Abstract
Throughout this talk, we will discuss tunneling in the context of the Schrödinger equation with a magnetic field. First, we'll look at the work of Helffer and Sjöstrand in the eighties in the case of electric potentials, and their relatively recent revival. Then, we'll explore work carried out over more than ten years - notably with V. Bonnaillie-Noël, S. Fournais, Y. Guedes Bonthonneau, F. Hérau, L. Morin and S. Vũ Ngọc - in a purely magnetic context. At the heart of the presentation, we will consider the magnetic Laplacian in dimension two. Under the assumption that the magnetic field is intense and has a generic double well, we'll exhibit an explicit asymptotic formula revealing that the gap between the two smallest eigenvalues is exponentially small, but non-zero. We'll also explain how the famous center-guide dynamics can be used to quantum tunnel through a magnetic barrier. We will show the first exponentially precise approximations of the eigenfunctions in the zone classically forbidden by magnetic field variations. We'll also highlight the essential differences between electrical and magnetic effects.
