On the relation between polynomial deformations of sl(2,R) and quasi-exactly solvability

TL;DR

This paper introduces a general method leveraging polynomial deformations of sl(2,R) Lie algebra to identify quasi-exact solvability in certain quantum Hamiltonians, demonstrated on specific models.

Contribution

It presents a novel approach using polynomial deformations of sl(2,R) to analyze quasi-exact solvability in quantum systems, expanding the tools available for such investigations.

Findings

Method successfully applied to sextic oscillator

Method demonstrated on second harmonic generation

Provides a systematic way to identify quasi-exact solvability

Abstract

A general method based on the polynomial deformations of the Lie algebra sl(2,R) is proposed in order to exhibit the quasi-exactly solvability of specific Hamiltonians implied by quantum physical models. This method using the finite-dimensional representations and differential realizations of such deformations is illustrated on the sextic oscillator as well as on the second harmonic generation.