MPEJ Volume 11, No. 5, 32 pp.
Received: Jul 14, 2004. Accepted:  Nov 20, 2005.

W. Jung
Gauge Transformations and Inverse Quantum Scattering with Medium-Range
Magnetic Fields

ABSTRACT: The time-dependent,  geometric method for high-energy limits and
inverse scattering is applied to nonrelativistic quantum particles in external
electromagnetic fields. Both the Schroedinger 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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