The quadratic contribution to the backscattering transform in the rotation invariant case (Q618992)

The authors consider the Schrödinger operator \(H_v = -\Delta + v\) in \({\mathbb R}^n\), where \(n \geq 3\) is odd. An analysis of the backscattering data for such Schrödinger operators motivates the introduction of the backscattering transform \(B:C^\infty_0({\mathbb R}^n; {\mathbb C})\to C^\infty ({\mathbb R}^n; {\mathbb C})\). This is an entire analytic mapping and \(Bv = \sum\limits_{N=1}^\infty B_N v\) where \(B_Nv\) is the \(N\)-th order term in the power series expansion at \(v = 0\). In the paper under review the authors give explicit formulas for \(B_2(v, w)\) where \(B_2\) is the symmetric bilinear operator that corresponds to the quadratic part of the backscattering transform and \(v\) and \(w\) are rotation invariant.