New 1-step extension of the Swanson oscillator and superintegrability of its two-dimensional generalization

TL;DR

This paper introduces a novel one-step extension of the Swanson oscillator using supersymmetric quantum mechanics techniques and demonstrates the superintegrability of its two-dimensional generalization.

Contribution

It presents a new one-step extension of the Swanson oscillator and establishes superintegrability for its two-dimensional extension.

Findings

Derived a one-step pseudo-Hermitian extension of the Swanson oscillator.

Established superintegrability of the two-dimensional generalized system.

Abstract

We derive a one-step extension of the well known Swanson oscillator that describes a specific type of pseudo-Hermitian quadratic Hamiltonian connected to an extended harmonic oscillator model. Our analysis is based on the use of the techniques of supersymmetric quantum mechanics and address various representations of the ladder operators starting from a seed solution of the harmonic oscillator given in terms of a pseudo-Hermite polynomial. The role of the resulting chain of Hamiltonians related via similarity transformation is then exploited. In the second part we write down a two dimensional generalization of the Swanson Hamiltonian and establish superintegrability of such a system.