Unified derivation of exact solutions for a class of quasi-exactly solvable models

Abstract

We present a unified treatment of exact solutions for a class of four quantum mechanical models, namely the anharmonic singular potential, the generalized quantum isotonic oscillator, the soft-core Coulomb potential, and the non-polynomially modified oscillator. We show that all four cases are reducible to the same basic ordinary differential equation, which is quasi-exactly solvable. A systematic and closed form solution to the basic equation is obtained via the Bethe ansatz method. Using the result, general exact expressions for the energies and the allowed potential parameters are given explicitly for each of the four cases in terms of the roots of a set of algebraic equations. A hidden sl(2) algebraic structure is also discovered in these models.

On the solvability of the generalized hyperbolic double-well models

“…1, to have no polynomial solutions, with the exception of the V 4 6 (x) subclass. To obtain the solution to our model, we employ the Bethe ansatz method [3][4][5] (and the Appendix for generalization) which has proven to be very effective in solving differential equations more general than the Heun class of equation.…”

Solvable potentials with exceptional orthogonal polynomials

“…The one-dimensional Schrödinger equation with the nonlinear oscillator potential (in the units = 2m = 1) − d 2 dx 2 + x 2 + 8(2x 2 − 1) (2x 2 + 1) 2 ψ n (x) = E n ψ n (x), ψ n (± ∞) = 0, −∞ < x < ∞, (1) has attracted attention over the past couple of decades [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15]. The special interest arose from the structure of the exact closed-form solutions given by E n = 2n − 3, ψ n (x) = e −x 2 /2 1 + 2x 2 P n (x), n = 0, 3, 4, 5, .…”

New quasi-exactly solvable class of generalized isotonic oscillators

“…However, the recently introduced functional Bethe ansatz method [15] (also see appendix for generalization) has proven very effective in obtaining exact closed-form polynomial solutions to many QES quantum mechanical model [16]- [18]. The aim of this paper is to obtain exact solutions of a class of quantum models based on the pseudo-Hermite EOP by means of the Bethe ansatz method [15].…”