A note on the partial data inverse problem for a nonlinear magnetic Schr\"odinger operator on Riemann surface

TL;DR

This paper demonstrates the recovery of a nonlinear magnetic Schrödinger potential on a Riemann surface from boundary measurements, using complex analytic methods and linearisation under certain conditions.

Contribution

It introduces a novel approach for inverse boundary problems for nonlinear magnetic Schrödinger operators on Riemann surfaces, leveraging complex analysis and linearisation techniques.

Findings

Potential can be uniquely recovered from boundary data.

Method applies to arbitrarily small boundary subsets.

Relies on analytic properties of the magnetic potential.

Abstract

We recover a nonlinear magnetic Schr\"odinger potential from measurement on an arbitrarily small open subset of the boundary on a compact Riemann surface. We assume that the magnetic potential satisfies suitable analytic properties, in which case the recovery can be obtained by a linearisation argument. The proof relies on the complex analytic methods introduced in [15].