In this paper, the SUSY partner Hamiltonians of the quasi-exactly solvable
(QES) sextic potential Vqes(x)=νx6+2νμx4+[μ2−(4N+3)ν]x2, N∈Z+, are revisited
from a Lie algebraic perspective. It is demonstrated that, in the variable τ=x2, the underlying sl2(R) hidden algebra of
Vqes(x) is inherited by its SUSY partner potential V1(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
V1 is derived explicitly. These odd-parity solutions appear in the form of
zero modes. The potential V1 can be represented as the sum of a polynomial
and rational parts. In particular, it is shown that the polynomial component is
given by Vqes with a different non-integer (cohomology) parameter
N1=N−23. 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
En=En(N) in the interval N∈[−1,3] are calculated for the three lowest
states n=0,1,2 of the system. The critical value Nc above which tunneling
effects (instanton-like terms) can occur is obtained as well. At N=0, the
non-algebraic sector of the spectrum of Vqes 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.Comment: 25 pages, 20 figure