Calderón inverse problem with partial data on Riemann surfaces

Abstract

On a fixed smooth compact Riemann surface with boundary $(M_0,g)$, we show that, for the Schrödinger operator ${\Delta }_g+V$ with potential $V\in C^{1,\alpha }(M_0)$ for some $\alpha >0$, the Dirichlet-to-Neumann map $N{|}_{\Gamma }$ measured on an open set $\Gamma \subset \partial M_0$ determines uniquely the potential $V$. We also discuss briefly the corresponding consequences for potential scattering at zero frequency on Riemann surfaces with either asymptotically Euclidean or asymptotically hyperbolic ends.