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

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 Schrodinger 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 Schrodinger 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 Schrodinger 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.

2. The BAM for the DWPs

In this section, we introduce the three DWPs that are discussed in this work and solve the corresponding Schrodinger 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]

V(x) = -[v.sub.1] [sech.sup.6] (x) - [v.sub.2] [sech.sup.4] (x) - [v.sub.3] [sech.sup.2] (x). (1)

The parameters [v.sub.1], [v.sub.2], and [v.sub.3] are real constants which under certain constraint conditions enable us to obtain the bound-state eigenenergies and associated wave functions exactly. In atomic units (m = h = c = 1), the Schrodinger equation with potential (1) is

(-[d.sup.2]/d[x.sup.2] - [v.sub.1] [sech.sup.6] (x) - [v.sub.2] [sech.sup.4] (x) - [v.sub.3] [sech.sup.2] (x)) x [psi] (x) = E[psi](x). (2)

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 [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:

[mathematical expression not reproducible], (3)

which, after substituting in (2), gives

[mathematical expression not reproducible], (4)

where

[lambda] = (3 + 2[square root of (-E)])[square root of ([v.sub.1])] + [v.sub.1] + [v.sub.2], [epsilon] = -[v.sub.1] - [v.sub.2] - [v.sub.3] - E + [square root of (-E)] - [square root of ([v.sub.1])]. (5)

In order to solve the present problem via BAM, we try to factorize the operator H as

H = [A.sup.+.sub.n][A.sub.n], (6)

such that [A.sub.n][[phi].sub.n](z) = 0. Now, we suppose that polynomial solution (Bethe ansatz) exists for (4) as

[mathematical expression not reproducible], (7)

with the distinct roots [z.sub.k] 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 [A.sub.n] must have the form

[A.sub.n] = d/dz - [n.summation over (k=1)] [1/[z - [z.sub.k]]], (8)

and then, the operator [A.sup.+.sub.n] has the following form:

[A.sup.+.sub.n] = -d/dz - (1/2z + [1 + [square root of (-E)]]/[z - 1] + [square root of ([v.sub.1])]) - [n.summation over (k=1)] 1/[z - [z.sub.k]]. (9)

By substituting (8) and (9) into (6), we have

[mathematical expression not reproducible]. (10)

The last term on the right of (10) is obviously a meromorphic function with simple poles at z = 0,1 and [z.sub.k]. Comparing the treatment of (10) with (4) at these points, we obtain the following relations for the unknown roots [z.sub.k] (the so-called Bethe ansatz equation), the energy eigenvalues, and the constraints on the potential parameters:

[n.summation over (j[not equal to]k)] 2/[[z.sub.k] - [z.sub.j]] + 1/2[z.sub.k] + [1 + [square root of (-[E.sub.n])]]/[[z.sub.k] - 1] + [square root of ([v.sub.1])] = 0, (11)

(3 + 2[square root of (-[E.sub.n])]) [square root of ([v.sub.1])] + [v.sub.1] + [v.sub.2] + 4n[square root of ([v.sub.1])] = 0, (12)

[mathematical expression not reproducible]. (13)

As examples of the above general solutions, we study the ground, first, and second excited states of the model in detail. For n = 0, by (12) and (3), we have the following relations:

[mathematical expression not reproducible], (14)

for the ground state energy and wave function, with the potential constraint given by

-[v.sub.1] - [v.sub.2] - [v.sub.3] - [E.sub.0] + [square root of (-[E.sub.0])] - [square root of ([v.sub.1])] = 0. (15)

For the first excited state n = 1, by (12) and (3), we have

[mathematical expression not reproducible], (16)

for the energy and wave function, respectively. Also, the constraint condition between the parameters of the potential is as

-[v.sub.1] - [v.sub.2] - [v.sub.3] - [E.sub.1] + 4[square root of ([v.sub.1])] [z.sub.1] + 5[square root of (-[E.sub.1])] - 5[square root of ([v.sub.1])] + 6 = 0, (17)

where the root [z.sub.1] is obtained from the Bethe ansatz equation (11) as

[mathematical expression not reproducible]. (18)

Similarly, for the second excited state n = 2, the energy, wave function, and the constraint condition between the potential parameters are given as

[mathematical expression not reproducible], (19)

where the two distinct roots [z.sub.1] and [z.sub.2] are obtainable from the Bethe ansatz equations

[mathematical expression not reproducible]. (20)

In 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 [v.sub.1] = 1, [v.sub.2] = -12, and [v.sub.3] = 15.255.

