Study of the generalized quantum isotonic nonlinear oscillator potential

TL;DR

This paper investigates a generalized quantum isotonic oscillator potential, presenting methods to find explicit solutions and high-accuracy eigenvalues for various parameter values, advancing understanding of this nonlinear quantum system.

Contribution

It introduces a new approach for obtaining quasi-polynomial solutions and applies the asymptotic iteration method to accurately compute eigenvalues for the generalized potential.

Findings

Explicit quasi-polynomial solutions derived

Eigenvalues computed with high accuracy for various parameters

Conditions on potential parameters established

Abstract

We study the generalized quantum isotonic oscillator Hamiltonian given by H=-d^2/dr^2+l(l+1)/r^2+w^2r^2+2g(r^2-a^2)/(r^2+a^2)^2, g>0. Two approaches are explored. A method for finding the quasi-polynomial solutions is presented, and explicit expressions for these polynomials are given, along with the conditions on the potential parameters. By using the asymptotic iteration method we show how the eigenvalues of this Hamiltonian for arbitrary values of the parameters g, w and a may be found to high accuracy.