Potentials of the Heun class: the triconfluent case

TL;DR

This paper explores a new class of quantum potentials that can be transformed into the triconfluent Heun equation, providing exact solutions for energy levels and eigenfunctions, thus extending analytical methods in quantum mechanics.

Contribution

It generalizes the transformation approach to include potentials reducible to the triconfluent Heun equation, offering new exactly solvable models in quantum mechanics.

Findings

Derived energy eigenvalues for the new potential class

Obtained explicit eigenfunctions and superpartners

Extended the class of analytically solvable Schrödinger equations

Abstract

Since the advent of quantum mechanics different approaches to find analytical solutions of the Schr\"odinger equation have been successfully developed. Here we follow and generalize the approach pioneered by Natanzon and others by which the Schr\"odinger equations can be transformed into another well-known equation for transcendental function (e.g., the hypergeometric equation). This sets a class of potentials for which this transformation is possible. Our generalization consists in finding potentials allowing the transformation of the Schr\"odinger equation into a triconfluent Heun equation. We find the energy eigenvalues of this class of potentials, the eigenfunction and the exact superpartners.