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

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.

accepted for publication in Physics Letters A

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• Pseudo-Hermitian extensions of the harmonic and isotonic oscillators
• Quantized Area of the Schwarzschild Black Hole: A non-Hermitian Perspective
• Exact solution for the time dependent non-Hermitian generalized Swanson oscillator
• Swanson Hamiltonian: non-PT-symmetry phase