Planar waveguide with "twisted" boundary conditions: small width

TL;DR

This paper analyzes the limiting behavior of a planar waveguide with twisted boundary conditions as its width approaches zero, establishing uniform resolvent convergence and providing estimates for the rate of convergence.

Contribution

It introduces a novel approach to studying the threshold effect in waveguides with twisted boundary conditions and derives the effective operator in the zero-width limit.

Findings

Effective operator identified as width tends to zero

Uniform resolvent convergence established

Convergence rates estimated

Abstract

We consider a planar waveguide with "twisted" boundary conditions. By twisting we mean a special combination of Dirichlet and Neumann boundary conditions. Assuming that the width of the waveguide goes to zero, we identify the effective (limiting) operator as the width of the waveguide tends to zero, establish the uniform resolvent convergence in various possible operator norms, and give the estimates for the rates of convergence. We show that studying the resolvent convergence can be treated as a certain threshold effect and we present an elegant technique which justifies such point of view.