A gap in the spectrum of the Neumann-Laplacian on a periodic waveguide

TL;DR

This paper investigates how small inclusions in a periodic waveguide create spectral gaps in the Laplacian spectrum, preventing certain frequencies from propagating, using asymptotic analysis of elliptic operators.

Contribution

It demonstrates the existence of spectral gaps in the Laplacian spectrum of a perturbed periodic waveguide with small inclusions, a novel result in spectral theory.

Findings

Spectral gaps appear when inclusion diameter is sufficiently small.

Asymptotic analysis confirms the opening of spectral gaps.

The study advances understanding of wave propagation in perturbed periodic structures.

Abstract

We will study the spectral problem related to the Laplace operator in a singularly perturbed periodic waveguide. The waveguide is a quasi-cylinder with contains periodic arrangement of inclusions. On the boundary of the waveguide we consider both Neumann and Dirichlet conditions. We will prove that provided the diameter of the inclusion is small enough in the spectrum of Laplacian opens spectral gaps, i.e. frequencies that does not propagate through the waveguide. The existence of the band gaps will verified using the asymptotic analysis of elliptic operators.