2.2. The Razavy Bistable Potential. Here, we consider the hyperbolic Razavy potential (also called the double sinh-Gordon (DSHG) potential) defined by [41]

V(x) = [([xi] cosh (2x)-M).sup.2], (21)

where [xi] is a real parameter. While the value of M is not restricted generally, but according to [8], the solutions of the first M states can be found exactly if M is a positive integer. This potential exhibits a double-well structure for M > [xi] with the two minima lying at cosh(2[x.sub.0]) = M/[xi]. Specifically, the Razavy potential can be considered as a realistic model for a proton in a hydrogen bond [42, 43]. The potential (21) has also been used by several authors, for studying the statistical mechanics of DSHG kinks theory [44]. The Schrodinger equation with potential (21) is

(-[d.sup.2]/d[x.sup.2] + [([xi] cosh (2x)-M).sup.2]) [psi](x) = E[psi](x). (22)

Exact and approximate solutions of the first M states for M = 1, 2, 3, 4, 5, 6, 7 have been obtained via different methods and can be found in [8, 35, 44]. In this and the next section, we extend the solutions of (22) to the general cases of arbitrarily M and obtain general exact expressions for the energies and wave functions using the BAM and QES methods. Using the change of variable z = [e.sup.2x] and the gauge transformation

[psi] (z) = [z.sup.(1-M)/2][e.sup.-([xi]/4)(z+1/z)][phi] (z), (23)

we obtain

H[phi](z) = 0, H = -4[z.sup.2] [d.sup.2]/d[z.sup.2] + (2[xi][z.sup.2] + (4M - 8)z - 2[xi]) d/dz + (-2[xi](M-1)z + [[xi].sup.2] + 2M - 1 - E). (24)

Now, we consider the polynomial solutions for (24) as

[mathematical expression not reproducible], (25)

where [z.sub.k] are unknown parameters to be determined by the Bethe ansatz equations. In this case, the operators [A.sub.M] and [A.sup.+.sub.M] are defined as

[A.sub.M] = d/dz - [M.summation over (k=2)] 1/[z - [z.sub.k]], [A.sup.+.sub.M] = -d/dz + [xi]/2 + [M - 2]/z - [xi]/2[z.sup.2] - [M.summation over (k=2)] [1/[z - [z.sub.k]]]. (26)

As a result, we have

[mathematical expression not reproducible]. (27)

Now, evaluating the residues at the two simple poles z = [z.sub.k] and z = 0, and comparing the results with (24), we obtain the following relations:

[E.sub.M] = [[xi].sup.2] + 2M - 1 + 2[xi] [M.summation over (k=2)] [z.sub.k], (28)

[M.summation over (j[not equal to]k)] [2/[[z.sub.k] - [z.sub.j]]] + [xi]/2[z.sup.2.sub.k] - [M - 2]/[z.sub.k] - [xi]/2 = 0, (29)

for the energy eigenvalues and the roots [z.sub.k], respectively. For example, for the ground state M = 1, from (28) and (23), we have the following relations for energy and wave function:

[E.sub.1] = [[xi].sup.2] + 1, [[psi].sub.1] (z) = [e.sup.-([xi]/4)(z+1/z)]. (30)

For the first excited state M = 2, from (28) and (23), we have

[E.sub.2] = [[xi].sup.2] + 3 + 2[xi]([z.sub.2]), [[psi].sub.2] (z) = [z.sup.-1/2][e.sup.-([xi]/4)(z+1/z)] (z - [z.sub.2]), (31)

where [z.sub.2] = [+ or -]1. Similarly, the second excited state solution corresponding to M = 3 is given by

[E.sub.3] = [[xi].sup.2] + 5 + 2[xi]([z.sub.2] + [z.sub.3]), [[psi].sub.3] (z) = [z.sup.-1][e.sup.-([xi]/4)(z+1/z)] ([z.sup.2] - ([z.sub.2] + [z.sub.3]) z + [z.sub.2][z.sub.3]), (32)

where the distinct roots [z.sub.2] and [z.sub.3] are obtained from the Bethe ansatz equation (29) as follows:

[mathematical expression not reproducible]. (33)

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 Figure 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 [E.sub.2] - [E.sub.1] < [E.sub.4] - [E.sub.3] < ... and therefore the energy levels are paired together.

