The QES sextic and Morse potentials: exact WKB condition and supersymmetry

TL;DR

This paper analyzes quasi-exactly solvable sextic and Morse potentials, deriving accurate WKB corrections and energies, revealing shape invariance and supersymmetry properties, and providing analytical approximations with high precision.

Contribution

It introduces analytical approximations for WKB corrections and energies of QES sextic and Morse potentials, demonstrating their shape invariance and supersymmetry features.

Findings

WKB correction $oldsymbol{ eq 0}$ for sextic potential, accurately computed for 50 states.

WKB correction $oldsymbol{= 0}$ for Morse potential, indicating shape invariance.

Analytical approximations achieve relative accuracy $oldsymbol{oxed{ extless} 10^{-3}}$ across studied parameters.

Abstract

In this paper, as a continuation of [Contreras-Astorga A., Escobar-Ruiz A. M. and Linares R., \textit{Phys. Scr.} {\bf99} 025223 (2024)] the one-dimensional quasi-exactly solvable (QES) sextic potential $V^{\rm(qes)}(x) = \frac{1}{2}(\nu\, x^{6} + 2\, \nu\, \mu\,x^{4} + \left[\mu^2-(4N+3)\nu \right]\, x^{2})$ is considered. In the cases $N=0,\frac{1}{4},\,\frac{1}{2},\,\frac{7}{10}$ the WKB correction $\gamma=\gamma(N,n)$ is calculated for the first lowest 50 states $n\in [0,\,50]$ using highly accurate data obtained by the Lagrange Mesh Method. Closed analytical approximations for both $\gamma$ and the energy $E=E(N,n)$ of the system are constructed. They provide a reasonably relative accuracy $|\Delta|$ with upper bound $\lesssim 10^{-3}$ for all the values of $(N,n)$ studied. Also, it is shown that the QES Morse potential is shape invariant characterized by a hidden $\mathfrak{sl}_2(\mathbb{R})$ Lie algebra and vanishing WKB correction $\gamma=0$.