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

TL;DR

This paper develops a geometric method to analyze inverse quantum scattering in the presence of medium-range magnetic fields, extending previous results to broader classes of vector potentials and gauge choices.

Contribution

It introduces a class of medium-range vector potentials, discusses their gauge properties, and generalizes inverse scattering results to these broader conditions.

Findings

Reconstruction of magnetic fields and potentials from high-energy scattering data.

Explicit gauge transformation behavior of the scattering operator.

Extension of inverse scattering techniques to 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 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.