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

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.


Introduction
Double well potentials (DWPs) are an important class of configurations 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 etc. 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][10][11][12][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][15][16][17][18][19] and the Lie algebraic approach [10][11][12][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][21][22], condensed matter physics [23][24][25], quantum cosmology [26][27][28][29][30][31][32] and so on, which their 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], the Wronskian method [36], etc.
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][38][39][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 sl(2) Lie algebra and the second order QES differential equations.

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.

The generalized Manning potential
First, we consider the three parameter generalized Manning potential as [36] Xie [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 Ref. [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 In Table 1, we report and compare our numerical results for the first three states. Also, in Fig. 1

The Razavy bistable potential
Here, we consider the hyperbolic Razavy potential (also called the double sinh-Gordon (DSHG) potential) defined by [41] where  is a real parameter. While the value of M is not restricted generally, but according to Ref. Razavy potential can be considered as a realistic model for a proton in a hydrogen bond [42,43]. The potential (23) has also been used by several authors, for studying the statistical mechanics of DSHG kinks theory [44]. The Schrödinger equation with potential (23) is Exact and approximate solutions of the first M states for has obtained via different methods and can be found in Refs. [8,35,44]. In this and the next section, we extend the solutions of Now, we consider the polynomial solutions for (26) as 1 , Now, evaluating the residues at the two simple poles k zz  and 0 z  , and comparing the results with (26), we obtain the following relations The results obtained for the first three levels are reported and compared in table 2. The Razavy potential and its energy levels splitting are plotted in Fig. 2. Also, the numerical results for the eigenvalues and energy levels splitting are presented in table 3. As can be seen, for a given M, the energy differences between the two adjacent levels satisfy the inequality and therefore the energy levels are paired together.

The hyperbolic Shifman potential
Now, we consider a hyperbolic potential introduced by Shifman as [9]  1 , Comparing the residues at the simple poles   is plotted in Fig. 3. In the next section, we intend to reproduce the results using the Lie algebraic approach in the framework of quasi-exact solvability.

The Lie algebraic approach for the DWPs
In the previous section, we applied the BAM to obtain the exact solutions of the systems. In this section, we solve the same models by using the Lie algebraic approach and show how the relation with the ) 2 ( sl Lie algebra underlies the solvability of them. To this aim, for each model, we show that the corresponding differential equation is an element of the universal enveloping algebra of (2) sl and thereby we obtain the exact solutions of the systems using the representation theory of (2) sl . The method is outlined in the Appendix.

The generalized Manning potential
Applying the results of Appendix, it is easy to verify that Eq. (4) can be written in the Lie algebraic where the constraints on the potential parameters is obtained from (58) as For the first excited state 1 n  , the energy equation is given by where the potential parameters satisfy the constraint Solutions of the second excited state corresponds to 2 n  are given as where the constraints on the potential parameters is obtained from (58) as follows 3   1  2  3  1   2   1  2  3  1  1   2   1   12   22   2  2  2   2   2  3  1  1   1  1  1   22   2   13 3  3  8  4 2  2   3 2  2  16  3 2  6  2   64  3 2  2  2  0.
Some numerical results are reported and compared with BAM results in table 1. As can be seen the results achieved by the two methods are identical.

The Razavy bistable potential
In this case, using the results of Appendix, the operator H of Eq. (26) is expressed as an element of the universal enveloping algebra of (2) sl as 2 0 Then, using the representation theory of (2) sl , the general condition for existence of a non-trivial solution is obtained as    The interested reader is referred to Refs. [9][10][11][12][13][14] for further details.   Table 3. Energy splitting of the first 12 states for the Razavy bistable potential with 2   .