A conditionally exactly solvable generalization of the inverse square root potential

Abstract

We present a conditionally exactly solvable singular potential for the one-dimensional Schrödinger equation which involves the exactly solvable inverse square root potential. Each of the two fundamental solutions that compose the general solution of the problem is given by a linear combination with non-constant coefficients of two confluent hypergeometric functions. Discussing the bound-state wave functions vanishing both at infinity and in the origin, we derive the exact equation for the energy spectrum which is written using two Hermite functions of non-integer order. In specific auxiliary variables this equation becomes a mathematical equation that does not refer to a specific physical context discussed. In the two-dimensional space of these auxiliary variables the roots of this equation draw a countable infinite set of open curves with hyperbolic asymptotes. We present an analytic description of these curves by a transcendental algebraic equation for the involved variables. The intersections of the curves thus constructed with a certain cubic curve provide a highly accurate description of the energy spectrum.

A conditionally integrable Schrödinger potential of a bi-confluent Heun class

“…The potential we have introduced belongs to the first Lemieux-Bose family which involves negative integer and half-integer power-law terms. This is a remarkable family the known members of which include, for instance, the second Stillinger [4] and the first Exton [7] potentials as well as the inverse square root potential [1], the potential by López-Ortega [11] and the recent generalization of the latter two potentials [12].…”

“…The next come the potentials for which the fundamental solutions present irreducible combinations of two Hermite functions. These are the two Exton potentials [7] the first of which involves the inverse square root potential [1] and its conditionally integrable generalization [12] which includes the López-Ortega supersymmetric pair of potentials [11].…”

“…The term "conditionally integrable" is conventionally used to indicate that the parameters involved in the potential are not varied independently or a potential parameter is fixed to a constant. Well known examples of such potentials discussed by many authors in the past are the two Stillinger confluent hypergeometric potentials [4] and the Dutt-Khare-Varshni ordinary hypergeometric potential [5] (for other examples see, e.g., [6][7][8][9][10][11][12]).…”

Exact solutions of the sextic oscillator from the bi-confluent Heun equation

“…the (confluent) hypergeometric function. Inspired by previous results [21,23], the solutions of the bi-confluent Heun equation (11) can be sought for as an expansion in terms of Hermite functions possessing shifted and scaled argument [15]:…”

“…Other examples involving truncated series solutions of the Heun equations in terms of functions of the hypergeometric class are presented in Refs. [20][21][22][23][24][25] The concept of solvability has also been extended, revealing further aspects of quantum mechanical potentials problems. In the case of conditionally exactly solvable (CES) potentials, exact solutions are obtained only for potentials in which some of the potential parameters are correlated, or are restricted to constant values.…”