# Local smoothing for the backscattering transform - Mathematics > Analysis of PDEs

Abstract: An analysis of the backscattering data for the Schr\-odinger operator in odddimensions $n\ge 3$ motivates the introduction of the backscattering transform$B: C 0^\infty {\mathbb R}^n;{\mathbb C}\to C^\infty {\mathbb R}^n; {\mathbbC}$. 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 $s\ge n-3-2$.

Author: Ingrid Beltita, Anders Melin

Source: https://arxiv.org/