Location: 383N
Time: Thursday, 2:15pm.
Abstract: We consider resonances for the operator $-\Delta +V\otimes \delta_{\partial\Omega}$ where $\Omega\subset\re^d$ is a bounded domain. This operator is a model for quantum corrals as well as other lossy systems. We give a bound on the size of the resonance free region for very general $\Omega$ and in the case that $\partial \Omega$ is strictly convex, we give a dynamical characterization of the resonance free region that is generically sharp. We describe how this characterization can be thought of as a Sabine Law in certain cases.
Abstract: I will try to describe joint work with M. Filoche, D. Jerison, and S. Mayboroda. The initial motivation (Filoche-Mayboroda) concerns the localization of eigenfunctions, say, for a Schrödinger operator with a complicated bounded potential, or the Laplacian on a complicated domain. What we do is try to find an automatic decomposition of the domain into small pieces, related to the given operator, and for this we minimize a variant of the Alt, Caffarelli, and Friedman free boundary problem, where we authorize a large number of phases (instead of 2). The results concern the regularity of the minimizers.