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arXiv · 0711.3699

Prepotential approach to exact and quasi-exact solvabilities

Choon-Lin Ho
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Abstract

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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Choon-Lin Ho. 2007-11-23. Prepotential approach to exact and quasi-exact solvabilities. https://doi.org/10.1016/j.aop.2008.04.010

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