Green's functions for first-order systems of ordinary differential equations without the unique continuation property

TL;DR

This paper develops a spectral theory framework for first-order differential systems with distributional coefficients where unique continuation of solutions may not hold, establishing the existence of Green's functions in this challenging setting.

Contribution

It introduces a method to describe symmetric restrictions and prove the existence of Green's functions for self-adjoint relations without relying on unique continuation.

Findings

Established symmetric restrictions of the maximal relation.

Proved existence of Green's functions despite lack of unique continuation.

Extended spectral theory to systems with distributional coefficients.

Abstract

This paper is a contribution to the spectral theory associated with the differential equation $Ju'+qu=wf$ 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. Under these hypotheses it may not be possible to uniquely continue a solution from one point to another, thus blunting the standard tools of spectral theory. Despite this fact we are able to describe symmetric restrictions of the maximal relation associated with $Ju'+qu=wf$ and show the existence of Green's functions for self-adjoint relations even if unique continuation of solutions fails.