A note on the spectrum of the Neumann Laplacian in periodic waveguides

TL;DR

This paper investigates the spectral properties of the Neumann Laplacian in thin periodic waveguides, revealing band structures and asymptotic behaviors as the waveguide's thickness diminishes.

Contribution

It provides new insights into the spectral band structure of the Neumann Laplacian in periodic waveguides, especially in the thin limit, using asymptotic analysis.

Findings

Spectral bands exhibit specific asymptotic behaviors as waveguide thickness decreases.

The band structure can be characterized through asymptotic analysis.

Results contribute to understanding wave propagation in thin periodic structures.

Abstract

We study the Neumann Laplacian $-\Delta^N$ restricted to a periodic waveguide. In this situation its spectrum $\sigma(-\Delta^N)$ presents a band structure. Our goal and strategy is to get spectral information from an analysis of the asymptotic behavior of these bands provided that the waveguide is sufficiently thin.