2.3. The Hyperbolic Shifman Potential. Now, we consider a hyperbolic potential introduced by Shifman as [9]

V(x) = [[a.sup.2]/2] [sinh.sup.2] (x) - a(n + 1/2) cosh (x), (34)

where the parameter a is a real constant. The Schrodinger equation for potential (34) is given by

(-[d.sup.2]/d[x.sup.2] + [[a.sup.2]/2] [sinh.sup.2] (x) - a(n + 1/2) cosh (x) - E) x [psi] (x) = 0. (35)

According to the asymptotic behaviours of the wave function at the origin and infinity, we consider the following transformations:

[psi] (x) = [e.sup.-a] cosh(x)[phi] (x), z = cosh (x). (36)

Therefore, the differential equation for [phi](x) reads

H[phi](z) = 0, H = (-[1/2] [z.sup.2] + 1/2) [d.sup.2]/d[z.sup.2] + (a[z.sup.2] - [1/2] z - a) d/dz - (naz + E). (37)

Now, by assuming

[mathematical expression not reproducible], (38)

and defining the operators [A.sub.n] and [A.sup.+.sub.n] as

[A.sub.n] = d/dz - [n.summation over (k=1)] 1/[z - [z.sub.k]], [A.sup.+.sub.n] = -d/dz - ([2a[z.sup.2] - z - 2a]/[1 - [z.sup.2]]) - [n.summation over (k=1)] 1/[z - [z.sub.k]], (39)

we obtain

[mathematical expression not reproducible]. (40)

Comparing the residues at the simple poles z = [+ or -]1 and z = [z.sub.k] with (37), we obtain the following set of equations for the energy and the zeros [z.sub.k]:

[E.sub.n] = a [n.summation over (k=1)] [z.sub.k] - [n.sup.2]/2, (41)

[n.summation over (j[not equal to]k)] [2/[[z.sub.j] - [z.sub.k]]] + [2a[z.sup.2.sub.k] - [z.sub.k] - 2a]/[1 - [z.sup.2.sub.k] = 0, (42)

respectively. Here, we obtain exact solutions of the first three levels. For n = 0, from (41) and (36), we get

[E.sub.0] = 0, [[psi].sub.0] (z) = [e.sup.-az], (43)

and for the first excited state n = 1,

[E.sub.1] = a[z.sub.1] - 1/2, [[psi].sub.1] (z) = [e.sup.-az] (z - [z.sub.1]), (44)

where the root [z.sub.1] is obtained from Bethe ansatz equation (42) as

[z.sub.1] = [1 [+ or -] [square root of (1 + 16[a.sup.2])]]/4a. (45)

Solutions of the second excited state corresponding to n = 2 are given as

[E.sub.2] = a([z.sub.1] + [z.sub.2]) - 2, [[psi].sub.2] (z) = [e.sup.-az] ([z.sup.2] - ([z.sub.1] + [z.sub.2]) z + [z.sub.1][z.sub.2]), (46)

where the roots [z.sub.1] and [z.sub.2] are obtained from (42) as

[mathematical expression not reproducible]. (47)

Here, we have taken the parameter a = 0.1. Our numerical results obtained for the first three levels are displayed and compared in Table 4. Also, the Shifman DWP for the parameter values a = 0.1 and n = 1 is plotted in Figure 3. In the next section, we intend to reproduce the results using the Lie algebraic approach in the framework of quasi-exact solvability.

3. 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 sl(2) 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 sl(2) and thereby we obtain the exact solutions of the systems using the representation theory of sl(2). The method is outlined in the Appendix.

