Abstract
We investigate the Schrödinger 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 ofsl(2). Then, we obtain the general exact solutions of the problem by employing the representation theory ofsl(2)Lie algebra.
Highlights
From the viewpoint of solvability, the spectral problems are divided into two main classes: the exactly solvable (ES) models and exactly nonsolvable models
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 sl(2) symmetry
The main advantage of our algebraic method is that we can quickly obtain the general solutions of the systems for any arbitrary n from (27), (29), and (31) without the cumbersome numerical and analytical procedures usually involved in obtaining the solutions for higher states
Summary
From the viewpoint of solvability, the spectral problems are divided into two main classes: the exactly solvable (ES) models and exactly nonsolvable models. In 1980s, the series of papers by Shifman, Ushveridze, and Turbiner was devoted to the introduction of an intermediate class between the ES and the exactly nonsolvable 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. Using the method given above, we show that exact solutions of the Schrodinger equation for the four models can be obtained in a unified treatment
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