Energy levels for $\mathcal{PT}$-symmetric deformation of the Mathieu equation

TL;DR

This paper introduces a non-Hermitian, $ ext{PT}$-symmetric deformation of the Mathieu equation, analyzing its spectral properties and phase transitions, revealing a richer spectral structure and the effects of boundary conditions and parameters.

Contribution

It presents a novel $ ext{PT}$-symmetric deformation of the Mathieu equation and explores its spectral phases and boundary effects, extending understanding of non-Hermitian quantum models.

Findings

Reproduces expected $ ext{PT}$-symmetric behaviors

Identifies a richer spectral structure

Analyzes boundary condition effects on $ ext{PT}$ breaking

Abstract

We propose a non-Hermitian deformation of the Mathieu equation that preserves $\mathcal{PT}$ symmetry and study its spectrum and the transition from $\mathcal{PT}$-unbroken to $\mathcal{PT}$-broken phases. We show that our model not only reproduces behaviors expected by the literature but also indicates the existence of a richer structure for the spectrum. We also discuss the influence of the boundary condition and the model parameters in the exceptional line that marks the $\mathcal{PT}$ breaking.