Dirichlet eigenvalues of cones in the small aperture limit

Thomas Ourmières-Bonafos (1, 2)

1 IRMAR - Institut de Recherche Mathématique de Rennes
2 BCAM - Basque Center for Applied Mathematics

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 fiber of the Dirichlet Laplacian. In order to do this, 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 eigenfunctions include simultaneously two scales.

Mots clés: Dirichlet Laplacian; Semi-classical Analysis; Eigenvalues and eigenfunctions

Domaines: Théorie spectrale [math.SP]

Thomas Ourmières-Bonafos. Dirichlet eigenvalues of cones in the small aperture limit. Journal of Spectral Theory, 2014, 4 (3), pp.485-513. ⟨10.4171/JST/77⟩. ⟨hal-00802302v2⟩