Calderon inverse Problem with partial data on Riemann Surfaces

TL;DR

This paper proves that on a Riemann surface, the potential in a Schrödinger operator can be uniquely determined from partial boundary data, with implications for scattering theory on certain non-compact surfaces.

Contribution

It establishes unique identifiability of the potential from partial boundary measurements on Riemann surfaces, extending Calderón's inverse problem to this setting.

Findings

Unique determination of potential from partial Dirichlet-to-Neumann data

Extension of inverse problem results to Riemann surfaces with boundary

Implications for potential scattering at zero frequency

Abstract

On a fixed smooth compact Riemann surface with boundary $(M_0,g)$, we show that for the Schr\"odinger operator $\Delta +V$ 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 0 frequency on Riemann surfaces with asymptotically Euclidean or asymptotically hyperbolic ends.