3.1. The Generalized Manning Potential. Applying the results of the Appendix, it is easy to verify that (4) can be written in the Lie algebraic form

[mathematical expression not reproducible], (48)

if the following condition (constraint of quasi-exact solvability) is fulfilled,

(3 + 2[square root of (-[E.sub.n])])[square root of ([v.sub.1])] + [v.sub.1] + [v.sub.2] = -4n[square root of ([v.sub.1])], (49)

which is the same result as (12). As a result, the operator H preserves the finite-dimensional invariant subspace [phi](z) = [[SIGMA].sup.n.sub.m=0] [a.sub.m][z.sup.m] spanned by the basis <1, z, [z.sup.2], ..., [z.sup.n]> and therefore the n + 1 states can be determined exactly. Accordingly, (48) can be represented as a matrix equation whose nontrivial solution exists if the following constraint is satisfied (Cramer's rule),

[mathematical expression not reproducible], (50)

which provides important constraints on the potential parameters. Also, from (3), the wave function is as

[mathematical expression not reproducible], (51)

where the expansion coefficients [a.sub.m] obey the following three-term recurrence relation:

([m.sup.2] + [3/2] m + 1/2) [p.sub.m+1] + (2 [square root of ([v.sub.1])]) [p.sub.m-1] - (m([chi] + m - 1) + [epsilon]/4) [p.sub.m] = 0, (52)

with boundary conditions [a.sub.-1] = 0 and [a.sub.n+1] = 0. Now, for comparison with the results of BAM obtained in the previous section, we study the first three states. For n = 0, from (49), the energy equation is as

(3 + 2[square root of (-[E.sub.0])])[square root of ([v.sub.1])] + [v.sub.1] + [v.sub.2] = 0, (53)

where the constraints on the potential parameters is obtained from (50) as

-[v.sub.1] - [v.sub.2] - [v.sub.3] - [E.sub.0] + [square root of (-[E.sub.0])] - [square root of ([v.sub.1])] = 0. (54)

For the first excited state n = 1, the energy equation is given by

(7 + 2[square root of (-[E.sub.1])]) [square root of ([v.sub.1])] + [v.sub.1] + [v.sub.2] = 0, (55)

where the potential parameters satisfy the constraint

[mathematical expression not reproducible]. (56)

Solutions of the second excited state corresponding to n = 2 are given as

(11 + 2 [square root of (-[E.sub.2])]) [square root of ([v.sub.1])] + [v.sub.1] + [v.sub.2] = 0, (57)

where the constraints on the potential parameters are obtained from (50) as follows:

[mathematical expression not reproducible]. (58)

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.

3.2. The Razavy Bistable Potential. In this case, using the results of the Appendix, the operator H of (24) is expressed as an element of the universal enveloping algebra of sl(2) as

H[phi](z) = 0, H = -[J.sup.+.sub.M][J.sup.-.sub.M] + [xi]/2 [J.sup.+.sub.M] + [J.sup.0.sub.M] + [xi]/2 [J.sup.-.sub.M] + ([M-1]/2 + [E + 1 - 2M - [[xi].sup.2]]/4), (59)

where M = n + 1 = 1, 2, 3,.... As a result of the above algebraization, we can use the representation theory of sl(2) which results in a general matrix equation for arbitrary M whose nontrivial solution condition gives the exact solutions of the system as

[mathematical expression not reproducible]. (60)

Also, from (23), the wave function of the system is as

[[psi].sub.M] (z) = [z.sup.(1-M)/2][e.sup.-([xi]/4)(z+1/z)] [M-1.summation over (m=0)] [a.sub.m][z.sup.m], (61)

where the coefficients [a.sub.m] satisfy the following recursion relation:

(m[xi]/2) [a.sub.m] + ([E - [[xi].sup.2] - 6m + 3]/4) [a.sub.m-1] + ([xi]) [a.sub.m-2] = 0, (62)

with initial conditions [a.sub.M] = 0 and [a.sub.-1] = 0. As examples of the general formula (60) and also, for comparison purpose, we study the first three levels. For the ground state M = 1, from (60), we have

[E.sub.0] = 1 + [[xi].sup.2]. (63)

For the first excited state, from (60), we obtain

[E.sub.2] = [[xi].sup.2] + 3 [+ or -] 2[xi]. (64)

In a similar way, for the second excited state M = 3, we have

1/64 ([[xi].sup.4] + (-2[E.sub.3] - 2)[[xi].sup.2] + [E.sup.2.sub.3] - 14[E.sub.3] + 45) x (-[[xi].sup.2] + [E.sub.3] - 5) = 0, (65)

which yields the energy as

[mathematical expression not reproducible]. (66)

The numerical results are reported and compared with BAM results in Table 2.

