2013/07/29 by Yernat M. Assylbekov, Hanming Zhou, Assylbekov, Yernat M. +1
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Numerical methods in inverse problems #Spectral Theory in Mathematical Physics #math.AP #math.DG
paper · pdf · doi:10.48550/arxiv.1307.7449
16 pages. arXiv admin note: substantial text overlap with arXiv:math/0611788 by other authors
arxiv created 2013/07/29 · openalex publication_date 2013/07/29 · arxiv updated 2013/07/30 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
In this paper, we consider a compact Riemannian manifold with boundary, endowed with a magnetic potential α 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 α 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.