Spectral Theory for Systems of Ordinary Differential Equations with Distributional Coefficients

TL;DR

This paper develops a spectral theory framework for first-order systems of differential equations with distributional coefficients, extending classical results to more general, possibly indefinite, cases.

Contribution

It introduces a spectral theory approach for systems with distributional coefficients without the usual definiteness condition, including construction of relations and spectral transformations.

Findings

Constructed minimal and maximal relations for the system

Established the existence of a spectral (Fourier) transformation

Analyzed special cases with regular endpoints and 2x2 systems

Abstract

We study the spectral theory for the first-order system $Ju'+qu=wf$ of differential equations on the real interval $(a,b)$ when $J$ is a constant, invertible skew-Hermitian matrix and $q$ and $w$ are matrices whose entries are distributions of order zero with $q$ Hermitian and $w$ non-negative. Also, we do not pose the definiteness condition customarily required for the coefficients of the equation. Specifically, we construct minimal and maximal relations, and study self-adjoint restrictions of the maximal relation. For these we determine Green's function and prove the existence of a spectral (or generalized Fourier) transformation. We have a closer look at the special cases when the endpoints of the interval $(a,b)$ are regular as well as the case of a $2\times2$ system. Two appendices provide necessary details on distributions of order zero and the abstract spectral theory for relations.