On inverse scattering at high energies for the multidimensional nonrelativistic Newton equation in electromagnetic field
-
A. Jollivet
Abstract
We consider the multidimensional nonrelativistic Newton equation in a static electromagnetic field
where V ∈ C2(, ℝ), B(x) is the n × n real antisymmetric matrix with elements Bi,k(x), Bi,k ∈ C1(
, 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 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.
© de Gruyter 2009
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Articles in the same Issue
- Numerical methods for solving inverse problems for time fractional diffusion equation with variable coefficient
- On inverse scattering at high energies for the multidimensional nonrelativistic Newton equation in electromagnetic field
- Representation formulae for solutions to direct and inverse degenerate in time first-order Cauchy problems in Banach spaces
- Recover implied volatility of underlying asset from European option price
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