Riesz bases and Jordan form of the translation operator in semi-infinite periodic waveguides

TL;DR

This paper analyzes the spectral properties of the translation operator in semi-infinite periodic waveguides, showing it can be represented by a Jordan matrix with a finite number of nontrivial blocks, and establishes Riesz bases for solutions and boundary traces.

Contribution

It introduces a Riesz basis of solutions for the wave equation in periodic waveguides and characterizes the translation operator as a Jordan matrix with finite nontrivial blocks.

Findings

Existence of a Riesz basis of solutions in the waveguide

Representation of the translation operator as a Jordan matrix with finite blocks

Riesz basis property of boundary traces on the waveguide boundary

Abstract

We study the propagation of time-harmonic acoustic or transverse magnetic (TM) polarized electromagnetic waves in a periodic waveguide lying in the semi-strip $(0,\infty)\times(0,L)$. It is shown that there exists a Riesz basis of the space of solutions to the time-harmonic wave equation such that the translation operator shifting a function by one periodicity length to the left is represented by an infinite Jordan matrix which contains at most a finite number of Jordan blocks of size $> 1$. Moreover, the Dirichlet-, Neumann- and mixed traces of this Riesz basis on the left boundary also form a Riesz basis. Both the cases of frequencies in a band gap and frequencies in the spectrum and a variety of boundary conditions on the top and bottom are considered.