Unified treatment of a class of spherically symmetric potentials: quasi-exact solution

In this paper, we investigate the Schr\"odinger equation for a class of spherically symmetric potentials in a simple and unified manner using the Lie algebraic approach within the framework of quasi-exact solvability. We illustrate that all models give rise to the same basic differential equation, which is expressible as an element of the universal enveloping algebra of sl(2). Then, we obtain the general exact solutions of the problem by employing the representation theory of sl(2) Lie algebra.


Introduction
From the viewpoint of solvability, the spectral problems are divided into two main classes, the exactly solvable (ES) models and exactly non-solvable models. A quantum model is called ES if for all its energy levels and corresponding wavefunctions, explicit expressions can be determined algebraically. These models are distinguished by the fact that there is a natural basis in the Hilbert space in which the infinite-dimensional Hamiltonian can be diagonalized with the help of algebraic methods. In the literature, considerable efforts have been devoted to obtaining the exact solutions of the relativistic and non-relativistic equations using different methods and techniques [1][2][3][4][5]. In contrast, the exactly non-solvable models are the spectral problems which their infinite-dimensional Hamiltonian cannot be diagonalized algebraically.
Unfortunately, there are only a small number of potentials for which the Schrödinger equation can be solved exactly, such as the harmonic oscillator [6,7], Pöschl-Teller [8,9], Coulomb [10][11][12], Morse [13,14], Rosen-Morse [15], Manning-Rosen [16,17], Tietz [18], etc [19][20][21][22]. In 1980's, the series of papers by Shifman, Ushveridze and Turbiner was devoted to the introduction of an intermediate class between the ES and the exactly non-solvable models for which a certain finite number of eigenvalues and eigenfunctions, but not the whole spectrum, can be calculated exactly by algebraic methods. They were called quasi-exactly solvable (QES) [23][24][25][26]. These models are distinguished by the fact that the Hamiltonian is expressible as a quadratic combination of the generators of a finite-dimensional Lie algebra of first order differential operators preserving a finite-dimensional subspace of functions and thereby can be represented as a block-diagonal matrix with at least one finite block. Thus, the problem reduces to diagonalizing this block and computing the corresponding eigenvalues and eigenfunctions, which always can be done. In this paper, using the Lie algebraic approach, we present a simple unified derivation and exact solution of the Schrödinger equation for a class of four spherically symmetric potentials within the framework of quasi-exact solvability. We demonstrate that all four cases are reducible to the same basic differential equation which can be solved exactly due to the existence of a hidden (2) sl symmetry. Then, with the aid of the representation theory of (2) sl , the general exact solution to the basic equation is determined.
This paper is organized as follows: in section 2, we briefly review the Lie algebraic approach of quasi-exact solvability. In section 3, we introduce the four systems and transform the corresponding equations into a same form that is suitable for (2) sl algebraization and can be expressed as an element of the universal enveloping algebra of (2) sl . Then, using the representation theory of (2) sl , we obtain the general exact solution to the basic equation in section 4. Also, the closed-form expressions for the eigenvalues and eigenfunctions as well as the allowed parameters of potentials are given for each of the systems. We end with conclusions in section 5.

Quasi-exact solvability through the sl(2) algebraization
The general problem of quantum mechanics is to solve the Schrödinger equation  [25,26]. According to Ref. [25], any onedimensional QES differential equation possesses a hidden Lie algebra (2) sl which is the only Lie algebra of first-order differential operators that possesses a finite-dimensional representation. This implies that they can be rewritten in terms of (2) sl generators with differential operators [23][24][25] n n With these properties, it is easy to verify that the most general second order differential operator in the enveloping algebra of In the next section, using the method given above, we show that exact solutions of the Schrödinger equation for the four models can be simply obtained in a unified treatment.

The four models and the corresponding differential equations
In this section, we introduce the models which will be the object of study in this paper. For each case, we illustrate that the corresponding Schrödinger equation is reducible to the basic differential equation of second order which is QES due to the existence of a hidden (2) sl algebraic structure.

Non-polynomial Potential
First, we consider the non-polynomial oscillator defined as [27] where 0   and  is a real constant. This potential appears in various branches of physics such as the zero-dimensional quantum field theory with nonlinear Lagrangian [28,29], quantum mechanics [30,31], laser physics [32,33], etc. This potential has been studied by a variety of methods including the analytic continued fractions [34], the supersymmetric quantum mechanics [35], the 1 N expansion method [27], the wavefunction ansatz method [12] and the Bethe ansatz method [36]. In atomic units ( 1 mc    ), the radial Schrödinger equation with potential (8) is

Screened Coulomb Potential
Here, we consider the screened Coulomb potential defined by [37] ( ) The corresponding radial Schrödinger equation is given by Several methods and techniques for solving this problem can be found in Refs. [12,[36][37][38][39].
Applying the transformation

Singular integer power potential
The problem of the singular power potentials, have been widely carried out in various branches of physics, in both classical and quantum mechanics [40][41][42][43]. Here, we consider the singular integer power potential as [44,45] From the asymptotic behaviour of the wavefunction, we consider the following transformation Substituting this into Eq. (17) and also replacing r by z, we get

Solutions of the basic differential equation for the four models
In the previous section, we have shown that our QES models, after the appropriate transformations, are expressible as second-order differential equations (11), (15), (19) and (23), respectively. These equations have the same basic structure where a , 0 c , 1 c , 0 b , 1 b and 2 b are real constants. Here, we intend to solve this equation using the Lie algebraic approach within the representation theory of ) 2 ( sl . More precisely, from Eq. (4), the general form of a one-dimensional QES differential equation is as follows [25] n n n n n n n n n n n n n if the following constraint on the coefficients holds 1

. (29) c nb 
Hence, we have shown that the differential operator H is an element of the universal enveloping algebra of (2) sl and thereby we can use the representation theory of (2) with the boundary conditions 1 0 n p   and 1 0 p   . Therefore, we have succeeded in obtaining the exact expressions for the energies, wavefunctions and the allowed values of the potential parameters for the 1 n  first states algebraically. The main advantage of our algebraic method is that we can quickly obtain the general solutions of the systems for any arbitrary n from Eqs. (27), (29) and (31) without the cumbersome numerical and analytical procedures usually involved in obtaining the solutions for higher states. In the following, we apply the above results to obtain explicit solutions for each of the four systems.

Non-polynomial Potential
In this case, from (11) with (25), we get The results obtained for the first three states of this model are displayed in table 1.

Screened Coulomb Potential
In this case, comparing (15) with (25) Then by Eqs. (29) and (14), we obtain the following relations for the energy eigenvalues and the corresponding wavefunctions The results for the ground, first, and second excited states of this model are reported in table 2. Relation between potential parameters and l The radial wavefunction

Singular integer power potential
In this case, from (19) and (25), we have Then from Eqs. (29) and (18), we obtain the following relations for energy and wavefunction

Conclusions
In this paper, we have studied the Schrödinger equation for a class of spherically symmetric potentials and illustrated that these models can be treated in a simple and unified manner in the Lie algebraic approach. We have shown that all these models give rise to the same basic differential equation, which is expressible as an element of the universal enveloping algebra of (2) sl . We have then obtained the general exact solutions of the basic equation within the framework of representation theory of (2) sl Lie algebra. Also, we have reported the explicit expressions for the energy, wavefunction and the constraint on the potential parameters for each of the systems. The advantage of our algebraic method is that we can quickly obtain the general solutions of the systems for any arbitrary n, without the cumbersome procedures of obtaining the solutions for higher states. This method is found to be computationally much simpler than other methods.