Bay Area Microlocal Analysis Seminar: Schrödinger equations with conormal potentials

Seminar | April 12 | 2:10-3 p.m. | 740 Evans Hall

 Jared Wunsch, Northwestern University

 Department of Mathematics

Consider the semiclassical Schrödinger equation $(h^2\Delta +V-E)u=0$, where, instead of being smooth, $V$ is allowed to be singular across a hypersurface. The singularity in the potential turns out to have very interesting consequences for the structure of solutions $u$; in effect, WKB solutions include not just contributions from classical propagation across the interface but also reflected singularities, in what amounts to a quantum diffraction effect (meaning one that is not visible at the level of classical Hamiltonian dynamics). I will discuss the propagation and reflection of semiclassical singularities in this setting, and also its consequences for the existence of quantum resonances in systems where trajectories escape to infinity under classical flow but not under the branched flow where we allow reflections. This is joint work with Oran Gannot.