Gauge Transformations and Inverse Quantum Scattering with Medium-Range Magnetic Fields
نویسنده
چکیده
The time-dependent, geometric method for high-energy limits and inverse scattering is applied to nonrelativistic quantum particles in external electromagnetic fields. Both the Schrödingerand the Pauli equations in R2 and R 3 are considered. The electrostatic potential A0 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 A0 are reconstructed from an X-ray transform, which is obtained from the highenergy limit of S. Here previous results by Arians are generalized to the medium-range situation. In a sequel paper, medium-range vector potentials are applied to relativistic scattering.
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