Quasi-Exact Solvability and the direct approach to invariant subspaces

Details Quasi-Exact Solvability and the direct approach to invariant subspaces Quasi-Exact Solvability and the direct approach to invariant subspaces

2004-01-21

arxiv.org/abs/nlin/0401030v1

J.Phys. A38 (2005) 2005-2020

Nonlinear Sciences

Exactly Solvable and Integrable Systems

17 pages, 3 figures

Scientific paper

10.1088/0305-4470/38/9/011

We propose a more direct approach to constructing differential operators that preserve polynomial subspaces than the one based on considering elements of the enveloping algebra of sl(2). This approach is used here to construct new exactly solvable and quasi-exactly solvable quantum Hamiltonians on the line which are not Lie-algebraic. It is also applied to generate potentials with multiple algebraic sectors. We discuss two illustrative examples of these two applications: an interesting generalization of the Lam\'e potential which posses four algebraic sectors, and a quasi-exactly solvable deformation of the Morse potential which is not Lie-algebraic.

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