Advances in High Energy Physics

Volume 2017 (2017), Article ID 8429863, 13 pages

https://doi.org/10.1155/2017/8429863

## Exact Solutions of a Class of Double-Well Potentials: Algebraic Bethe Ansatz

Department of Physics, University of Guilan, Rasht 41635-1914, Iran

Correspondence should be addressed to M. Baradaran

Received 5 September 2017; Revised 15 November 2017; Accepted 19 November 2017; Published 26 December 2017

Academic Editor: Marc de Montigny

Copyright © 2017 M. Baradaran and H. Panahi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP^{3}.

#### Abstract

Applying the Bethe ansatz method, we investigate the Schrödinger 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 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.

#### 1. Introduction

Double-well potentials (DWPs) are an important class of configurations which have been extensively used in many fields of physics and chemistry for the description of the motion of a particle under two centers of force. Recently, solutions of the Schrödinger equation with DWPs have found applications in the Bose–Einstein condensation [1], molecular systems [2], quantum tunneling effect [3, 4], microscopic description of Tunneling Systems [5], and so forth. Some well-known DWPs in the literature are the quartic potential [6], the sextic potential [7], the Manning potential [2], and the Razavy potential [8]. In addition, it has been found that with some special constraints on the parameters of these potentials, a finite part of the energy spectrum and corresponding eigenfunctions can be obtained as explicit expressions in a closed form. In other words, these systems are quasi-exactly solvable (QES) [9–13]. DWPs in the framework of QES systems have received a great deal of attention. This is due to the pioneering work of Razavy, who proposed his well-known potential for describing the quantum theory of molecules [8]. The fundamental idea behind the quasi-exact solvability is the existence of a hidden dynamical symmetry. QES systems can be studied by two main approaches: the analytical approach based on the Bethe ansatz [14–19] and the Lie algebraic approach [10–13]. These techniques are of great importance because only a few number of problems in quantum mechanics can be solved exactly. Therefore, these approaches can be applied as accurate and efficient techniques to study and solve the new problems that arise in different areas of physics such as quantum field theory [20–22], condensed matter physics [23–25], and quantum cosmology [26–32], whose exact solutions are hard to obtain or are impossible to find. In the literature, DWPs have been studied by using various techniques such as the WKB approximation [33, 34], asymptotic iteration method (AIM) [35], and the Wronskian method [36]. On the other hand, it is well known that the tunnel splitting which is the differences between the adjacent energy levels is the characteristic of the energy spectrum for the DWPs [37–40]. In this paper, we apply two different methods to solve the Schrödinger equation for three QES DWPs, the Bethe ansatz method (BAM) and the Lie algebraic method, and show that the results of the two methods are consistent. Also, we provide some numerical results of the bistable Razavy potential to display the energy levels splitting explicitly.

This paper is organized as follows: in Section 2, we introduce the QES DWPs and obtain the exact solutions of the corresponding Schrödinger equations using the BAM. Also, general exact expressions for the energies and the wave functions are obtained in terms of the roots of the Bethe ansatz equations. In Section 3, we solve the same problems using the Lie algebraic approach within the framework of quasi-exact solvability and therein we make a comparison between the solutions obtained by the BAM and QES method. We end with conclusions in Section 4. In the Appendix, we review the connection between Lie algebra and the second-order QES differential equations.

#### 2. The BAM for the DWPs

In this section, we introduce the three DWPs that are discussed in this work and solve the corresponding Schrödinger equations via the factorization method in the framework of algebraic Bethe ansatz [15]. The general exact expressions for the energies, the wave functions, and the allowed values of the potential parameters are obtained in terms of the roots of the Bethe ansatz equations.

##### 2.1. The Generalized Manning Potential

First, we consider the three-parameter generalized Manning potential as [36]The parameters , , and are real constants which under certain constraint conditions enable us to obtain the bound-state eigenenergies and associated wave functions exactly. In atomic units (), the Schrödinger equation with potential (1) isXie [36] has studied this problem and obtained exact solutions of the first two states in terms of the confluent Heun functions. In this paper, we intend to extend the results of [36] by determining general exact expressions for the energies, wave functions, and the allowed values of the potential parameters, using the factorization method in the framework of the Bethe ansatz. To this end, and for the purpose of extracting the asymptotic behaviour of the wave function, we consider the following transformations:which, after substituting in (2), giveswhereIn order to solve the present problem via BAM, we try to factorize the operator assuch that . Now, we suppose that polynomial solution (Bethe ansatz) exists for (4) as with the distinct roots that are interpreted as the wave function nodes and can be determined by the Bethe ansatz equations. As a result, it is evident that the operator must have the formand then, the operator has the following form: By substituting (8) and (9) into (6), we haveThe last term on the right of (10) is obviously a meromorphic function with simple poles at and . Comparing the treatment of (10) with (4) at these points, we obtain the following relations for the unknown roots (the so-called Bethe ansatz equation), the energy eigenvalues, and the constraints on the potential parameters:As examples of the above general solutions, we study the ground, first, and second excited states of the model in detail. For , by (12) and (3), we have the following relations: for the ground state energy and wave function, with the potential constraint given byFor the first excited state , by (12) and (3), we have for the energy and wave function, respectively. Also, the constraint condition between the parameters of the potential is aswhere the root is obtained from the Bethe ansatz equation (11) asSimilarly, for the second excited state , the energy, wave function, and the constraint condition between the potential parameters are given aswhere the two distinct roots and are obtainable from the Bethe ansatz equationsIn Table 1, we report and compare our numerical results for the first three states. Also, in Figure 1, we draw the potential (1) for the possible values of the parameters , , and .