The inverse Calderón problem for Schrödinger operators on Riemann surfaces

Abstract

We show that on a smooth compact Riemann surface with boundary $\left(M_0,g\right)$ the Dirichlet-to-Neumann map of the Schrödinger operator ${\Delta }_g+V$ determines uniquely the potential $V$.

This seemingly analytical problem turns out to have connections with ideas in symplectic geometry and differential topology. We will discuss how these geometrical features arise and the techniques we use to treat them.

This is joint work with Colin Guillarmou of ENS. The speaker is partially supported by NSF Grant No. DMS-0807502 during this work.