Algebra of 2D periodic operators with local and perpendicular defects

TL;DR

This paper develops an algebraic framework for 2D periodic operators with defects, providing an algorithm to compute their spectrum, including both continuous and discrete parts, with the latter being more complex.

Contribution

It introduces an algebraic approach and an algorithm for spectrum calculation of 2D periodic operators with local and perpendicular defects, addressing both continuous and discrete spectral components.

Findings

Spectrum can be computed via algebraic operations on matrix-valued functions.

Continuous spectrum is accessible through simple algebraic operations and integrations.

Discrete spectrum computation is significantly more complex.

Abstract

We show that 2D periodic operators with local and perpendicular defects form an algebra. We provide an algorithm of finding spectrum for such operators. While the continuous spectral components can be computed by simple algebraic operations on some matrix-valued functions and few number of integrations, the discrete part is much more complicated.