On spectral and pseudospectral functions of first-order symmetric systems

TL;DR

This paper characterizes all spectral and pseudospectral functions of first-order symmetric systems using a Nevanlinna boundary parameter, extending previous boundary problem parametrizations to more general systems.

Contribution

It provides a comprehensive parametrization of spectral and pseudospectral functions for general first-order symmetric systems via Nevanlinna boundary parameters.

Findings

Parametrization of spectral functions using Nevanlinna boundary parameters

Extension of boundary problem parametrizations to non-Hamiltonian systems

Unified framework for spectral and pseudospectral functions

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.