BOUNDARY AND SCATTERING RIGIDITY PROBLEMS IN THE PRESENCE OF A MAGNETIC FIELD AND A POTENTIAL

作者：Assylbekov Yernat M; Zhou Hanming^*

来源：Inverse Problems and Imaging, 2015, 9(4): 935-950.

DOI：10.3934/ipi.2015.9.935

• Boundary rigidity
• magnetic field
• potential
• gauge invariance
• action

In this paper, we consider a compact Riemannian manifold with boundary, endowed with a magnetic potential a 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. 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 up to natural obstructions, even under the assumption that the boundary restriction of the system is given, and we provide some counterexamples. By reducing an MP-system to the corresponding magnetic system and applying the results of [6] on simple magnetic systems, we prove rigidity results for metrics in a given conformal class, for simple real analytic MP-systems and for simple two-dimensional MP-systems.

• 出版日期2015-11