Generalized quantum isotonic nonlinear oscillator in d dimensions

TL;DR

This paper provides an exact solution to the d-dimensional Schrödinger equation with a generalized isotonic nonlinear-oscillator potential, revealing specific parameter conditions for solvability and explicit bound-state solutions.

Contribution

It introduces a supersymmetric analysis for the generalized isotonic nonlinear oscillator in multiple dimensions and derives explicit energy spectra and wavefunctions under certain conditions.

Findings

Exact solutions for energy levels and wavefunctions are obtained.

The potential is solvable when g=2 and ( a^2)^2 = B^2 + ( + (d-2)/2)^2.

Explicit formulas are provided for all bound states.

Abstract

We present a supersymmetric analysis for the d-dimensional Schroedinger equation with the generalized isotonic nonlinear-oscillator potential V(r)={B^2}/{r^{2}}+\omega^{2} r^{2}+2g{(r^{2}-a^{2})}/{(r^{2}+a^{2})^{2}}, B\geq 0. We show that the eigenequation for this potential is exactly solvable provided g=2 and (\omega a^2)^2 = B^2 +(\ell +(d-2)/2)^2. Under these conditions, we obtain explicit formulae for all the energies and normalized bound-state wavefunctions.