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

• URL: http://arxiv.org/abs/2409.18311v1
• Date: Thu, 26 Sep 2024 21:42:32 GMT
• Title: The QES sextic and Morse potentials: exact WKB condition and supersymmetry
• Authors: Alonso Contreras-Astorga, A. M. Escobar-Ruiz,
• Abstract summary: The one-dimensional quasi-exactly solvable (QES) sextic potential $Vrm(qes)(x) is considered. The WKB correction $gamma=gamma(N,n)$ is calculated for the first lowest 50 states.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• 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 $\gamma=0$.

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