Exact solutions of a class of double-well potentials: Algebraic Bethe ansatz

TL;DR

This paper derives exact solutions for certain double-well potentials using Bethe ansatz and Lie algebra methods, providing explicit energy spectra and wave functions for these quasi-exactly solvable models.

Contribution

It presents a unified approach to solving three specific double-well potentials exactly, combining Bethe ansatz and Lie algebra techniques.

Findings

Exact energy expressions in terms of algebraic roots

Wave functions explicitly constructed

Energy level splitting demonstrated numerically

Abstract

In this paper, applying the Bethe ansatz method, we investigate the Schr\"odinger equation for the three quasi-exactly solvable double-well potentials, namely the generalized Manning potential, the Razavy bistable potential and the hyperbolic Shifman potential. General exact expressions for the energies and the associated wave functions are obtained in terms of the roots of a set of algebraic equations. Also, we solve the same problems using the Lie algebraic approach of quasi-exact solvability through the sl(2) algebraization and show that the results are the same. The numerical evaluation of the energy spectrum is reported to display explicitly the energy levels splitting.