A method for computation of scattering amplitudes and Green functions of whole axis problems

Abstract

A method for the computation of scattering data and of the Green function for the one-dimensional Schrödinger operator $H:=-\frac{d^2}{dx^2}+q(x)$ with a decaying potential is presented. It is based on representations for the Jost solutions in the case of a compactly supported potential obtained in terms of Neumann series of Bessel functions (NSBF), an approach recently developed in arXiv:1508.02738. The representations are used for calculating a complete orthonormal system of generalized eigenfunctions of the operator $H$ which in turn allow one to compute the scattering amplitudes and the Green function of the operator $H-λ$ with $λ\in\mathbb{C}$.