A singular Lambert-W Schr\"odinger potential exactly solvable in terms of the confluent hypergeometric functions

TL;DR

This paper introduces two exactly solvable one-dimensional Schr"odinger potentials involving the Lambert-W function, with solutions expressed via confluent hypergeometric functions, including a singular potential with finite bound states.

Contribution

The paper presents new Lambert-W based potentials for the Schr"odinger equation with solutions in terms of confluent hypergeometric functions, including a novel singular potential.

Findings

Exact solutions are expressed through confluent hypergeometric functions.

One potential is singular, behaving as inverse square root near the origin.

The singular potential supports only a finite number of bound states.

Abstract

We introduce two potentials explicitly given by the Lambert-W function for which the exact solution of the one-dimensional stationary Schr\"odinger equation is written through the first derivative of a double-confluent Heun function. One of these potentials is a singular potential that behaves as the inverse square root in the vicinity of the origin and vanishes exponentially at the infinity. The exact solution of the Schr\"odinger equation for this potential is given through fundamental solutions each of which presents an irreducible linear combination of two confluent hypergeometric functions. Since the potential is effectively a short-range one it supports only a finite number of bound states.