On the completeness of the system of root vectors for first-order systems

TL;DR

This paper establishes the completeness of root functions for first-order systems with broad classes of boundary conditions, introducing weak regularity as a key criterion, and extends results to Dirac equations with irregular conditions.

Contribution

It introduces the concept of weakly regular boundary conditions, expanding the class of boundary conditions for which completeness is proven, and provides the first results on completeness for general first order systems.

Findings

Root functions are complete and minimal under weakly regular boundary conditions.

Weak regularity is necessary for completeness in some cases.

Results apply to Dirac and Dirac-type equations with irregular boundary conditions.

Abstract

The paper is concerned with the completeness problem of root functions of general boundary value problems for first order systems of ordinary differential equations. Namely, we introduce and investigate the class of \emph{weakly regular boundary conditions}. We show that this class is much broader than the class of {\em regular boundary conditions} introduced by G.D. Birkhoff and R.E. Langer. Our main result states that the system of root functions of a boundary value problem is complete and minimal provided that the boundary conditions are weakly regular. Moreover, we show that in some cases \emph{the weak regularity} of boundary conditions \emph{is also necessary} for the completeness. Also we investigate the completeness for $2\times 2$ Dirac and Dirac type equations subject to irregular or even to degenerate boundary conditions. We emphasize that our results are the first results on the completeness problem for general first order systems even in the case of regular boundary conditions.