Lie symmetry and the Bethe ansatz solution of a new quasi-exactly solvable double-well potential

TL;DR

This paper introduces a new quasi-exactly solvable double-well potential and derives its energy levels and wave functions using Bethe ansatz and Lie algebraic methods, confirming their consistency with each other and prior results.

Contribution

It presents a novel quasi-exactly solvable double-well potential and applies two different analytical methods to solve it, demonstrating their agreement.

Findings

Exact energy expressions obtained

Wave functions explicitly derived

Results agree with previous methods

Abstract

In this paper, we study the Schr\"odinger equation with a new quasi-exactly solvable double-well potential. Exact expressions for the energies, the corresponding wave functions and the allowed values of the potential parameters are obtained using two different methods, the Bethe ansatz method and the Lie algebraic approach. Some numerical results are reported and it is shown that the results are in good agreement with each other and with those obtained previously via a different method.