Recovery of singularities from fixed angle scattering data for biharmonic operator in dimensions two and three

TL;DR

This paper demonstrates that in two and three dimensions, the main singularities of a potential function in a biharmonic operator can be reconstructed from fixed angle scattering data, improving inverse problem techniques.

Contribution

It introduces a method to recover the singularities of the potential from fixed angle scattering data for a biharmonic operator in low dimensions.

Findings

The difference between the Born approximation and the potential function is smoother in Sobolev scale.

Main singularities of the potential are recoverable from fixed angle scattering amplitude.

The method applies to dimensions two and three for the biharmonic operator.

Abstract

The inverse fixed angle problem for operator $\Delta^2 u + V(x,|u|) u$ is considered in dimensions $n=2,3$. We prove that the difference between an inverse fixed angle Born approximation and the function $V(\cdot,1)$ is smoother than the function $V$ itself in some Sobolev scale. This allows us to conclude that the main singularities of the perturbation $V$ can be reconstructed from the knowledge of the scattering amplitude with some fixed incident angle.