A gap in the continuous spectrum of a cylindrical waveguide with a periodic perturbation of the surface

TL;DR

This paper proves that small periodic surface perturbations in a cylindrical waveguide can create spectral gaps in the Laplace operator's continuous spectrum, especially under long period and small caverns.

Contribution

It demonstrates the conditions under which periodic surface perturbations open spectral gaps in cylindrical waveguides, extending understanding of waveguide spectral properties.

Findings

Spectral gaps can be opened by small periodic surface perturbations.

Long period and small caverns favor gap opening.

Theoretical proof of gap formation in the Laplace operator spectrum.

Abstract

It is proved that small periodic singular perturbation of a cylindrical waveguide surface may open a gap in the continuous spectrum of the Dirichlet problem for the Laplace operator. If the perturbation period is long and the caverns in the cylinder are small, the gap certainly opens.