Boundary and scattering rigidity problems in the presence of a magnetic field and a potential

TL;DR

This paper investigates boundary and scattering rigidity problems for magnetic potential and potential systems on simple Riemannian manifolds, establishing their equivalence and demonstrating the limitations of data at a single energy level for unique determination.

Contribution

It introduces the concept of $ ext{MP}$-systems, proves the equivalence of boundary and scattering rigidity problems for these systems, and provides new rigidity results under various conditions.

Findings

Boundary and scattering rigidity problems are equivalent for simple $ ext{MP}$-systems.

Single energy level data is insufficient for unique determination of $ ext{MP}$-systems.

Rigidity results are established for conformal class metrics, real analytic systems, and 2D systems.

Abstract

In this paper, we consider a compact Riemannian manifold with boundary, endowed with a magnetic potential $\alpha$ and a potential $U$. For brevity, this type of systems are called $\MP$-systems. On simple $\MP$-systems, we consider both the boundary rigidity problem and scattering rigidity problem, see the introduction for details. We show that these two problems are equivalent on simple $\MP$-systems. Unlike the cases of geodesic or magnetic systems, knowing boundary action functions or scattering relations for only one energy level is insufficient to uniquely determine a simple $\MP$-system, even under the assumption that we know the restriction of the system on the boundary $\p M$, and we provide some counterexamples. These problems can only be solved up to an isometry and a gauge transformations of $\alpha$ and $U$. We prove rigidity results for metrics in a given conformal class, for simple real analytic $\MP$-systems and for simple two-dimensional $\MP$-systems.