Singular point-like perturbations of the Bessel operator in a Pontryagin space
type: Item
instance of: scholarly article
title: Singular point-like perturbations of the Bessel operator in a Pontryagin space
zbMATH Open document ID: 0960.47021
author: Yuri Shondin
author: A. Dijksma
published in: Journal of Differential Equations
publication date: 2000-08-27T00:00:00Z
full work available at URL: 2000JDiffEqDijksma.pdf
review text: This paper is elaborated in an (introduction plus) 6 sections, namely: 1. Pontryagin space realization of singular perturbations of a nonnegative operator. 2. A model for singular perturbations of a nonnegative operator. 3. Point-like perturbation of the Bessel operator. 4. Spectral properties of \(H^t\). 5. Eigenfunction expansions. 6. Scattering matrices. To produce a general but very concise presentation of the content, it's useful to cite the authors' abstract: ``The spectral problem for the Bessel equation of order \(\nu\) on \((0,\infty)\) in the case \(0< \nu<1\) is closely related to the Nevanlinna functions \[ Q(z)= -\pi(- z)^\nu/(2\sin\pi\nu). \] If \(\nu> 1\) and \(\nu\neq 2,3,\dots\), this function belongs to the generalized Nevanlinna class \(N_m\), \(m= [{v+1\over 2}]\). A natural question appears: To what spectral problem does this function correspond? We answer this and related questions using Pontryagin space operator realizations of suitable singular point-like perturbations of the Bessel operator. In this paper we discuss the spectra of these realizations and we derive eigenfunction expansions via related wave operators''.
reviewed by: Trandafir Balan
Mathematics Subject Classification ID: 47B50
Mathematics Subject Classification ID: 46C20
zbMATH DE Number: 1496034
zbMATH Keywords: eigenfunction expansions
zbMATH Keywords: scattering matrices
zbMATH Keywords: Pontryagin space realization
zbMATH Keywords: singular perturbations
zbMATH Keywords: nonnegative operator
zbMATH Keywords: Bessel operator
zbMATH Keywords: Nevanlinna functions
zbMATH Keywords: generalized Nevanlinna class
zbMATH Keywords: wave operators
MaRDI profile type: Publication
OpenAlex ID: W2160592470
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DOI: 10.1006/JDEQ.1999.3755
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description: scientific article; zbMATH DE number 1496034