A Unified Treatment of Quasi-Exactly Solvable Potentials II: Eckart Type Potentials

TL;DR

This paper extends the unified framework for quasi-exactly solvable potentials, focusing on Eckart type potentials, by solving the Schrödinger equation and relating different potentials through variable transformations.

Contribution

It introduces a unified method to derive and connect various quasi-exactly solvable potentials, including Eckart, Hultén, Rosen-Morse, Coulomb, and harmonic oscillator.

Findings

Eigenvalues and eigenstates expressed via orthogonal polynomials

Unified approach to relate different potentials

Explicit solutions for Eckart type potentials

Abstract

This work continues to study the set of quasi exactly solvable potentials related to the Eckart, Hult\'{e}n, Rosen-Morse, Coulomb and the harmonic oscillator potentials. We solve the Schr\"{o}dinger equation for each potential and obtain the eigenvalues and eigenstates in terms of the orthogonal polynomials. We present a unified approach to obtain the potentials from each other by suitable change of variables.