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

Abstract

We consider a magnetic Schrödinger operator (∇ X ) * ∇ 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. Cauchy-Riemann operators on MNext we will introduce the Cauchy Riemann operators ∂,∂ as mappings of kforms to (k + 1)-forms, k = 0, 1.On functions their action is defined by ∂f := π 1,0 df,∂f := π 0,1 df.In holomorphic coordinates this is simply ∂f := ∂ z f dz,∂f := ∂zf dz,