Unified derivation of exact solutions for a class of quasi-exactly solvable models

TL;DR

This paper unifies the exact solutions of four quantum models by reducing them to a common quasi-exactly solvable differential equation and solving it using the Bethe ansatz method, revealing an underlying algebraic structure.

Contribution

It provides a unified framework and explicit solutions for four quantum models, demonstrating their reducibility to a single quasi-exactly solvable equation and uncovering a hidden algebraic symmetry.

Findings

Explicit energy expressions for all models

Unified differential equation for diverse models

Discovery of hidden $sl(2)$ algebraic structure

Abstract

We present a unified treatment of exact solutions for a class of four quantum mechanical models, namely the singular anharmonic potential, the generalized quantum isotonic oscillator, the soft-core Coulomb potential, and the non-polynomially modified oscillator. We show that all four cases are reducible to the same basic ordinary differential equation, which is quasi-exactly solvable. A systematic and closed form solution to the basic equation is obtained via the Bethe ansatz method. Using the result, general exact expressions for the energies and the allowed potential parameters are given explicitly for each of the four cases in terms of the roots of a set of algebraic equations. A hidden $sl(2)$ algebraic structure is also discovered in these models.