Prepotential approach to exact and quasi-exact solvabilities

Details Prepotential approach to exact and quasi-exact solvabilities Prepotential approach to exact and quasi-exact solvabilities

2007-11-23

arxiv.org/abs/0711.3699v3

Physics

Mathematical Physics

11 pages, no figures. New paragraphs added in the Introduction and Summary sections. New references added. Version to appear i

Scientific paper

Exact and quasi-exact solvabilities of the one-dimensional Schr\"odinger equation are discussed from a unified viewpoint based on the prepotential together with Bethe ansatz equations. This is a constructive approach which gives the potential as well as the eigenfunctions and eigenvalues simultaneously. The novel feature of the present work is the realization that both exact and quasi-exact solvabilities can be solely classified by two integers, the degrees of two polynomials which determine the change of variable and the zero-th order prepotential. Most of the well-known exactly and quasi-exactly solvable models, and many new quasi-exactly solvable ones, can be generated by appropriately choosing the two polynomials. This approach can be easily extended to the constructions of exactly and quasi-exactly solvable Dirac, Pauli, and Fokker-Planck equations.

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