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

  • A. Jollivet
Published/Copyright: July 6, 2009
Journal of Inverse and Ill-posed Problems
From the journal Volume 17 Issue 5

Abstract

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

where VC2(, ℝ), B(x) is the n × n real antisymmetric matrix with elements Bi,k(x), Bi,kC1(, ℝ) (and B satisfies the closure condition), and β|j1|(1 + |x|)–(α + |j1|) for x, 1 ≤ |j1| ≤ 2, 0 ≤ |j2| ≤ 1, |j2| = |j1| – 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.

Received: 2007-07-19
Published Online: 2009-07-06
Published in Print: 2009-July

© de Gruyter 2009

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