Non-local divergence-free currents for the account of symmetries in two-dimensional wave scattering

TL;DR

This paper introduces a systematic method using non-local divergence-free currents to incorporate symmetries of the potential into the analysis of two-dimensional wave scattering, improving understanding of symmetry effects on the scattered wave.

Contribution

It develops a novel approach employing symmetry-induced non-local currents to account for potential symmetries in 2D wave scattering, including near-field effects.

Findings

Symmetry-induced currents are divergence-free and relate directly to potential symmetries.

The approach provides symmetry conditions for wave expansion coefficients in angular momentum basis.

It offers a comprehensive description of scattering that includes near-field and symmetry effects.

Abstract

We explore wave-mechanical scattering in two spatial dimensions assuming that the corresponding potential is invariant under linear symmetry transforms such as rotations, reflections and coordinate exchange. Usually the asymptotic scattering conditions do not respect the symmetries of the potential and there is no systematic way to predetermine their imprint on the scattered wave field. Here we show that symmetry induced, non-local, divergence-free currents can be a useful tool for the description of the consequences of symmetries on higher dimensional wave scattering, focusing on the two-dimensional case. The condition of a vanishing divergence of these non-local currents, being in one-to-one correspondence with the presence of a symmetry in the scattering potential, provides a systematic pathway to to take account if the symmetries in the scattering solution. It leads to a description of the scattering process which is valid in the entire space including the near field regime. Furthermore, we argue that the usual asymptotic representation of the scattering wave function does not account for insufficient account for a proper description of the underlying potential symmetries. Within our approach we derive symmetry induced conditions for the coefficients in the wave field expansion with respect to the angular momentum basis in two dimensions, which determine the transition probabilities between different angular momentum states.