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

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.

15 pages

References in corpus (2)

• On occurrence of spectral edges for periodic operators inside the Brillouin zone
• Absolute continuity in periodic thin tubes and strongly coupled leaky wires

Cited by in corpus (4)

• An effective Hamiltonian for the eigenvalue asymptotics of a Robin Laplacian with a large parameter
• Opening up and control of spectral gaps of the Laplacian in periodic domains
• Creation of spectral bands for a periodic domain with small windows
• Spectral gaps in a double-periodic perforated Neumann waveguide