Bender-Dunne Orthogonal Polynomials and Quasi-Exact Solvability

Details Bender-Dunne Orthogonal Polynomials and Quasi-Exact Solvability Bender-Dunne Orthogonal Polynomials and Quasi-Exact Solvability

1996-01-18

arxiv.org/abs/hep-th/9601088v1

Physics

High Energy Physics

High Energy Physics - Theory

14 pages, LaTeX

Scientific paper

The paper is devoted to the further study of the remarkable classes of orthogonal polynomials recently discovered by Bender and Dunne. We show that these polynomials can be generated by solutions of arbitrary quasi - exactly solvable problems of quantum mechanichs both one-dimensional and multi-dimensional. A general and model-independent method for building and studying Bender-Dunne polynomials is proposed. The method enables one to compute the weight functions for the polynomials and shows that they are the orthogonal polynomials in a discrete variable $E_k$ which takes its values in the set of exactly computable energy levels of the corresponding quasi-exactly solvable model. It is also demonstrated that in an important particular case, the Bender-Dunne polynomials exactly coincide with orthogonal polynomials appearing in Lanczos tridiagonalization procedure.

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