Recovering singularities of a potential from singularities of scattering data (Q1311704)

The authors consider recovery of a compactly supported potential \(q\) on \(\mathbb{R}^ n\), \(n\geq 3\), of the Schrödinger equation \(-\Delta+q\) from the backscattering data \(A(\omega, -\omega,\lambda)\), \(\omega\in S^{n- 1}\), \(\lambda\in\mathbb{R}\), where \(A\) is the scattering amplitude (far field pattern) of this Schrödinger operator. Recently \textit{G. Eskin} and \textit{J. Ralston} [Commun. Math. Phys. 124, No. 2, 169-215 (1989; Zbl 0706.35136)] proved that the map from \(q\) to the backscattering data is a locally invertible for generic (``almost all'') \(q\), however still there is no global uniqueness result for \(q\). By using the associated wave equation and microlocal analysis the authors prove uniqueness of recovery of a smooth surface \(S\) and normal jumps of \(q\) and its derivatives across \(S\).