Inverse scattering for the magnetic Schr\"odinger operator on surfaces with Euclidean ends

TL;DR

This paper establishes that, for surfaces with Euclidean ends, the scattering matrix at a fixed frequency uniquely determines the magnetic connection and potential, advancing inverse scattering theory on non-compact surfaces.

Contribution

It proves a fixed frequency inverse scattering result for the magnetic Schrödinger operator on surfaces with Euclidean ends, identifying the gauge class and potential from scattering data.

Findings

Scattering matrix determines the gauge class of the connection.

Scattering matrix determines the zeroth order potential.

Results hold under suitable decay conditions.

Abstract

We prove a fixed frequency inverse scattering result for the magnetic Schr\"odinger operator (or connection Laplacian) on surfaces with Euclidean ends. We show that, under suitable decaying conditions, the scattering matrix for the operator determines both the gauge class of the connection and the zeroth order potential.