Quasi-exact solvability beyond the SL(2) algebraization

Abstract

We present evidence to suggest that the study of one dimensional quasi-exactly solvable (QES) models in quantum mechanics should be extended beyond the usual $\sla(2)$ approach. The motivation is twofold: We first show that certain quasi-exactly solvable potentials constructed with the $\sla(2)$ Lie algebraic method allow for a new larger portion of the spectrum to be obtained algebraically. This is done via another algebraization in which the algebraic hamiltonian cannot be expressed as a polynomial in the generators of $\sla(2)$. We then show an example of a new quasi-exactly solvable potential which cannot be obtained within the Lie-algebraic approach.