3.3. The Hyperbolic Shifman Potential. Comparing (37) with the results of the Appendix, it is seen that the operator H can be expressed in the Lie algebraic form

H[phi](z) = 0, H = 1/2 ([J.sup.+.sub.n] [J.sup.-.sub.n] + [J.sup.-.sub.n] [J.sup.-.sub.n]) - a ([J.sup.+.sub.n] + [J.sup.-.sub.n]) - ([n + 1]/2) [J.sup.0.sub.n] - (E + [n (n + 1)]/4). (67)

Then, using the representation theory of sl(2), the general condition for existence of a nontrivial solution is obtained as

[mathematical expression not reproducible], (68)

which gives the same results as those obtained by BAM in the previous section. For example, for n = 0, from (68), the ground state energy of the system is [E.sub.0] = 0. For n = 1, the energy equation is as follows:

2[E.sup.2.sub.1] + [E.sub.1] - 2[a.sup.2] = 0, (69)

which yields

[E.sub.1] = -1 [+ or -] [square root of (16[a.sup.2] + 1)]]/4. (70)

Likewise, for n = 2, the second excited state energy of the system is obtained from the following relation:

-[E.sup.3.sub.2] - [5/2] [E.sup.2.sub.2] + (4[a.sup.2] - 1) [E.sub.2] + 6[a.sup.2] = 0. (71)

The numerical results are reported and compared with BAM results in Table 4. The wave function of the system from (36) is given as

[[psi].sub.n] (z) = [e.sup.-az] [n.summation over (m=0)] [a.sub.m][z.sup.m], (72)

where the expansion coefficients [a.sub.m]'s satisfy the recursion relation

(- (m + 1) a) [a.sub.m-2] - (E + [m - 1]/2) [a.sub.m-1] - (ma) [a.sub.m] + (m(m + 1)/2) [a.sub.m+1] = 0, (73)

with boundary conditions [a.sub.-2] = 0, [a.sub.-1] = 0, and [a.sub.n+1] = 0.

4. Conclusions

Using the Bethe ansatz method, we have solved the Schrodinger equation for a class of QES DWPs and obtained the general expressions for the energies and the wave functions in terms of the roots of the Bethe ansatz equations. In addition, we have solved the same problems using the Lie algebraic approach within the framework of quasi-exact solvability and obtained the exact solutions using the representation theory of sl(2) Lie algebra. It was found that the results of the two methods are consistent. Also, we have provided some numerical results for the Razavy potential to display the energy splitting explicitly. The main advantage of the methods we have used is that we determine the general exact solutions of the systems and thus, we can quickly calculate the solution of any arbitrary state, without cumbersome procedures and without difficulties in obtaining the solutions of the higher states.

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

Appendix

The Lie Algebraic Approach of Quasi-Exact Solvability

In this Appendix, we outline the connection between sl(2) Lie algebra and the second-order QES differential equations. A differential equation is said to be QES if it lies in the universal enveloping algebra of a QES Lie algebra of differential operators [12]. In the case of one-dimensional systems, sl(2) Lie algebra is the only algebra of first-order differential operators which possesses finite-dimensional representations, whose generators [12],

[J.sup.+.sub.n] = -[z.sup.2] [d/dz] + nz, [J.sup.0.sub.n] = z [d/dz] - n/2, [J.sup.-.sub.n] = d/dz, (A.1)

obey the sl(2) commutation relations as

[[J.sup.+.sub.n], [J.sup.-.sub.n]] = 2[J.sup.0.sub.n], [[J.sup.0.sub.n], [J.sup.[+ or -].sub.n]] = [+ or -][J.sup.[+ or -].sub.n] (A.2)

and leave invariant the finite-dimensional space

[P.sub.n+1] = <1, z, [z.sup.2], ..., [z.sup.n]>. (A.3)

Hence, the most general one-dimensional second-order differential equation H can be expressed as a quadratic combination of the sl(2) generators as

H = [summation over (a,b=0,[+ or -])] [C.sub.ab][J.sup.a][J.sup.b] + [summation over (a=0,[+ or -])] [C.sub.a][J.sup.a] + C. (A.4)

On the other hand, the operator (A.4) as an ordinary differential equation has the following differential form:

