The third exactly solvable hypergeometric quantum-mechanical potential

TL;DR

This paper introduces a new exactly solvable hypergeometric potential in quantum mechanics, providing explicit solutions and analyzing transmission properties, expanding the set of known solvable models.

Contribution

It presents the third independent exactly solvable hypergeometric potential with unique shape properties and explicit solutions involving hypergeometric functions.

Findings

Derived a compact formula for the reflection coefficient.

Provided explicit solutions for the Schrödinger equation with the new potential.

Characterized the potential as an asymmetric step-barrier with variable height and steepness.

Abstract

We introduce the third independent exactly solvable hypergeometric potential, after the Eckart and the P\"oschl-Teller potentials, which is proportional to an energy-independent parameter and has a shape that is independent of this parameter. The general solution of the Schr\"odinger equation for this potential is written through fundamental solutions each of which presents an irreducible combination of two Gauss hypergeometric functions. The potential is an asymmetric step-barrier with variable height and steepness. Discussing the transmission above such a barrier, we derive a compact formula for the reflection coefficient.