Fixed angle scattering: recovery of singularities and its limitations

TL;DR

This paper demonstrates that fixed angle scattering data can recover the main singularities of a potential in multiple dimensions, with optimal regularity gains, and explores the limitations of this recovery process.

Contribution

It establishes the optimal regularity gain in recovering potential singularities from fixed angle scattering and constructs examples showing the limits of this method.

Findings

Main singularities are contained in the Born approximation.

The potential difference can be more regular than the original potential.

Maximum derivative gain depends on the potential's regularity and dimension.

Abstract

We prove that in dimension $n \ge 2$ the main singularities of a complex potential $q$ having a certain a priori regularity are contained in the Born approximation $q_\theta$ constructed from fixed angle scattering data. Moreover, ${q-q_\theta}$ can be up to one derivative more regular than $q$ in the Sobolev scale. In fact, this result is optimal, we construct a family of compactly supported and radial potentials for which it is not possible to have more than one derivative gain. Also, these functions show that for $n>3$, the maximum derivative gain can be very small for potentials in the Sobolev scale not having a certain a priori level of regularity which grows with the dimension.