Journal of Inverse and Ill-posed Problems
Editor-in-Chief: Kabanikhin, Sergey I.
IMPACT FACTOR increased in 2015: 0.987
Rank 59 out of 312 in category Mathematics and 93 out of 254 in Applied Mathematics in the 2015 Thomson Reuters Journal Citation Report/Science Edition
SCImago Journal Rank (SJR) 2015: 0.583
Source Normalized Impact per Paper (SNIP) 2015: 1.106
Impact per Publication (IPP) 2015: 0.712
Mathematical Citation Quotient (MCQ) 2015: 0.43
On inverse scattering at high energies for the multidimensional nonrelativistic Newton equation in electromagnetic field
1Department of Applied Physics and Applied Mathematics, Columbia University, New York NY, 10027, USA. Email: (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
- Published Online:
We consider the multidimensional nonrelativistic Newton equation in a static electromagnetic field
where V ∈ C 2(, ℝ), B(x) is the n × n real antisymmetric matrix with elements Bi,k(x), Bi,k ∈ C 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 P∇V 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.