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Inverse scattering problem for a two dimensional random potential. (English) Zbl 1203.35292

The setup of the stochastic scattering problem consists in the two-dimensional Schrödinger equation with outgoing radiation condition, where the potential is assumed to be a random generalized function supported in a bounded and simply connected domain in the plane. It is assumed also that its covariance operator is a classical pseudodifferential operator. It is proved that the backscattered field, obtained from a single realization of the potential \(q\), determines uniquely the principal symbol of the covariance operator of \(q\). The main tools used to this end are a combination of harmonic and microlocal analysis with stochastic methods.

MSC:

35R30 Inverse problems for PDEs
35R60 PDEs with randomness, stochastic partial differential equations
35J10 Schrödinger operator, Schrödinger equation
35P25 Scattering theory for PDEs
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
35A27 Microlocal methods and methods of sheaf theory and homological algebra applied to PDEs
81U40 Inverse scattering problems in quantum theory
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