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Journal of Inverse and Ill-posed Problems

Editor-in-Chief: Kabanikhin, Sergey I.


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1569-3945
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On inverse scattering at high energies for the multidimensional nonrelativistic Newton equation in electromagnetic field

A. Jollivet1

1Department of Applied Physics and Applied Mathematics, Columbia University, New York NY, 10027, USA. Email:

Citation Information: Journal of Inverse and Ill-posed Problems. Volume 17, Issue 5, Pages 441–476, ISSN (Online) 1569-3945, ISSN (Print) 0928-0219, DOI: https://doi.org/10.1515/JIIP.2009.029, July 2009

Publication History

Received:
2007-07-19
Published Online:
2009-07-06

Abstract

We consider the multidimensional nonrelativistic Newton equation in a static electromagnetic field

where VC 2(, ℝ), B(x) is the n × n real antisymmetric matrix with elements Bi,k(x), Bi,kC 1(, ℝ) (and B satisfies the closure condition), and β |j1|(1 + |x|)–(α + |j1|) for x, 1 ≤ |j 1| ≤ 2, 0 ≤ |j 2| ≤ 1, |j 2| = |j 1| – 1, i, k = 1, . . . , n and some α > 1. We give estimates and asymptotics for scattering solutions and scattering data for the equation (∗) for the case of small angle scattering. We show that at high energies the velocity valued component of the scattering operator uniquely determines the X-ray transforms PV and PBi,k (on sufficiently rich sets of straight lines). Applying results on inversion of the X-ray transform P we obtain that for n ≥ 2 the velocity valued component of the scattering operator at high energies uniquely determines (∇V, B). We also consider the problem of recovering (∇V, B) from our high energies asymptotics found for the configuration valued component of the scattering operator. Results of the present work were obtained by developing the inverse scattering approach of Novikov [Ark. Mat. 37: 141–169, 1999] for (∗) with B ≡ 0 and of Jollivet [J. Math. Phys. 47: 062902, 2006] for the relativistic version of (∗). We emphasize that there is an interesting difference in asymptotics for scattering solutions and scattering data for (∗) on the one hand and for its relativistic version on the other.

Key words.: Newton equation in electromagnetic field; scattering data; inverse problems

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