Singular Isotonic Oscillator, Supersymmetry and Superintegrability

TL;DR

This paper extends a supersymmetric quantum mechanics method to singular isotonic oscillators with two singularities and applies it to certain superintegrable systems, providing explicit solutions where polynomial algebra methods fail.

Contribution

It introduces a novel approach to construct isospectral partners for singular oscillators with multiple singularities and applies it to complex superintegrable systems.

Findings

Constructed isospectral partners for singular oscillators with multiple singularities.

Derived explicit solutions involving parabolic cylinder functions.

Extended supersymmetric methods to 2D superintegrable systems with higher-order integrals.

Abstract

In the case of a one-dimensional nonsingular Hamiltonian $H$ and a singular supersymmetric partner $H_a$, the Darboux and factorization relations of supersymmetric quantum mechanics can be only formal relations. It was shown how we can construct an adequate partner by using infinite barriers placed where are located the singularities on the real axis and recover isospectrality. This method was applied to superpartners of the harmonic oscillator with one singularity. In this paper, we apply this method to the singular isotonic oscillator with two singularities on the real axis. We also applied these results to four 2D superintegrable systems with second and third-order integrals of motion obtained by Gravel for which polynomial algebras approach does not allow to obtain the energy spectrum of square integrable wavefunctions. We obtain solutions involving parabolic cylinder functions.