# A new exactly integrable hypergeometric potential for the Schr\"odinger equation

## TL;DR

This paper introduces a novel exactly solvable hypergeometric potential in the Schrödinger equation, allowing explicit solutions and analysis of quantum particle transmission over a variable barrier.

## Contribution

The paper presents a new exactly integrable potential expressed via hypergeometric functions, expanding solvable models in quantum mechanics.

## Key findings

- Explicit solutions for the Schrödinger equation with the new potential
- Derived formulas for reflection and transmission coefficients
- Analysis of quantum particle transmission over variable barriers

## Abstract

We introduce a new exactly integrable potential for the Schr\"odinger equation for which the solution of the problem may be expressed in terms of the Gauss hypergeometric functions. This is a potential step with variable height and steepness. We present the general solution of the problem, discuss the transmission of a quantum particle above the barrier, and derive explicit expressions for the reflection and transmission coefficients.

---
Source: https://tomesphere.com/paper/1803.05363