H[phi](z) = 0, H = -[P.sub.4] (z) [[d.sup.2]/d[z.sup.2]] + [P.sub.3] (z) [d/dz] + [P.sub.2] (z), (A.5)

where [P.sub.l] are polynomials of at most degree l. Generally, this operator does not have the form of Schrodinger operator but can always be turned into a Schrodinger-like operator

[??] = [e.sup.-A(x)]H[e.sup.A(x)] = -[1/2] [[d.sup.2]/d[x.sup.2]] + [(A').sup.2] - A" + [P.sub.2] (z (x)), (A.6)

using the following transformations:

[mathematical expression not reproducible]. (A.7)

The interested reader is referred to [9-14] for further details.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

References

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M. Baradaran and H. Panahi

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

Correspondence should be addressed to M. Baradaran; marzie.baradaran@yahoo.com

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

Academic Editor: Marc de Montigny

Caption: Figure 1: The generalized Manning DW potential with [v.sub.1] = 1, [v.sub.2] = -12, and [v.sub.3] = 15.255.

Caption: Figure 2: The Razavy bistable potential and the energy levels splitting with [xi] = 2 and M = 12.

Caption: Figure 3: The Shifman DWP with a = 0.1 and n = 1.
Table 1: Solutions of the first three states for the generalized
Manning DWP with [v.sub.1] = 1 and the possible values of [v.sub.2],
where z = [tanh.sup.2](x).

                 Energy   [v.sub.3]   Energy   [v.sub.3]
                 (BAM)      (BAM)     (QES)      (QES)
n    [v.sub.2]    (12)      (13)       (49)      (50)

0       -6         -1         6         -1         6
1       12          4      15.2554       4      15.2554
                           26.7446              26.7446
2      -18         -9      26.8458      -9      26.8458
                           41.1214              41.1214
                           66.0329              66.0329

     Energy
n     [36]                Wave function (BAM)

0      -1      [mathematical expression not reproducible]
1      --      [mathematical expression not reproducible]
               [mathematical expression not reproducible]
2      --      [mathematical expression not reproducible]
               [mathematical expression not reproducible]
               [mathematical expression not reproducible]

Table 2: Solutions of the first three states for the Razavy bistable
potential with [xi] = 2, where z = [e.sup.2x].

     Energy     Energy
     (BAM)      (QES)     Energy
M     (28)       (60)      [44]           Wave function (BAM)

1      5          5         5      [[psi].sub.1](z) = [e.sup.-(1/2)
                                                      (z+1/z)]

2      3          3         3      [[psi].sub.2](z) = [z.sup.-1/2]
       11         11        11                        [e.sup.-(1/2)
                                                      (z+1/z)](z + 1)

                                                      [z.sup.-1/2]
                                                      [e.sup.-(1/2)
                                                      (z+1/z)](z - 1)

     2.7538     2.7538      --     [[psi].sub.3](z) = [z.sup.-1]
3      9          9                                   [e.sup.-(1/2)
    19.2462    19.2462                                (z+1/z)]
                                                      ([z.sup.2] +
                                                      (1.5616)z + 1)

                                                      [z.sup.-1]
                                                      [e.sup.-(1/2)
                                                      (z+1/z)]
                                                      ([z.sup.2] - 1)

                                                      [z.sup.-1]
                                                      [e.sup.-(1/2)
                                                      (z+1/z)]
                                                      ([z.sup.2] -
                                                      (2.5616)z +
                                                      0.9999)

Table 3: Energy splitting of the first 12 states for the Razavy
bistable potential with [xi] = 2.

