Plane waveguides with corners in the small angle limit

TL;DR

This paper analyzes the spectral properties of infinite V-shaped waveguides with small opening angles, providing asymptotic formulas for eigenvalues and eigenfunctions as the angle approaches zero.

Contribution

It introduces multi-scale asymptotic analysis for eigenpairs in cornered waveguides and explores their relation to a one-dimensional model and triangular domains.

Findings

Eigenvalues scale with the cube root of the angle.

Eigenfunctions exhibit multi-scale behavior in the triangular and straight parts.

Asymptotic formulas are derived for the lowest eigenvalues and eigenfunctions.

Abstract

The plane waveguides with corners considered here are infinite V-shaped strips with constant thickness. They are parametrized by their sole opening angle. We study the eigenpairs of the Dirichlet Laplacian in such domains when this angle tends to 0. We provide multi-scale asymptotics for eigenpairs associated with the lowest eigenvalues. For this, we investigate the eigenpairs of a one-dimensional model which can be viewed as their Born-Oppenheimer approximation. We also investigate the Dirichlet Laplacian on triangles with sharp angles. The eigenvalue asymptotics involve powers of the cube root of the angle, while the eigenvector asymptotics include simultaneously two scales in the triangular part, and one scale in the straight part of the guides.