On spectral and pseudospectral functions of first-order symmetric systems

Abstract

We consider 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 $a$. A distribution matrix-valued function $\Si(s), \; s\in\bR,$ is called a spectral (pseudospectral) function of such a system if the corresponding Fourier transform is an isometry (resp. partial isometry) from $\LI$ into $L^2(\Si)$. The main result is a parametrization of all spectral and pseudospectral functions of a given system by means of a Nevanlinna boundary parameter $\tau$. Similar parameterizations for various classes of boundary problems have earlier been obtained by Kac and Krein, Fulton, Langer and Textorius, Sakhnovich and others.