Bound states in waveguides with complex Robin boundary conditions

TL;DR

This paper investigates the spectral properties of the Laplacian in a waveguide with complex Robin boundary conditions, establishing conditions for the existence and asymptotic behavior of discrete eigenvalues under perturbations.

Contribution

It provides a new analysis of how boundary condition perturbations affect the discrete spectrum in non-self-adjoint waveguides, including asymptotic expansions in weak coupling.

Findings

Discrete spectrum exists when perturbations oppose unperturbed boundary conditions

Spectrum remains purely essential and real under constant boundary conditions

First-order asymptotic expansion of eigenvalues in weak coupling regime

Abstract

We consider the Laplacian in a tubular neighbourhood of a hyperplane subjected to non-self-adjoint $\mathcal{PT}$-symmetric Robin boundary conditions. Its spectrum is found to be purely essential and real for constant boundary conditions. The influence of the perturbation in the boundary conditions on the threshold of the essential spectrum is studied using the Birman-Schwinger principle. Our aim is to derive a sufficient condition for existence, uniqueness and reality of discrete eigenvalues. We show that discrete spectrum exists when the perturbation acts in the mean against the unperturbed boundary conditions and we are able to obtain the first term in its asymptotic expansion in the weak coupling regime.