Recovery of Singularities of a Multidimensional Scattering Potential

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

“…As opposite to backscattering or fixed angle scattering data, this problem is overdetermined: the full data, A(θ , θ, k), have 2n − 1 degrees of freedom, while the unknown q has only n degrees. In [27][28][29][30] the authors deal with this overdeterminacy by taking as Born approximation the spherical average, with respect to the incident angle, of the fixed angle Born approximation. They proved that, for real potentials q, the singularities are better recovered with this approximation (the way of measuring the singularities is stated in different categories, Lebesgue and Sobolev spaces or Hölder continuity).…”

“…However, many new difficulties arise, starting from the proof of the existence of the scattering solutions (see [9,16,18,24,37] for some results concerning the complex case), which often can be just proved for large wave number k. Since large values of k control the values for large frequencies of the Fourier transform of q, it is then a natural question whether the singularities of the actual complex potential q are the same as the singularities of the Born approximation (mathematical basis for diffraction tomography, see [1]). The question of recovery of singularities using the approach of [27][28][29][30] has not been addressed for complex potentials. We want to point out that in the fixed angle or backscattering approximation, allowing q to be complex is not a major difficulty; the key estimates proven in the above cited references are valid for the complex case.…”

Some inverse problems for Schrödinger operator with Kato potential

“…We prove the result about the spectrum for such operators and consider some familiar methods of reconstruction of the function q(x) in the cases n = 2, 3, which use either the high energy data from the scattering amplitude or the resolvent kernel of H (see, for example, [2,5,13,17,20,22,23,24,28]) and we consider also a new method for the reconstruction unknown potential.…”

“…In the work [17] of Päivärinta and Serov these theorems were proved for potentials having local singularities in the case n ≥ 3 (see also [25]). Ford [6] proved Theorem 4 in the case n = 3 for the same potentials as in the present work with some additional restriction on the value of the potential q(x).…”

“…Using the definition (0.5) of the scattering amplitude and the LippmannSchwinger equation (0.4), as in the articles [17] and [18] for k > 0 we can similarly get the following representation:…”