Quasi-exact treatment of the relativistic generalized isotonic oscillator

Abstract

We investigate the pseudospin symmetry case of a spin-\documentclass[12pt]{minimal}\begin{document}$\frac{1}{2}$\end{document}12 particle governed by the generalized isotonic oscillator, by presenting quasi-exact polynomial solutions of the Dirac equation with pseudospin symmetry vector and scalar potentials. The resulting equation is found to be quasi-exactly solvable owing to the existence of a hidden sl(2) algebraic structure. A systematic and closed form solution to the basic equation is obtained using the Bethe ansatz method. Analytic expression for the energy is obtained and the wavefunctions are derived in terms of the roots to a set of Bethe ansatz equations.