Small perturbations of canonical systems
type: Item
instance of: scholarly article
title: Small perturbations of canonical systems
zbMATH Open document ID: 0969.34070
DOI: 10.1007/BF01200125
published in: Integral Equations and Operator Theory
publication date: 2001-09-25T00:00:00Z
review text: The author considers a singular two-dimensional canonical system \(Jy' = -zHy\) on \([0,\infty)\) such that at \(\infty\) Weyl's limit point case holds. Here, \(H\) is a measurable, real and nonnegative definite matrix function, called Hamiltonian. From results of L. de Branges it follows that the correspondence between canonical systems and their Titchmarsh-Weyl coefficients is a bijection between the class of all Hamiltonians with \(\text{tr }H = 1\) and the class of Nevanlinna functions. In this note the author shows how the Hamiltonian \(H\) of a canonical system changes if its Titchmarsh-Weyl coefficient or the corresponding spectral measure undergoes certain small perturbations. This generalizes results of \textit{H. Dym} and \textit{N. Kravitsky} [Integral Equations Oper. Theory 1, 270-277 (1978; Zbl 0394.45002)] for so-called vibrating strings, in particular a generalization of a construction principle of \textit{I. M. Gelfand} and \textit{B. M. Levitan} [Am. Math. Soc., Translat., Ser. II 1, 253-304 (1955; Zbl 0066.33604)] can be shown.
Mathematics Subject Classification ID: 34L05
Mathematics Subject Classification ID: 34B20
Mathematics Subject Classification ID: 34A55
Mathematics Subject Classification ID: 47E05
Mathematics Subject Classification ID: 47B25
zbMATH DE Number: 1542411
zbMATH Keywords: vibrating strings
zbMATH Keywords: singular two-dimensional canonical system
zbMATH Keywords: Weyl's limit point
zbMATH Keywords: Titchmarsh-Weyl coefficients
zbMATH Keywords: Hamiltonians
zbMATH Keywords: Nevanlinna functions
author: Henrik Winkler
MaRDI profile type: Publication
cites work: Q5828277
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full work available at URL: https://doi.org/10.1007/bf01200125
OpenAlex ID: W2066472832
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description: scientific article; zbMATH DE number 1542411