On generalized resolvents and characteristic matrices of first-order symmetric systems

TL;DR

This paper characterizes all generalized resolvents and characteristic matrices of first-order symmetric systems with regular and singular endpoints, extending classical results to more general non-Hamiltonian systems.

Contribution

It provides a comprehensive parametrization of generalized resolvents and characteristic matrices for non-Hamiltonian symmetric systems using boundary conditions.

Findings

All generalized resolvents are described via boundary problems with boundary conditions.

Characteristic matrices are parametrized explicitly in terms of boundary conditions.

Results extend Strauß' classical theory to broader classes of systems.

Abstract

We study general (not necessarily Hamiltonian) first-order symmetric system $J y'-B(t)y=\D(t) f(t)$ on an interval $\cI=[a,b) $ with the regular endpoint $a$ and singular endpoint $b$. It is assumed that the deficiency indices $n_\pm(\Tmi)$ of the corresponding minimal relation $\Tmi$ in $\LI$ satisfy $n_-(\Tmi)\leq n_+(\Tmi)$. We describe all generalized resolvents $y=R(\l)f, \; f\in\LI,$ of $\Tmi$ in terms of boundary problems with $\l$-depending boundary conditions imposed on regular and singular boundary values of a function $y$ at the endpoints $a$ and $b$ respectively. We also parametrize all characteristic matrices $\Om(\l)$ of the system immediately in terms of boundary conditions. Such a parametrization is given both by the block representation of $\Om(\l)$ and by the formula similar to the well-known Krein formula for resolvents. These results develop the \u{S}traus' results on generalized resolvents and characteristic matrices of differential operators.