Colorado State University
Inverse Born Approximation in Two Dimensions
By Valery Serov
From Department of Mathematical Sciences, University of Oulu and
Department of Mathematics, University of Washington
When April 20, 2006
2:00 pm sharp
Where Room E105, Engineering Building
Abstract This work deals with the inverse scattering problem for the two-dimensional Schrödinger operator. The inverse problem which is studied here can be formulated as follows: to reconstruct the quantum mechanical potential (or its points of singularities) from the far field measurements (scattering amplitude) of a set of scattering solutions of the Schrödinger equation. A widely applied approximate method of estimating the potential is to use the Born approximation of the scattering solutions. Four famous inverse scattering problems are considered, namely general scattering, backscattering, fixed angle scattering and fixed energy scattering. The obvious advantage of this approximation is: within the Born approximation, the scattering amplitude is simply the Fourier transform of the unknown potential. The main part of these considerations consists in getting sharp enough estimates for the first nonlinear term. These estimates allow to conclude that all singularities and jumps of the unknown potential can be obtained exactly by the inverse Born approximation. Especially, for the potentials from L^p-spaces the approximation agrees with the true potential up to the continuous function. In particular, if the potential is the characteristic function of a bounded domain this domain is uniquely determined by the inverse Born approximation.
Further Information Jennifer Mueller
The Colloquium counts as Seminar Credit for Mathematics Students.