Optimization of eigenvalues for the magnetic Laplacian
Spectral Methods in Mathematical Physics
07 February 16:30 - 17:30
Corentin Lena - Stockholm University
I will present an ongoing investigation, in collaboration with Pedro Antunes, of planar domains that optimize eigenvalues of the magnetic Laplacian. More specifically, I will consider the Dirichlet and Neumann realizations of a Schrödinger operator with an homogeneous magnetic field B. It has been shown in 1996 by L. Erdös that a disc minimizes the first Dirichlet eigenvalue among all domains of given area, for all values of B. Recent work of S. Fournais and B. Helffer suggests that the disc should also maximize the first Neumann eigenvalue, under the additional constraint of simple connectedness.
I will review those results, and describe a numerical approach based on boundary variation that we applied to the problem. I will then discuss preliminary results about higher eigenvalues and the influence of B.
University of Zurich, UZH