Stability for a magnetic Schr\"odinger operator on a Riemann surface with boundary

TL;DR

This paper establishes a stability estimate for magnetic Schr"odinger operators on Riemann surfaces with boundary, linking Cauchy data to electric potential, magnetic field, and connection holonomy, with implications for inverse problems.

Contribution

It provides the first $ ext{log} ext{log}$-type stability estimates for both electric and magnetic potentials on Riemann surfaces with boundary, including holonomy stability.

Findings

Proved $ ext{log} ext{log}$-type stability for electric potential and magnetic field.

Established stability for the holonomy of the connection 1-form.

Applicable to inverse boundary value problems on Riemann surfaces.

Abstract

We consider a magnetic Schr\"odinger operator $(\nabla^X)^*\nabla^X+q$ 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 $X$.