Generalized Master Function Approach to Quasi-Exactly Solvable Models

TL;DR

This paper introduces a generalized master function framework to systematically derive all second-order differential equations that are quasi-exactly solvable, unifying known models and revealing polynomial solution structures.

Contribution

It presents a new generalized master function approach of order up to four, enabling the derivation of all quasi-exactly solvable second-order differential equations and their polynomial solutions.

Findings

All known quasi-exactly solvable models are encompassed by these equations.

Polynomial solutions exhibit factorization properties.

Roots of polynomial Pn(E) correspond to eigenvalues.

Abstract

By introducing the generalized master function of order up to four together with corresponding weight function, we have obtained all quasi-exactly solvable second order differential equations. It is shown that these differntial equations have solutions of polynomial type with factorziation properties, that is polynomial solutions Pm(E) can be factorized in terms of polynomial Pn(E) for m not equal to n. All known quasi-exactly quantum solvable models can be obtained from these differential equations, where roots of polynomial Pn(E) are corresponding eigen-values.