Talks and Abstracts (tentative)

Jake Fillman: The unitary almost-Mathieu operator

We introduce a family of unitary operators via two-dimensional quantum walks subject to orthogonal magnetic fields. The resulting operators exhibit many striking characteristics such as fractal spectra, Aubry-André duality, and a metal-insulator phase transition.

David Krejcirik: Criticality transition for powers of the discrete Laplacian

We reveal a feature of discrete Schroedinger operators which has no continuous counterpart: The power of the discrete Laplacian on the half-line stops to satisfy any Hardy-type inequality for all sufficiently large exponents (explicitly determined). This is joint work with Borbala Gerhat and Frantisek Stampach.

Nicolas Raymond: On the tunnelling effect with pure magnetic fields

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.