Quasi-exact Treatment of the Relativistic Generalized Isotonic Oscillator

TL;DR

This paper provides a quasi-exact polynomial solution to the Dirac equation for a relativistic isotonic oscillator under pseudospin symmetry, revealing a hidden algebraic structure and deriving energy and wavefunctions.

Contribution

It introduces a novel quasi-exact solution method for the relativistic isotonic oscillator using Bethe ansatz and uncovers a hidden $sl(2)$ algebraic structure.

Findings

Analytic energy expressions derived

Wavefunctions expressed via Bethe ansatz roots

Revealed hidden $sl(2)$ algebraic structure

Abstract

We investigate the pseudospin symmetry case of a spin-1/2 particle governed by the generalized isotonic oscillator, by presenting quasi exact polynomial solutions to 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 obtain and the wavefunction is derived in terms of the roots to a set of Bethe ansatz equations.