Recovery of the singularities of a potential from backscattering data in general 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 $q_B$ constructed from backscattering data. This is archived using a new explicit formula for the multiple dispersion operators in the Fourier transform side. We also show that $q-q_B$ can be up to one derivative more regular than $q$ in the Sobolev scale. On the other hand, we construct counterexamples showing that in general it is not possible to have more than one derivative gain, sometimes even strictly less, depending on the a priori regularity of $q$.

33 pages, 2 figures

References in corpus (2)

• Inverse backscattering for the Schrödinger equation in 2D
• Fixed angle scattering: recovery of singularities and its limitations

Cited by in corpus (3)

• The Born approximation in the three-dimensional Calderón problem
• The Born approximation in the three-dimensional Calderón problem II: Numerical reconstruction in the radial case
• Stable factorization of the Calderón problem via the Born approximation