Semi-exact solutions of the Razavy potential

TL;DR

This paper derives semi-exact solutions for the symmetric Razavy quantum potential using confluent Heun functions, analyzing how wave functions and energy levels vary with potential parameters.

Contribution

It provides a method to find exact solutions for the Razavy potential and explores the effects of parameter changes on wave functions and energy spectra.

Findings

Wave functions are expressed via confluent Heun functions.

Parity symmetry is broken as the potential parameter increases.

Energy levels decrease with increasing potential parameter.

Abstract

In this work we study the quantum system with the symmetric Razavy potential and show how to find its exact solutions. We find that the solutions are given by the confluent Heun functions. The eigenvalues have to be calculated numerically. The properties of the wave functions depending on $m$ are illustrated graphically for a given potential parameter $\xi$. We find that the even and odd wave functions with definite parity are changed to odd and even wave functions when the potential parameter $m$ increases. This arises from the fact that the parity, which is a defined symmetry for very small $m$, is completely violated for large $m$. We also notice that the energy levels $\epsilon_{i}$ decrease with the increasing potential parameter $m$.