Wiener-Hopf factorisation on unit circle: some examples from discrete scattering

TL;DR

This paper explores Wiener-Hopf factorization on an annulus for discrete scattering problems, providing examples where matrix kernels can sometimes be reduced to scalar functions, and discusses connections to classical scattering problems.

Contribution

It presents specific examples of Wiener-Hopf factorization on an annulus related to discrete scattering, including cases where matrix problems reduce to scalar ones, and links to classical scattering issues.

Findings

Scalar functions often suffice for factorization

Some matrix kernels can be reduced to scalar problems

Connections to classical scattering problems

Abstract

I discuss some problems featuring scattering due to discrete edges on certain structures. These problems stem from linear difference equations and the underlying basic issue can be mapped to Wiener-Hopf factorization on an annulus in the complex plane. In most of these problems, the relevant factorization involves a scalar function, while in some cases a nxn matrix kernel, with n>=2, appears. For the latter, I give examples of two non-trivial cases where it can be further reduced to a scalar problem but in general this is not the case. Some of the problems that I have presented in this paper can be also interpreted as discrete analogues of well-known scattering problems, notably a few of which are still open, in Wiener-Hopf factorization on an infinite strip in complex plane.