A method for computation of scattering amplitudes and Green functions of whole axis problems
This work provides a computational method for scattering problems in quantum mechanics, but it is incremental as it extends an existing NSBF approach to whole-axis problems.
The paper presents a method for computing scattering amplitudes and Green functions for the one-dimensional Schrödinger operator with a decaying potential, using Neumann series of Bessel functions. The approach enables calculation of a complete orthonormal system of generalized eigenfunctions, leading to efficient computation of scattering data.
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 $λ\in\mathbb{C}$.