Quasi-exact solvability and intertwining relations

Details Quasi-exact solvability and intertwining relations Quasi-exact solvability and intertwining relations

2004-10-06

arxiv.org/abs/hep-th/0410064v2

Physics

High Energy Physics

High Energy Physics - Theory

16 pages, typos corrected, minor changes, 2 references added

Scientific paper

We emphasize intertwining relations as a universal tool in constructing one-dimensional quasi-exactly solvable operators and offer their possible generalization to the multidimensional case. Considered examples include all quasi-exactly solvable operators with invariant subspaces of monomials. We show that the simplest case of generalized intertwining relations allows to naturally relate quasi-exactly solvable operators with two invariant monomial subspaces to a nonlinear parasupersymmetry of second order. Quantum-mechanical systems with linear and nonlinear supersymmetry are discussed from the viewpoint of quasi-exact solvability. We construct such a general system with a cubic supersymmetry and argue that quantum-mechanical systems with nonlinear supersymmetry of fourth order and higher are generally not quasi-exactly solvable. Besides, we construct two examples of quasi-exactly solvable operators with invariant subspaces which cannot be reduced to monomial spaces.

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