# Quasi-exactly solvable extended trigonometric P\"oschl-Teller potentials with position-dependent mass

## TL;DR

This paper develops quasi-exactly solvable position-dependent mass Schr"odinger equations based on deformed supersymmetry and trigonometric P"oschl-Teller potentials, providing new solvable models with potential applications in physics.

## Contribution

It introduces a method to construct quasi-exactly solvable P"oschl-Teller potentials with position-dependent mass using a generating function approach within a deformed supersymmetric framework.

## Key findings

- Constructed infinite families of solvable equations with known eigenstates.
- Extended classical potentials while maintaining solvability.
- Provided explicit superpotentials and partner potentials.

## Abstract

Infinite families of quasi-exactly solvable position-dependent mass Schr\"odinger equations with known ground and first excited states are constructed in a deformed supersymmetric background. The starting points consist in one- and two-parameter trigonometric P\"oschl-Teller potentials endowed with a deformed shape invariance property and, therefore, exactly solvable. Some extensions of them are considered with the same position-dependent mass and dealt with by a generating function method. The latter enables to construct the first two superpotentials of a deformed supersymmetric hierarchy, as well as the first two partner potentials and the first two eigenstates of the first potential from some generating function $W_+(x)$ [and its accompanying function $W_-(x)$]. The generalized trigonometric P\"oschl-Teller potentials so obtained are thought to have interesting applications in molecular and solid state physics.