Bound-states for generalized trigonometric and hyperbolic P\"oschl-Teller potentials

TL;DR

This paper introduces a new method to solve the Schrödinger equation for generalized P"oschl-Teller potentials, revealing bound states through finite series solutions involving Jacobi polynomials, expanding the class of solvable quantum potentials.

Contribution

It applies the tridiagonal representation approach to find bound states for generalized P"oschl-Teller potentials, which are not traditionally exactly solvable.

Findings

Derived finite series solutions for bound states

Expressed solutions in terms of Jacobi polynomials

Extended solvability to new classes of potentials

Abstract

We use the "tridiagonal representation approach" to solve the time-independent Schr\"odinger equation for the bound states of generalized versions of the trigonometric and hyperbolic P\"oschl-Teller potentials. These new solvable potentials do not belong to the conventional class of exactly solvable problems. The solutions are finite series of square integrable functions written in terms of the Jacobi polynomial.