The third five-parametric hypergeometric quantum-mechanical potential

TL;DR

This paper introduces a new five-parametric hypergeometric quantum potential that can model both short-range wells and asymmetric step barriers, with solutions expressed via hypergeometric functions.

Contribution

It presents the third such potential with a shape independent of a parameter, expanding the class of exactly solvable quantum potentials.

Findings

Potential can model different physical scenarios (well or barrier).

Solutions are expressed in terms of hypergeometric functions.

Potential shape remains invariant under parameter changes.

Abstract

We introduce the third five-parametric ordinary hypergeometric energy-independent quantum-mechanical potential, after the Eckart and P\"oschl-Teller potentials, which is proportional to an arbitrary variable parameter and has a shape that is independent of that parameter. Depending on an involved parameter, the potential presents either a short-range singular well (which behaves as inverse square root at the origin and vanishes exponentially at infinity) or a smooth asymmetric step-barrier (with variable height and steepness). The general solution of the Schr\"odinger equation for this potential, which is a member of a general Heun family of potentials, is written through fundamental solutions each of which presents an irreducible linear combination of two Gauss ordinary hypergeometric functions.