Physics – Mathematical Physics
Scientific paper
2004-12-30
MPEJ vol 11, No 5, December 2005, 32 pp.
Physics
Mathematical Physics
30+2 pages, updated with minor changes, new appendix contains a preview of the sequel paper
Scientific paper
The time-dependent, geometric method for high-energy limits and inverse scattering is applied to nonrelativistic quantum particles in external electromagnetic fields. Both the Schr"odinger- and the Pauli equations in R^2 and R^3 are considered. The electrostatic potential A_0 shall be short-range, and the magnetic field B shall decay faster than |x|^{-3/2} . A natural class of corresponding vector potentials A of medium range is introduced, and the decay and regularity properties of various gauges are discussed, including the transversal gauge, the Coulomb gauge, and the Griesinger vector potentials. By a suitable combination of these gauges, B need not be differentiable. The scattering operator S is not invariant under the corresponding gauge transformations, but experiences an explicit transformation. Both B and A_0 are reconstructed from an X-ray transform, which is obtained from the high-energy limit of S . Here previous results by Arians and Nicoleau are generalized to the medium-range situation. In a sequel paper, medium-range vector potentials are applied to relativistic scattering.
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