Methods for Generating Quasi-Exactly Solvable Potentials

TL;DR

This paper introduces three versatile methods for creating quasi-exactly solvable potentials in quantum mechanics, enabling analytical solutions for a finite set of eigenstates, thus expanding the toolkit beyond existing approaches.

Contribution

It presents three novel, general methods—polynomial ansatz, point canonical transformations, and supersymmetric quantum mechanics—for generating quasi-exactly solvable potentials.

Findings

Methods produce richer sets of solvable potentials.

Each method is applicable to a broad class of problems.

Results extend current literature on solvable quantum systems.

Abstract

We describe three different methods for generating quasi-exactly solvable potentials, for which a finite number of eigenstates are analytically known. The three methods are respectively based on (i) a polynomial ansatz for wave functions; (ii) point canonical transformations; (iii) supersymmetric quantum mechanics. The methods are rather general and give considerably richer results than those available in the current literature.