Inverse Problems and Imaging (IPI)

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

Pages: 935 - 950, Volume 9, Issue 4, November 2015      doi:10.3934/ipi.2015.9.935

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Yernat M. Assylbekov - Department of Mathematics, University of Washington, Seattle, WA 98195-4350, United States (email)
Hanming Zhou - DPMMS, Centre for Mathematical Sciences, Cambridge CB3 0WB, United Kingdom (email)

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 $\mathcal{MP}$-systems. On simple $\mathcal{MP}$-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 $\mathcal{MP}$-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 $\mathcal{MP}$-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 $\mathcal{MP}$-systems and for simple two-dimensional $\mathcal{MP}$-systems.

Keywords:  Boundary rigidity, magnetic field, potential, gauge invariance, action.
Mathematics Subject Classification:  Primary: 53C24; Secondary: 35R30.

Received: February 2015;      Revised: June 2015;      Available Online: October 2015.