On a particular form of a symmetric P\"oschl-Teller potential

TL;DR

This paper derives a new exact closed-form representation of solutions to the Schrödinger equation with a symmetric Pöschl-Teller potential, expressing hypergeometric functions as finite trigonometric combinations, and provides explicit integral formulas.

Contribution

It introduces a novel closed-form expression for hypergeometric functions related to a specific Pöschl-Teller potential, expanding analytical tools for quantum mechanics solutions.

Findings

Explicit closed-form solutions in terms of trigonometric functions.

New integral formulas involving hypergeometric and trigonometric functions.

Enhanced understanding of eigenfunctions for the symmetric Pöschl-Teller potential.

Abstract

We show that solutions of the Schr\"odinger equation with a symmetric P\"oschl-Teller potential of a particular form can be expressed in terms of a closed combination (not series) of trigonometric functions. Using some properties of the eigenfunctions of the Schr\"odinger equation and their inner product we determine a new exact representation of the hypergeometric function with certain values of parameters in terms of a closed combination of trigonometric functions. We also obtain new results in an explicit closed form for integrals with the hypergeometric function and with the specific combination of trigonometric functions.