Plane waveguides with corners in the small angle limit

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...

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Bibliographic Details
Published in:Journal of mathematical physics 2012-12, Vol.53 (12), p.1
Main Authors: Dauge, Monique, Raymond, Nicolas
Format: Article
Language:English
Subjects:
Analysis of PDEs
Approximation
Asymptotic methods
Asymptotic properties
Born-Oppenheimer approximation
Corners
Dirichlet problem
Eigenvalues
Exact sciences and technology
Laplace transforms
Mathematical methods in physics
Mathematical models
Mathematical Physics
Mathematics
Physics
Planes
Sciences and techniques of general use
Waveguides
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Summary: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.
ISSN:0022-2488
1089-7658
DOI:10.1063/1.4769993