
With the results given here and in [18], rational extensions of all well-known one-dimensional solvable quantal systems based on sinusoidal coordinates have been generated by the prepotential approach. One would like to apply the same approach to find rational extensions of the other solvable models based on non-sinusoidal coordinates, following the work of the third paper in [25]. Unfortunately, such way of rational extensions only lead to quasi-exactly solvable systems, because the energy quantum number n will appear in the x-dependent terms in V (x). A non-trivial generalization of the present approach may be in order, which we hope to report in the near future.