Discrete convolution operators in positive characteristic: a variation on the Floquet-Bloch Theory

TL;DR

This paper introduces a variation of Floquet-Bloch theory tailored for matrix convolution operators on lattices over fields with positive characteristic, extending classical and discrete Floquet theories to new algebraic settings.

Contribution

It develops a novel Floquet theory variation for convolution operators in positive characteristic, expanding the mathematical framework for periodic difference equations.

Findings

Extended Floquet theory to positive characteristic fields

Analyzed spectral properties of convolution operators

Provided new tools for periodic lattice problems

Abstract

The classical Floquet theory deals with Floquet-Bloch solutions of periodic PDEs (see e.g., P. Kuchment. Floquet Theory for Partial Differential Equations. Basel: Birkhauser, 1993). Peter Kuchment developed as well a discrete version of this theory for difference vector equations on lattices, including the Floquet theory on infinite periodic graphs. Here we propose a variation on this theory for matrix convolution operators acting on vector functions on lattices with values in a field of positive characteristic.