Criterion for discreteness of the spectrum of a singular canonical system
type: Item
instance of: scholarly article
title: Criterion for discreteness of the spectrum of a singular canonical system
zbMATH Open document ID: 0878.47032
DOI: 10.1007/BF01077059
publication date: 1996-07-09T00:00:00Z
review text: The discreteness of the spectrum of the system \[ J{d\over dt} x= z{\mathcal K}(t)x, x(0)=\begin{pmatrix} 1\\ 0\end{pmatrix},\text{ where }{\mathcal K}=\begin{pmatrix} a(t) b(t)\\ c(t) d(t)\end{pmatrix}, J=\begin{pmatrix} 0 &-1\\ 1 & 0\end{pmatrix}, t\in I\subset\mathbb{R},\tag{\(*\)} \] with \(I\) interval, \({\mathcal K}(t)\geq 0\) for any \(t\in I\) and \(z\) is a complex parameter is studied. After some definitions involving the properties of the interval \(I\) and some kind of functions, one theorem is given for the discreteness of the spectrum of the system \((*)\). Another theorem gives necessary and sufficient conditions between the regularity of the solution of \((*)\) and the discreteness of the spectrum of the system.
Mathematics Subject Classification ID: 47E05
Mathematics Subject Classification ID: 47A10
Mathematics Subject Classification ID: 34L05
zbMATH DE Number: 898981
zbMATH Keywords: discreteness of the spectrum
author: Israel S. Kac
reviewed by: S. Totaro
MaRDI profile type: Publication
cites work: Q5539946
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published in: Functional Analysis and Its Applications
description: scientific article; zbMATH DE number 898981