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 $\gamma=0$.
Categories: quant-ph math-ph math.MP