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

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.


Introduction
A quantum mechanical system is exactly solvable (ES) if all the eigenvalues and corresponding eigenfunctions can be determined exactly through algebraic means. These quantum systems play an important role in various branches of physics. However, such systems are rare and the Schrödinger equation cannot be solved exactly to obtain the whole spectrum except for a limited number of potentials, such as the harmonic oscillator, Coulomb and Pöschl-Teller potentials [1][2][3][4][5]. A review of early works in this area can be found in Refs. [6][7][8]. In contrast, a quantum system is called quasi-exactly solvable (QES) if only a finite part of the spectrum can be found exactly [9][10][11][12]. During the last decades, the QES models have received a great deal of attention because of their wide applications in quantum mechanics [13][14][15][16][17][18][19][20][21]. These models are distinguished by the fact that their infinite-dimensional Hamiltonian can be reduced to a block diagonal matrix with at least a finite-dimensional block, in which its eigenvalues and eigenfunctions can always be determined by diagonalizing the corresponding matrix. On the other hand, during the last decades, panahi@guilan.ac.ir -t mail: -Corresponding author E 1 a great deal of attention has been given to the study of the Schrödinger equation with QES doublewell potential (DWP) including the quartic potential [22], the sextic potential [23] and the Razavy potential [24]. In the literature, there are two distinct approaches for investigating the QES systems: the Lie algebraic approach [9][10][11] and the analytical approach [12,15] which is based on the Bethe ansatz method (BAM). The interested reader is referred to Refs. [4,[25][26][27][28][29] and references therein for more detailed information regarding the application of the wave function ansatz method in physical problems. In this paper, applying the analytical approach of quasi-exact solvability, we investigate the Schrödinger equation for a new type of one-dimensional QES DWP proposed by Chen et.al. [17]. They studied the problem and obtained solutions of the first two states by using two methods, the confluent Heun functions and the Wronskian method [17]. Within the present study, through the BAM, we are going to extend the results of Ref. [17] by finding general exact expressions for the energies, the wave functions and the special constraints on the potential parameters. Also, we solve the same problem using the Lie algebraic approach and illustrate how the relation with the In section 3, we solve the same problems using the Lie algebraic approach and demonstrate that the system possesses a hidden (2) sl algebraic symmetry which is responsible for quasi-exact solvability. Some numerical results, obtained by the BAM and QES methods are reported and discussed in section 4. Finally, in section 5, we present the conclusions.

The analytical method based on the Bethe ansatz for the QES DWP
We begin with the one-dimensional three parameter QES DWP proposed by Chen et al. [17] as , .
In 1935, Manning used this symmetric double minima potential to study the vibrational normal modes of the ND3 and NH3 molecules [30]. This application is possible because the nitrogen atom in these molecules has two equilibrium positions on either side of the D2 and H2 planes. In the following, we show that under certain constraints on the potential parameters 1 v , 2 v and 3 v , a finite number of the energy eigenvalues and eigenfunctions of the corresponding Schrödinger operator can be obtained exactly in explicit form. In Fig. 2, we draw the potential (1) for the allowed values of potential parameters. In atomic units ( 1 mc    ), the Schrödinger equation with potential (1) is written as Chen et al. [17] In order to apply the BAM to the present problem, we suppose that (5)  Substituting Eq. (7) into Eq. (5), and after some algebra, we obtain Comparing the left and right hand sides of Eq. (8), we obtain the following relations     (1 cosh ( )) , for the ground state energy and the corresponding wave function, where the potential parameters from Eq. (11) satisfy For 1 n  , by Eqs. (10) and (4), we obtain the first excited state energy and wave function as respectively. In this case, the constraint on the potential parameters is given by   where the root 1 z of the wave function is obtainable from the Bethe ansatz equation (9) as Analogusly, for the second excited state for the energy and wave function, respectively, where the potential parameters satisfy the constraint condition    . In Table 1, we have reported and compared our numerical results for the first four states. As an additional comment on the treatment of the wave functions, from Eqs. (13), (15) and (18) it can be seen that except the ground state wave function, the higher excited states are mathematically meaningless for large x, since as This property can be explained well by the asymptotic behavior of the Heun function which is only convergent within the circle 1 z  . The interested reader is referred to Ref. [17] for details of the problem.

The Lie algebraic approach for the QES DWP
In the previous section, we have investigated the Schrödinger equation for the QES DWP and obtained the general exact solutions of the system within the framework of the Bethe ansatz. In this section, we solve the same problem by using the Lie algebraic approach of quasi-exact solvability and obtain the exact solutions through the (2) sl algebraization. A differential equation is said to be QES if it is an element of the universal enveloping algebra of a finite-dimensional QES Lie algebra of differential operators [10]. In one dimension, the Lie algebra ) 2 ( sl is the only algebra of differential operators with finite-dimensional representations [11]. The usual realization of the ) 2 ( sl Lie algebra is given by the following differential operators [10,11]

Conclusions
Using the Bethe ansatz method, we have solved the Schrödinger equation for a new QES DWP and obtained the general exact expressions for the energies and the corresponding wave functions as well as the allowed values of the potential parameters in terms of the roots of the Bethe ansatz equations. In addition, we have solved the same problem using the Lie algebraic approach within the framework of quasi-exact solvability and obtained the exact solutions using the representation theory of (2) sl Lie algebra. Also, we have reported some numerical results and shown that the results are in good agreement with each other and with those obtained previously by using a different method.