August  2010, 4(3): 397-409. doi: 10.3934/ipi.2010.4.397

Reconstructing the metric and magnetic field from the scattering relation

1. 

Kazakh British Technical University, 59 Tole Bi st., Almaty, 050000, Kazakhstan

2. 

Department of Mathematics, University of Washington, Seattle, WA 98195-4350

Received  August 2009 Revised  May 2010 Published  July 2010

We develop a method for reconstructing the conformal factor of a Riemannian metric and the magnetic field on a surface from the scattering relation associated to the corresponding magnetic flow. The scattering relation maps a starting point and direction of a magnetic geodesic into its end point and direction. The key point in the reconstruction is the interplay between the magnetic ray transform, the fiberwise Hilbert transform on the circle bundle of the surface, and the Laplace-Beltrami operator of the underlying Riemannian metric.
Citation: Nurlan Dairbekov, Gunther Uhlmann. Reconstructing the metric and magnetic field from the scattering relation. Inverse Problems and Imaging, 2010, 4 (3) : 397-409. doi: 10.3934/ipi.2010.4.397
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