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.

Submitted to the proceedings of the 2005 Dubna workshop on superintegrability

References in corpus (4)

• The Darboux transformation and algebraic deformations of shape-invariant potentials
• Supersymmetry and algebraic deformations
• Quasi-Exact Solvability and the direct approach to invariant subspaces
• A Novel Multi-parameter Family of Quantum Systems with Partially Broken N-fold Supersymmetry

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• Quasi-exactly solvable relativistic soft-core Coulomb models
• Approximate solution for Fokker-Planck equation
• Polynomial deformations of $sl(2)$ and unified algebraic framework for solutions of a class of spin models