QUASI-EXACT SOLVABILITY BEYOND THE sl(2) ALGEBRAIZATION

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 sl(2) approach. The motivation is twofold: We first show that certain quasi-exactly solvable potentials constructed with the sl(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 sl(2). We then show an example of a new quasi-exactly solvable potential which cannot be obtained within the Lie-algebraic approach.