A Schr¨odinger Potential Involving x 2/3 and Centrifugal-Barrier terms Conditionally Integrable in Terms of the Confluent Hypergeometric Functions

Abstract

The solution of the one-dimensional Schr¨odinger equation for a potential involving an attractive ∼ x 2/3 and a repulsive centrifugal-barrier ∼ x −2 terms is presented in terms of the non-integer-order Hermite functions. The potential belongs to one of the five biconfluent Heun families. This is a conditionally integrable potential in that the strength of the centrifugal-barrier term is fixed to 91~2/(72m). The general solution of the problem is composed using fundamental solutions each of which presents an irreducible linear combination of two Hermite functions of a scaled and shifted argument. The potential presents an infinitely extended confining well defined on the positive semi-axis and sustains infinitely many bound states.