Waveguide with non-periodically alternating Dirichlet and Robin conditions: homogenization and asymptotics

TL;DR

This paper analyzes the spectral properties of a magnetic Schrödinger operator in a strip with non-periodic alternating boundary conditions, establishing homogenization results, convergence rates, and asymptotic expansions for the spectrum.

Contribution

It provides new homogenization results and asymptotic expansions for the spectral bands of the operator with non-periodic boundary condition alternations.

Findings

Uniform resolvent convergence in various norms.

Improved convergence estimates using boundary correctors.

Asymptotic expansions for spectral band functions and spectrum bottom.

Abstract

We consider a magnetic Schroedinger operator in a planar infinite strip with frequently and non-periodically alternating Dirichlet and Robin boundary conditions. Assuming that the homogenized boundary condition is the Dirichlet or the Robin one, we establish the uniform resolvent convergence in various operator norms and we prove the estimates for the rates of convergence. It is shown that these estimates can be improved by using special boundary correctors. In the case of periodic alternation, pure Laplacian, and the homogenized Robin boundary condition, we construct two-terms asymptotics for the first band functions, as well as the complete asymptotics expansion (up to an exponentially small term) for the bottom of the band spectrum.