We consider the observability problem for the Schrödinger equation on the torus. It is well known that the Schrödinger equation (with a scalar potential) is observable from any nonempty open set. However, this is no longer the case in the presence of a magnetic potential. As the first-order perturbation, the long-time dynamics of the semiclassical Schrödinger equation in the high-frequency regime differs significantly from that of the equation with a scalar potential. In particular, high-frequency solutions may concentrate along critical sets of averaged magnetic potential, leading to non-observability away from certain closed geodesics.
In this talk, I will first review the microlocal approach for the observability problem with a scalar potential as a warm-up. I will then explain a natural new geometric condition that is sufficient, and almost necessary, for observability in the case of a magnetic potential. This is joint work with Kévin Le Balc’h (Inria Paris) and Jingrui Niu (HIT).
