New estimates of the Green–Faddeev function and recovering of singularities in the two-dimensional Schrödinger operator with fixed energy

Abstract

We prove that in two-dimensional potential scattering the singularities of the unknown compactly supported potential can be obtained exactly by the Born approximation corresponding to the scattering data with fixed positive energy. The proof is based on the new estimates for the Green–Faddeev function in the weighted spaces Lpσ(R2), 1 < p ≤ ∞. These estimates allow us to consider potentials with stronger singularities than in previous publications.