On spectral and pseudospectral functions of first-order symmetric systems

Abstract

We consider first-order symmetric system $Jy'-B(t)y=\Delta(t)f(t)$ on an interval $\mathcal I=[a,b)$ with the regular endpoint $a$. A distribution matrix-valued function $\Sigma(s)$, $s\in\mathbb R$, is called a pseudospectral function of such a system if the corresponding Fourier transform is a partial isometry with the minimally possible kernel. The main result is a parametrization of all pseudospectral functions of a given system by means of a Nevanlinna boundary parameter $\tau$. Similar parameterizations for regular systems have earlier been obtained by Arov and Dym, Langer and Textorius, A. Sakhnovich.