Stability for a magnetic Schrödinger operator on a Riemann surface with boundary

Abstract

We consider a magnetic Schrödinger operator $({\nabla }^X{)}^{\ast }{\nabla }^X+q$ on a compact Riemann surface with boundary and prove a $\log \log$-type stability estimate in terms of Cauchy data for the electric potential and magnetic field under the assumption that they satisfy appropriate a priori bounds. We also give a similar stability result for the holonomy of the connection 1-form $X$.