The d-bar approach to approximate inverse scattering at fixed energy in three dimensions

Details The d-bar approach to approximate inverse scattering at fixed energy in three dimensions The d-bar approach to approximate inverse scattering at fixed energy in three dimensions

2005-05-10

arxiv.org/abs/math-ph/0505033v1

International Mathematics Research Papers 2005:6 (2005) 287-349

Physics

Mathematical Physics

Scientific paper

10.1155/IMRP.2005.287

We develop the d-bar approach to inverse scattering at fixed energy in dimensions $d\ge 3$ of [Beals, Coifman 1985] and [Henkin, Novikov 1987]. As a result we propose a stable method for nonlinear approximate finding a potential $v$ from its scattering amplitude $f$ at fixed energy $E>0$ in dimension $d=3$. In particular, in three dimensions we stably reconstruct n-times smooth potential $v$ with sufficient decay at infinity, $n>3$, from its scattering amplitude $f$ at fixed energy $E$ up to $O(E^{-(n-3-\epsilon)/2})$ in the uniform norm as $E\to +\infty$ for any fixed arbitrary small $\epsilon >0$ (that is with almost the same decay rate of the error for $E\to +\infty$ as in the linearized case near zero potential).

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