Dirichlet eigenvalues of cones in the small aperture limit

Thomas Ourmières-Bonafos (1)

1 IRMAR - Institut de Recherche Mathématique de Rennes

We are interested in finite cones of fixed height 1 parametrized by their opening angle. We study the eigenpairs of the Dirichlet Laplacian in such domains when their apertures tend to 0. We provide multi-scale asymptotics for eigenpairs associated with the lowest eigenvalues of each fibers of the Dirichlet Laplacian. To do so, we investigate the family of their one-dimensional Born-Oppenheimer approximations. The eigenvalue asymptotics involves powers of the cube root of the aperture, while the eigenvectors include simultaneously two scales.

Domaines: Théorie spectrale [math.SP]

Thomas Ourmières-Bonafos. Dirichlet eigenvalues of cones in the small aperture limit. 2013. ⟨hal-00802302v1⟩