M           E           AE

1           5
2           3           [E.sub.2]-[E.sub.1] = 8
            11
3      2.753788749      [E.sub.2]-[E.sub.1] = 6.246211251
            9
       19.24621125
4      4.071796770      [E.sub.2]-[E.sub.1] = 4.345197986
       8.416994756      [E.sub.4]-[E.sub.3] = 11.65480201
       17.92820323
       29.58300524
5      6.541491983      [E.sub.2]-[E.sub.1] = 2.458508017
            9           [E.sub.4]-[E.sub.3] = 10.53070532
       18.46929468
            29
       41.98921334
6      9.410470052      [E.sub.2]-[E.sub.1] = 1.001031138
       10.41150119      [E.sub.4]-[E.sub.3] = 8.85309320
       21.29169652      [E.sub.6]-[E.sub.5] = 14.14587566
       30.14478972
       42.29783343
       56.44370909
7      12.00197331      [E.sub.2]-[E.sub.1] = 0.28239331
       12.28436662      [E.sub.4]-[E.sub.3] = 6.39511677
       26.60488323      [E.sub.6]-[E.sub.5] = 13.25910659
            33
       44.45652679
       57.71563338
       72.93661667
8    14.2739943644243   [E.sub.2]-[E.sub.1] = 0.05985777
     14.3338521331318   [E.sub.4]-[E.sub.3] = 3.64123808
     33.7279836821029   [E.sub.6]-[E.sub.5] = 12.08376806
     37.3692217631865   [E.sub.8]-[E.sub.7] = 16.21513610
     48.7528515817037
     60.8366196369519
     75.2451703717691
     91.4603064667298
9    16.4065712724492   [E.sub.2]-[E.sub.1] = 0.01029220
     16.4168634672976   [E.sub.4]-[E.sub.3] = 1.46545772
     41.3677353353592   [E.sub.6]-[E.sub.5] = 10.13015038
     42.8331930589016   [E.sub.8]-[E.sub.7] = 15.40382383
     55.7615205763811
     65.8916709587647
     79.4544486908676
     94.8582725150361
     112.009724124943
10   18.4886350611369   [E.sub.2]-[E.sub.1] = 0.00149855
     18.4901336128637   [E.sub.4]-[E.sub.3] = 0.40230849
     48.4784563542366   [E.sub.6]-[E.sub.5] = 7.12561407
     48.8807648437805   [E.sub.8]-[E.sub.7] = 14.43392092
     65.7434045220649   [E.sub.10]-[E.sub.9] = 18.0366580
     72.8690185914347
     85.7451955766933
     100.179116457372
     116.544308485868
     134.580966494549
11     20.54831894      [E.sub.2]-[E.sub.1] = 0.00018969
       20.54850863      [E.sub.4]-[E.sub.3] = 0.08261653
       55.03511334      [E.sub.6]-[E.sub.5] = 3.82023271
       55.11772987      [E.sub.8]-[E.sub.7] = 13.01121477
       77.68896865      [E.sub.10]-[E.sub.9] = 17.2568387
       81.50920136
       94.52213923
       107.5333540
       123.0343674
       140.2912061
       159.1710925
12     22.59494691      [E.sub.2]-[E.sub.1] = 0.00002127
       22.59496818      [E.sub.4]-[E.sub.3] = 0.01380242
       61.34425227      [E.sub.6]-[E.sub.5] = 1.40632980
       61.35805469      [E.sub.8]-[E.sub.7] = 10.5294253
       89.87448537      [E.sub.10]-[E.sub.9] = 16.3641941
       91.28081517      [E.sub.12]-[E.sub.11] = 19.6862271
       106.4782162
       117.0076415
       131.6165721
       147.9807662
       166.0915272
       185.7777543

Table 4: Solutions of the first three states for the Shifman DWP with
a = 0.1, where z = cosh(x).

    Energy    Energy
     (BAM)     (QES)    Energy
n    (41)      (68)       [9]             Wave function (BAM)

0      0         0         0      [[psi].sub.0] (z) = [e.sup.-(1/10)z]

1   0.0193    0.0193    0.0193    [[psi].sub.1] (z) = [e.sub.-(1/10)z]
    -0.5193   -0.5193   -0.5193                       (z - 5.1926)

                                                      [e.sup.-(1/10)z]
                                                      (z + 0.1926)

2   -2.0067   -2.0067   -2.0067   [[psi].sup.2] (z) = [e.sup.-(1/10)z]
    0.05457   0.05457   0.05457                       ([z.sup.2] +
    -0.5479   -0.5479   -0.5479                       (0.0670)z -
                                                      0.4949)

                                                      [e.sup.-(1/10)z]
                                                      ([z.sup.2] -
                                                      (20.5457)z +
                                                      55.9707)

                                                      [e.sup.-(1/10)z]
                                                      ([z.sup.2] -
                                                      (14.5213)z -
                                                      4.4757)
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Title Annotation:Research Article
Author:Baradaran, M.; Panahi, H.
Publication:Advances in High Energy Physics
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Geographic Code:1USA
Date:Jan 1, 2018
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