The fixed angle scattering problem with a first order perturbation

TL;DR

This paper addresses the inverse scattering problem for magnetic and electric potentials, demonstrating unique recovery from finitely many measurements, extending previous results to first order perturbations using wave equation techniques.

Contribution

It extends fixed angle scattering results to Hamiltonians with first order perturbations, showing unique determination of coefficients with minimal measurements under symmetry conditions.

Findings

Coefficients are uniquely determined by 2n measurements up to gauge.

Full first order term can be recovered for related gauge-free equations.

Reduced measurements are possible if coefficients have symmetries.

Abstract

We study the inverse scattering problem of determining a magnetic field and electric potential from scattering measurements corresponding to finitely many plane waves. The main result shows that the coefficients are uniquely determined by $2n$ measurements up to a natural gauge. We also show that one can recover the full first order term for a related equation having no gauge invariance, and that it is possible to reduce the number of measurements if the coefficients have certain symmetries. This work extends the fixed angle scattering results of Rakesh and M. Salo to Hamiltonians with first order perturbations, and it is based on wave equation methods and Carleman estimates.