On the spectral theory for first-order systems without the unique continuation property

TL;DR

This paper investigates the spectral properties of first-order differential systems lacking the unique continuation property, characterizing their solution sets and operator adjoints despite the absence of uniqueness.

Contribution

It provides a detailed analysis of the solution structure and operator adjoints for first-order systems without unique continuation, extending spectral theory in this context.

Findings

The minimal operator's adjoint equals the maximal operator.

Solutions may not be unique or exist for all initial conditions.

The set of solutions is explicitly characterized.

Abstract

We consider 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. In this situation it may happen that there is no existence and uniqueness theorem for balanced solutions of a given initial value problem. We describe the set of solutions the equation does have and establish that the adjoint of the minimal operator is still the maximal operator, even though unique continuation of balanced solutions fails.