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

In this paper, as a continuation of [Contreras-Astorga A., Escobar-Ruiz A. M. and Linares R., $\mathit{\text{Phys.}}\mathit{\text{Scr.}}$ {\bf99} 025223 (2024)] the one-dimensional quasi-exactly solvable (QES) sextic potential $V^{(\mathrm{q}\mathrm{e}\mathrm{s})}(x)=\frac{1}{2}(\nu x^6+2\nu \mu x^4+[{\mu }^2-(4N+3)\nu ]x^2)$ 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 $\mathrm{\mid }\mathrm{\Delta }\mathrm{\mid }$ 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 ${\mathfrak{s}\mathfrak{l}}_2(\mathbb{R})$ Lie algebra and vanishing WKB correction $\gamma =0$.