Prepotential approach to exact and quasi-exact solvabilities

TL;DR

This paper introduces a unified prepotential approach to classify and construct exactly and quasi-exactly solvable quantum models, providing a systematic way to generate known and new solvable potentials.

Contribution

The work presents a novel classification scheme using two polynomials that determine solvability, unifying and extending the construction of solvable quantum models.

Findings

Most known solvable models are generated by the polynomial classification.

New quasi-exactly solvable models are constructed within this framework.

The approach can be extended to other equations like Dirac, Pauli, and Fokker-Planck.

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.