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

Abstract

We propose a non-Hermitian deformation of the Mathieu equation that preserves $\mathcal{P}\mathcal{T}$ symmetry and study its spectrum and the transition from $\mathcal{P}\mathcal{T}$-unbroken to $\mathcal{P}\mathcal{T}$-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 $\mathcal{P}\mathcal{T}$ breaking.