New approaches to inverse scattering (Q2832509)

The author presents a survey of different reconstruction processes occurring in scattering theory. He starts considering Cormack's scattering and more specifically a family of confocal paraboloids \(Z(p,\omega )\) of revolution with focal point at the origin in the Euclidean space \(E^{n}\). He presents the formulas which allow reconstructing a function \(f\in C^{1}(E^{2})\) from the integrals \(\mathrm{R}f(p,\omega )=\int_{Z(p,\omega )}fdS\) or \(\mathrm{M} f(p,\omega )=\int_{Z(p,\omega )}\frac{fdS}{\left| x/\left| x\right| +\omega \right| }\). For the proof, he introduces the resolved function which generates the family \(Z(p,\omega )\) and he analyzes its properties. He finally uses earlier results he proved. The next section of the paper is devoted to the description of Compton's recovering process in the case of hyperbolic spaces. The author again presents the formulas which allow reconstructing a function \(f\) with compact support on the hyperbolic ball from its hyperbolic integral transform Rg, distinguishing between the cases \(n\) odd or even. In the case of Bragg's scattering with a smooth (0,2)-tensor field \(g\) with compact support in \( E\setminus \Gamma \) where \(E\) is the Euclidean space of dimension 3 and \( \Gamma \subset E\) is a piecewise smooth curve, the author proves how to reconstruct the Saint-Venant tensor \(Vg(x)\) from the ray integrals \(Xg(y;v)\) . The author then considers the case of monochromatic waves moving in a non-homogeneous medium with a smooth refraction coefficient \(\mathbf{n}\). He first recalls the stability results concerning the Helmholtz operator \( (\Delta +(2\pi \widetilde{\mathbf{n}}\omega )^{2}\widetilde{\mathbf{n}} ^{2}\omega ^{2})\widetilde{u}=0\) that he proved in [J. Anal. Math. 91, 247--268 (2003; Zbl 1078.78008)] using Gaussian beam solutions to \((\Delta +(2\pi \mathbf{n}\omega )^{2})u=0\). Here \(\widetilde{\mathbf{n}}\) is a perturbation of \(\mathbf{n}\). He derives from these special solutions a phase contrast imaging process. The paper ends with considerations on inverse source problems and the time-reversal algorithm. The author here recalls the definition of oscillatory polynomials and of hyperbolic cavities and their properties.