Fixed Angle Scattering: Recovery of Singularities and Its Limitations

Abstract

We prove that in dimension n ≥ 2 the main singularities of a complex potential q having a certain a priori regularity are contained in the Born approximation q θ constructed from fixed angle scattering data. Moreover, q − q θ can be up to one derivative more regular than q in the Sobolev scale. In fact, this result is optimal, we construct a family of compactly supported and radial potentials for which it is not possible to have more than one derivative gain. Also, these functions show that for n > 3, the maximum derivative gain can be very small for potentials in the Sobolev scale not having a certain a priori level of regularity which grows with the dimension.

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 .…”

“…holds for any sphere S ρ ⊂ R n of radius ρ and (Lebesgue) measure σ ρ , see [18,Proposition A.1] for an elementary proof of this result.…”

“…Let us introduce the backscattering inverse problem more rigorously (see, for example, [11, chapter V] for an introduction to Schrödinger scattering theory). We follow a similar exposition to that of [18].…”

“…The case of full data has been studied in [20][21][22] (real potentials) and [1,18] (complex potentials) and the case of fixed angle data in [31] in dimension 2, and [28] in dimension n ≥ 2. The regularity gain has been improved recently in [18]. Analogous problems have been formulated to study the recovery of singularities of live loads in Navier elasticity, see [3,4].…”

The Fixed Angle Scattering Problem with a First-Order Perturbation

“…The one-dimensional case is quite classical, see [6,15]. In dimensions n ≥ 2, uniqueness has been proved for small or generic potentials [1,23], recovery of singularities results are given in [18,20], and uniqueness of the zero potential is considered in [2]. Recently, in [19,21] it was proved that measurements for two opposite fixed angles uniquely determine a potential q ∈ C ∞ c (R n ).…”