Absolute continuity and band gaps of the spectrum of the Dirichlet Laplacian in periodic waveguides

TL;DR

This paper investigates the spectral properties of the Dirichlet Laplacian in periodic waveguides, demonstrating absolute continuity under thinness conditions and identifying the existence and location of spectral gaps.

Contribution

It establishes conditions for absolute continuity of the spectrum and proves the existence of spectral gaps in periodic waveguides.

Findings

Spectrum is absolutely continuous in sufficiently thin waveguides.

At least one spectral gap exists and its location is determined.

Results apply to the Dirichlet Laplacian in periodic waveguides.

Abstract

Consider the Dirichlet Laplacian operator $-\Delta^D$ in a periodic waveguide $\Omega$. On the condition that $\Omega$ is sufficiently thin, we show that its spectrum $\sigma(-\Delta^D)$ is absolutely continuous (in each finite region). In addition, we ensure the existence of at least one gap in $\sigma(-\Delta^D)$ and locate it.