The SUSY partners of the QES sextic potential revisited

Abstract

In this paper, the SUSY partner Hamiltonians of the quasi-exactly solvable (QES) sextic potential $V^{\mathrm{qes}}(x)=\nu x^6+2\nu \mu x^4+\left[{\mu }^2-(4N+3)\nu \right]x^2$, $N\in {\mathbb{Z}}^{+}$, are revisited from a Lie algebraic perspective. It is demonstrated that, in the variable $\tau =x^2$, the underlying ${\mathfrak{sl}}_2(\mathbb{R})$ hidden algebra of $V^{\mathrm{qes}}(x)$ is inherited by its SUSY partner potential $V_1(x)$ only for $N=0$. At fixed $N>0$, the algebraic polynomial operator $h(x,{\partial }_x;N)$ that governs the $N$ exact eigenpolynomial solutions of $V_1$ is derived explicitly. These odd-parity solutions appear in the form of zero modes. The potential $V_1$ can be represented as the sum of a polynomial and rational parts. In particular, it is shown that the polynomial component is given by $V^{\mathrm{qes}}$ with a different non-integer (cohomology) parameter $N_1=N-\frac{3}{2}$. A confluent second-order SUSY transformation is also implemented for a modified QES sextic potential possessing the energy reflection symmetry. By taking $N$ as a continuous real constant and using the Lagrange-mesh method, highly accurate values ($\sim 20$ s. d.) of the energy $E_n=E_n(N)$ in the interval $N\in [-1,3]$ are calculated for the three lowest states $n=0,1,2$ of the system. The critical value $N_c$ above which tunneling effects (instanton-like terms) can occur is obtained as well. At $N=0$, the non-algebraic sector of the spectrum of $V^{\mathrm{qes}}$ is described by means of compact physically relevant trial functions. These solutions allow us to determine the effects in accuracy when the first-order SUSY approach is applied on the level of approximate eigenfunctions.

25 pages, 20 figures