Local smoothing for the backscattering transform

TL;DR

This paper investigates the backscattering transform associated with the Schrödinger operator in odd dimensions, establishing its analyticity and providing estimates for its power series components in Sobolev spaces.

Contribution

It introduces the backscattering transform for Schrödinger operators and proves its entire analyticity in certain Sobolev spaces, along with estimates for its power series terms.

Findings

The backscattering transform is entire analytic in specified Sobolev spaces.

Estimates for the N-th order terms in the transform's power series are established.

The transform's analyticity extends to distributions when s ≥ (n-3)/2.

Abstract

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 $s\ge (n-3)/2$.