On the analytical solutions of the quasi-exactly solvable Razavy type potential V(x) = Vo (sinh**4(x)-k sinh**2(x) )

TL;DR

This paper explores analytical solutions for a specific quasi-exactly solvable Razavy potential using polynomial solutions of the Confluent Heun Equation, revealing limitations in the method when tuning parameters.

Contribution

It demonstrates how to derive analytical solutions for the Razavy potential and highlights the divergence issue when tuning the parameter k.

Findings

Analytical solutions constructed for the Razavy potential.

Divergence of energy eigenvalues as parameter k approaches -1.

Method's limitations due to divergence when tuning parameters.

Abstract

In order to show how stringent the restrictions posed on analytical solutions of quasi-exactly solvable potentials are, we construct analytical solutions for the Razavy type potential V(x) = Vo (sinh**4(x)-k sinh**2(x) ) based on the polynomial solutions of the related Confluent Heun Equation, where the free parameter k allows to tune energy eigenvalues, a desirable feature in different theories. However, we show that with the described method, the energy eigenvalues found diverge when k (goes to) -1, a feature caused solely by the procedure.