Local smoothing for the backscattering transform

Details Local smoothing for the backscattering transform Local smoothing for the backscattering transform

2007-12-22

arxiv.org/abs/0712.3865v1

Mathematics

Analysis of PDEs

22 pages

Scientific paper

An analysis of the backscattering data for the Schr\"odinger operator in odd dimensions $n\ge 3$ motivates the introduction of the backscattering transform $B: C_0^\infty ({\mathbb R}^n;{\mathbb C})\to C^\infty ({\mathbb R}^n; {\mathbb C})$. This is an entire analytic mapping and we write $ Bv = \sum_1^\infty B_Nv $ where $B_Nv$ is the $N$:th order term in the power series expansion at $v=0$. In this paper we study estimates for $B_Nv$ in $H_{(s)}$ spaces, and prove that $Bv$ is entire analytic in $v \in H_{(s)}\cap \Cal E'$ when $s\ge (n-3)/2$.

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