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

TL;DR

This paper introduces a new method using Neumann series of Bessel functions to compute scattering data and Green functions for one-dimensional Schrödinger operators with decaying potentials, enhancing analytical and numerical capabilities.

Contribution

It develops a novel approach based on NSBF representations for calculating scattering amplitudes and Green functions for whole axis problems in quantum mechanics.

Findings

Provides explicit representations for Jost solutions using NSBF

Enables computation of complete eigenfunction systems

Facilitates calculation of scattering amplitudes and Green functions

Abstract

A method for the computation of scattering data and of the Green function for the one-dimensional Schr\"{o}dinger 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 $H-\lambda$ with $\lambda\in\mathbb{C}$.