Reconstruction of singularities from full scattering data by new estimates of bilinear Fourier multipliers

Abstract

We prove that the singularities of a complex valued potential q in the Schrödinger hamiltonian + q can be reconstructed from the linear Born approximation for full scattering data by averaging in the extra variables. We prove that, with this procedure, the accuracy in the reconstruction improves the previously known accuracy obtained from fixed angle or backscattering data. In particular, for q ∈ W α,2 for α ≥ 0, in 2D we recover the main singularity of q with an accuracy of one derivative; in 3D the accuracy is > 1/2, increasing with α. This gives a mathematical basis for diffraction tomography. The proof is based on some new estimates for multidimensional bilinear Fourier multipliers of independent interest.

Recovery of the singularities of a potential from backscattering data in general dimension

“…Indeed, Theorems 1 and 2 leave a gap of up to 1/2 derivative when max(m, 0) ≤ β < (n − 1)/2 between the positive and negative results. A similar situation is found in the fixed angle and full data scattering problems, where analogous results to Theorems 1 and 2 have been proved in [18] (see [1] for the positive results in the case of full data scattering). In backscattering, this gap has been partially closed in dimension 3 by the mentioned result in [27] and in dimension 2 in [2], where a uniform 1 − derivative gain has been obtained using a weaker regularity scale than the Sobolev scale W α,2 .…”

Fixed Angle Scattering: Recovery of Singularities and Its Limitations

“…Since then, this approach has received a great amount of attention in all the three different scattering problems mentioned. Due to its radial symmetry properties the most studied cases are the backscattering problem (see, among others [2,6,11,15,16,19] and [8,10] for a different approach) and the full data scattering problem (see [12][13][14] for real potentials and [1] for complex potentials). In the case of fixed angle scattering we mention [17] for results in dimension n ≥ 2, [20] in n = 2, and [3] where the techniques introduced in [17] are applied to fixed angle scattering in elasticity.…”

“…We now introduce with more detail the fixed angle and full data scattering inverse problems, following [1] (see, for example, [7, chapter V] for a more general introduction to scattering). Consider the scattering solution u s (k, θ, x), k ∈ (0, ∞), θ ∈ S n−1 , of the stationary Schrödinger equation which satisfies…”

Improved bound for the bilinear Bochner–Riesz operator

“…The similar bilinear operators given by multipliers with different types of singularities also have been of interest and studied by several authors. We refer the reader to [2,3,14,17,23,33,43] and references therein for further relevant literature.…”