Exact solution of the Schr\"odinger equation for a short-range exponential potential with inverse square root singularity

TL;DR

This paper presents an exact analytical solution to the Schrödinger equation for a novel short-range potential with an inverse square root singularity, expanding the class of solvable quantum models.

Contribution

It introduces a new exactly solvable potential within the Heun family, with solutions expressed via hypergeometric functions, and derives the exact energy spectrum.

Findings

Exact spectrum equation derived for the potential

Potential supports bound states with specific energy levels

Solution expressed through hypergeometric functions

Abstract

We introduce an exactly integrable singular potential for which the solution of the one-dimensional stationary Schr\"odinger equation is written through irreducible linear combinations of the Gauss hypergeometric functions. The potential, which belongs to a general Heun family, is a short-range one that behaves as the inverse square root in the vicinity of the origin and vanishes exponentially at the infinity. We derive the exact spectrum equation for the energy and discuss the bound states supported by the potential.