Afternoon on magnetic spectral problems

14.00-15.00 Jacob Shapiro, Princeton University

15.00-15.30 Coffee Break

15.30-16.30 Nicolas Raymond, Université d’Angers

Titles and Abstracts

Jacob Shapiro

Title: Quantum Tunneling and Its Absence in Deep Wells and Strong Magnetic Fields

Abstract: New results on quantum tunneling between deep potential wells, in the presence of a strong constant magnetic field are presented. This includes a family of double well potentials containing examples for which the low-energy eigenvalue splitting vanishes, and hence quantum tunneling is eliminated. Further, by deforming within this family, the magnetic ground state can be made to transition from symmetric to anti-symmetric. For generic double wells in a certain regime, tunneling is not suppressed, and a lower bound for the eigenvalue splitting is given. Joint with Fefferman and Weinstein.

Nicolas Raymond

Title: On the tunnelling effect with pure magnetic fields.

Abstract: We will consider the magnetic Laplacian in dimension two. Under the assumption that the magnetic field has a generic double well, we will describe the smallest eigenvalues of this operator in the limit of the intense field. By establishing an explicit asymptotic formula, we will explain why the gap between the two smallest eigenvalues is exponentially small, but non-zero. We will see in particular how the famous center-guide dynamics allows to cross at the quantum level, by tunneling, a magnetic barrier. This is the first extension to a purely magnetic framework of the considerations of Helffer and Sjöstrand dating back to the 80s in the case of electric potentials. We will also underline the essential differences between the two phenomena - electric and magnetic, by evoking the works started more than 10 years ago in this direction. This is a collaboration with S. Fournais, Y. Guedes Bonthonneau and L. Morin.