The Lambert-W step-potential - an exactly solvable confluent hypergeometric potential

TL;DR

This paper introduces an exactly solvable asymmetric step-potential based on the Lambert-W function, providing explicit solutions to the Schrödinger equation and analyzing quantum reflection phenomena.

Contribution

It presents a new exactly solvable potential in quantum mechanics using confluent hypergeometric functions, expanding the class of analytically solvable models.

Findings

The Lambert-W potential yields smaller reflection coefficients compared to traditional step potentials.

The potential is a four-parametric sub-potential of a more general five-parametric solvable potential.

Solutions involve confluent hypergeometric functions and derivatives of bi-confluent Heun functions.

Abstract

We present an asymmetric step-barrier potential for which the one-dimensional stationary Schr\"odinger equation is exactly solved in terms of the confluent hypergeometric functions. The potential is given in terms of the Lambert -function, which is an implicitly elementary function also known as the product logarithm. We present the general solution of the problem and consider the quantum reflection at transmission of a particle above this potential barrier. Compared with the abrupt-step and hyperbolic tangent potentials, which are reproduced by the Lambert potential in certain parameter and/or variable variation regions, the reflection coefficient is smaller because of the lesser steepness of the potential on the particle incidence side. Presenting the derivation of the Lambert potential we show that this is a four-parametric sub-potential of a more general five-parametric one also solvable in terms of the confluent hypergeometric functions. The latter potential, however, is a conditionally integrable one. Finally, we show that there exists one more potential the solution for which is written in terms of the derivative of a bi-confluent Heun function.