Pseudo-Hermitian extensions of the harmonic and isotonic oscillators

TL;DR

This work presents a systematic construction of pseudo-Hermitian extensions for the harmonic and isotonic oscillators by coupling to imaginary gauge fields, yielding real, equidistant spectra despite non-Hermiticity. By mapping to Hermitian counterparts through Dyson maps and metrics, the authors derive exact position-space wavefunctions and spectra, and they elucidate intertwining and supersymmetric structures in this non-Hermitian setting. They extend the approach to general potentials and demonstrate how gauge couplings preserve spectral reality while enabling straightforward solution of the transformed Hermitian problems. The results provide a rigorous framework for exactly solvable non-Hermitian models with potential implications in optics and related fields, and they reveal rich connections between Swanson-type oscillators, Laguerre/Hermite structures, and isotonic variants.

Abstract

In this work, we describe certain pseudo-Hermitian extensions of the harmonic and isotonic oscillators, both of which are exactly-solvable models in quantum mechanics. By coupling the dynamics of a particle moving in a one-dimensional potential to an imaginary-valued gauge field, it is possible to obtain certain pseudo-Hermitian extensions of the original (Hermitian) problem. In particular, it is pointed out that the Swanson oscillator arises as such an extension of the quantum harmonic oscillator. For the pseudo-Hermitian extensions of the harmonic and isotonic oscillators, we explicitly solve for the wavefunctions in the position representation and also explore their intertwining relations.