Non-selfadjoint operators and small random perturbations

The spectrum of non-selfadjoint operators can be highly unstable even under very small perturbations. This phenomenon is referred to as "pseudospectral effect". Traditionally this pseudosepctral effect was considered a drawback since it can be the source of immense numerical errors, as shown for instance in the works of L. N. Trefethen. However, this pseudospectral effect can also be the source of many new insights. A line of works by Hager, Bordeaux-Montrieux, Sjöstrand, Christiansen and Zworski exploits the pseudospectral effect to show that the (discrete) spectrum of a large class of non-selfadjoint pseudo-differential operators subject to a small random perturbation follows a Weyl law with probability close to one. In this talk we will discuss some recent results on the macroscopic and microscopic distribution of eigenvalues as well as eigenvector localization and delocalization phenomena of non-selfadjoint operators subject to small random perturbations.