We introduce two potentials explicitly given by the Lambert-W function for which the exact solution of the one-dimensional stationary Schrödinger 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 Schrödinger 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.
- confluent hypergeometric function
- double-confluent Heun equation
- integrable potentials
- Stationary Schrödinger equation
ASJC Scopus subject areas
- Nuclear and High Energy Physics
- Astronomy and Astrophysics