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

Yernat M. Assylbekov; Hanming  Zhou

doi:10.3934/ipi.2015.9.935

Article Contents

2015, Volume 9, Issue 4: 935-950. Doi: 10.3934/ipi.2015.9.935

This issue Previous Article Next Article The inverse electromagnetic scattering problem by a mixed impedance screen in chiral media

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

Yernat M. Assylbekov^1, and
Hanming Zhou^2,

1.
Department of Mathematics, University of Washington, Seattle, WA 98195-4350
2.
DPMMS, Centre for Mathematical Sciences, Cambridge CB3 0WB

Received: January 31, 2015

Revised: May 31, 2015

Published: October 2015

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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.

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Mathematics Subject Classification: Primary: 53C24; Secondary: 35R30.

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