Solving for the low-rank tensor components of a scattering wave function

TL;DR

This paper introduces an iterative low-rank tensor decomposition method to efficiently approximate outgoing wave solutions of high-dimensional Schrödinger equations in atomic and molecular breakup reactions, demonstrated in 2D and 3D cases.

Contribution

It presents a novel iterative approach for low-rank approximation of scattering wave functions in high-dimensional problems, applicable to 2D matrices and 3D tensors.

Findings

Effective low-rank approximations for scattering waves in 2D and 3D

Iterative method converges to accurate solutions

Applicable to complex atomic and molecular breakup scenarios

Abstract

Atomic and molecular breakup reactions, such as multiple-ionisation, are described by a driven Schr\"odinger equation. This equation is equivalent to a high-dimensional Helmholtz equation and it has solutions that are outgoing waves, emerging from the target. We show that these waves can be described by a low-rank approximation. For 2D problems this it a matrix product of two low-rank matrices, for 3D problems it is a low-rank tensor decomposition. We propose an iterative method that solves, in an alternating way, for these low-rank components of the scattered wave. We illustrate the method with examples in 2D and 3D.