PT-symmetric waveguides

TL;DR

This paper investigates PT-symmetric waveguides with Robin boundary conditions, analyzing their spectral properties, eigenvalue emergence, and asymptotic behavior under boundary perturbations.

Contribution

It introduces a new model of PT-symmetric waveguides with Robin boundary conditions and provides spectral analysis, including eigenvalue asymptotics and conditions for their existence.

Findings

Essential spectrum remains positive and unaffected by boundary perturbations.

Small boundary perturbations can produce real eigenvalues near the spectrum threshold.

Explicit asymptotic formulas for eigenvalues and eigenfunctions are derived.

Abstract

We introduce a planar waveguide of constant width with non-Hermitian PT-symmetric Robin boundary conditions. We study the spectrum of this system in the regime when the boundary coupling function is a compactly supported perturbation of a homogeneous coupling. We prove that the essential spectrum is positive and independent of such perturbation, and that the residual spectrum is empty. Assuming that the perturbation is small in the supremum norm, we show that it gives rise to real weakly-coupled eigenvalues converging to the threshold of the essential spectrum. We derive sufficient conditions for these eigenvalues to exist or to be absent. Moreover, we construct the leading terms of the asymptotic expansions of these eigenvalues and the associated eigenfunctions.