Quasi-exact solvability beyond the SL(2) algebraization

Details Quasi-exact solvability beyond the SL(2) algebraization Quasi-exact solvability beyond the SL(2) algebraization

2006-01-24

arxiv.org/abs/nlin/0601053v2

Nonlinear Sciences

Exactly Solvable and Integrable Systems

Submitted to the proceedings of the 2005 Dubna workshop on superintegrability

Scientific paper

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.

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