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

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          Abstract

          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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          Author and article information

          Journal
          10 May 2005
          Article
          10.1155/IMRP.2005.287
          math-ph/0505033
          9d17dd98-03a0-42e0-b6f2-4f223265091e
          History
          Custom metadata
          International Mathematics Research Papers 2005:6 (2005) 287-349
          math-ph math.MP
          ccsd ccsd-00004900

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