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Recovering singularities of a potential from singularities of scattering data. (English) Zbl 0790.35112
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 G. Eskin and 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 \(S\).
Reviewer: V.Isakov (Wichita)

MSC:
35R30 Inverse problems for PDEs
35P25 Scattering theory for PDEs
81U40 Inverse scattering problems in quantum theory
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