The SUSY partners of the QES sextic potential revisited

Abstract

Abstract In this paper, the SUSY partner Hamiltonians of the quasi-exactly solvable (QES) sextic potential V qes ( x ) = ν x 6 + 2 ν μ x 4 + μ 2 − ( 4 N + 3 ) ν x 2 , N ∈ Z + , are revisited from a Lie algebraic perspective. It is demonstrated that, in the variable z = x 2, the underlying sl 2 ( R ) hidden algebra of V 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, ∂ 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 qes with a different non-integer (cohomology) parameter N 1 = N − 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 (∼20 s. d.) of the energy E n = E n (N) in the interval N ∈ [ − 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 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.