Sharp boundary behaviour or eigenvalues for Aharonov-Bohm operators with varying poles

par Manon Nys (Università degli Studi di Torino)

mardi 10 janv. 2017, 14:00 → 15:00 Europe/Paris

Fokko Du Cloux (Université Claude Bernard Lyon 1 - Campus de la Doua, Bâtiment Braconnier)

Joint work with Laura Abatangelo, Veronica Felli and Benedetta Noris. Let Ω be an open, simply connected set in R2. We perturb the problem by introducing a singular magnetic field, i.e. for every a = (a1, a2) ∈ Ω, we let A_a be a singular magnetic potential, corresponding to a singular magnetic field at a, orthog- onal to the plane and vanishing in Ω \ {a}. We are interested in the spectrum of an Aharonov-Bohm operator. We know that such a spectrum is composed of a positive and diverging sequence of eigenvalues of finite multiplicity. From previous results, we already know that the map a a→ λ^a_k has a continuous extension up to the boundary, i.e. λ^_ak → λ_k as a → ∂Ω, where λ_k is the k-th eigenvalue of the Dirichlet Laplacian. We will give the exact asymptotic estimate when the pole a converges to a point of the boundary. This exact estimate depends on the number of nodal lines ending at the boundary point of the limit eigenfunction, that is the eigenfunction of the Dirichlet Laplacian. The proof relies on the construction of a suitable limit profile (which gives a good approximation of the magnetic eigenfunctions at the limit) and makes use of an Almgren type monotonicity formula adapted to magnetic eigenfunctions.