On the spectral theory of systems of first order equations with periodic distributional coefficients

TL;DR

This paper develops a Floquet theorem for first-order systems with periodic distributional coefficients and explores the spectral theory of related equations for specific low-dimensional cases, extending classical results to distributional settings.

Contribution

It introduces a Floquet theorem for systems with periodic distributional coefficients and analyzes spectral properties for 1D and 2D cases with distributional potentials.

Findings

Established Floquet theorem for systems with periodic distributional coefficients.

Analyzed spectral theory for scalar and 2x2 systems with distributional potentials.

Extended classical spectral results to include distributional coefficients.

Abstract

We establish a Floquet theorem for a first-order system of differential equations $u'=ru$ where $r$ is an $n\times n$-matrix whose entries are periodic distributions of order $0$. Then we investigate, when $n=1$ and $n=2$, the spectral theory for the equation $Ju'+qu=wf$ on $\mathbb R$ when $J$ is a real, constant, invertible, skew-symmetric matrix and $q$ and $w$ are periodic matrices whose entries are real distributions of order $0$ with $q$ symmetric and $w$ non-negative.