Quantum waveguides with small periodic perturbations: gaps and edges of Brillouin zones

TL;DR

This paper studies how small periodic perturbations in quantum waveguides can create spectral gaps, providing asymptotic analysis of gap edges and quasi-momentum values, with applications to multi-dimensional structures.

Contribution

It introduces a method to analyze spectral gaps caused by small periodic perturbations in quantum waveguides, including asymptotic formulas for gap edges.

Findings

Perturbations can open spectral gaps under certain conditions.

Asymptotic expressions for the edges of spectral gaps are derived.

Examples in 2D and 3D structures illustrate the theory.

Abstract

We consider small perturbations of the Laplace operator in a multi-dimensional cylindrical domain by second order differential operators with periodic coefficients. We show that under certain non-degeneracy conditions such perturbations can open a gap in the continuous spectrum and give the leading asymptotic terms for the gap edges. We also estimate the values of quasi-momentum at which the spectrum edges are attained. The general machinery is illustrated by several new examples in two- and three-dimensional structures.