A conditionally exactly solvable generalization of the inverse square root potential

TL;DR

This paper introduces a new exactly solvable quantum potential related to the inverse square root potential, providing explicit solutions and a detailed analysis of the energy spectrum using hypergeometric and Hermite functions.

Contribution

It presents a novel conditionally exactly solvable potential extending the inverse square root potential and derives explicit energy spectrum equations using special functions.

Findings

Exact solutions for the wave functions are expressed via confluent hypergeometric functions.

The energy spectrum is determined by roots of an equation involving Hermite functions of non-integer order.

The energy levels correspond to intersections of algebraic and transcendental curves in auxiliary variables.

Abstract

We present a conditionally exactly solvable singular potential for the one-dimensional Schr\"odinger 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.