Example of periodic Neumann waveguide with gap in spectrum

TL;DR

This paper studies the spectral properties of a periodic Neumann waveguide with attached protuberances, demonstrating the existence of spectral gaps under certain conditions as the size of the protuberances becomes small.

Contribution

It proves the existence of spectral gaps in a periodic waveguide with protuberances, extending understanding of waveguide spectral properties with complex geometries.

Findings

Spectral gaps appear in the waveguide spectrum for small protuberance sizes.

Conditions on the weight function and protuberance dimensions influence gap formation.

The spectrum contains at least one gap under specified geometric and weight conditions.

Abstract

In this note we investigate spectral properties of a periodic waveguide $\Omega^\varepsilon$ ($\varepsilon$ is a small parameter) obtained from a straight strip by attaching an array of $\varepsilon$-periodically distributed identical protuberances having "room-and-passage" geometry. In the current work we consider the operator $\mathcal{A}^\varepsilon=-\rho^\varepsilon\Delta_{\Omega^\varepsilon}$, where $\Delta_{\Omega^\varepsilon}$ is the Neumann Laplacian in $\Omega^\varepsilon$, the weight $\rho^\varepsilon$ is equal to $1$ everywhere except the union of the "rooms". We will prove that the spectrum of $\mathcal{A}^\varepsilon$ has at least one gap as $\varepsilon$ is small enough provided certain conditions on the weight $\rho^\varepsilon$ and the sizes of attached protuberances hold. (Dedicated to Pavel Exner's 70th birthday)