Plane waveguides with corners in the small angle limit

Monique Dauge, Nicolas Raymond

The plane waveguides with corners considered here are infinite V-shaped strips with constant thickness. They are parametrized by their sole opening angle $\theta$. We study the eigenpairs of the Dirichlet Laplacian in such domains when their 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 $\theta$ and the zeros $z_{{\sf A}}(n)$ of the reverse Airy function: $$ \mu_{{\sf Gui},n}(\theta)\underset{\theta\to 0}{\sim} \sum_{j\ge0}\gamma^\Delta_{j,n}\theta^{j/3} \quad \mbox{with} \ \ \gamma^\Delta_{0,n}=\frac{1}{4}, \ \ \gamma^\Delta_{1,n}=0, \ \ \mbox{and}\ \ \gamma^\Delta_{2,n}=2(4\pi\sqrt{2})^{-2/3}z_{{\sf A}}(n), \quad n\ge1, $$ while the eigenvector asymptotics include simultaneously two scales in the triangular part, and one scale in the straight part of the guides.

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25 janvier 2012

Journal of Mathematical Physics 53 (2012) 123529. DOI: 10.1063/1.4769993

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