On selfadjoint dilation of the dissipative extension of a direct sum differential operator
type: Item
instance of: scholarly article
title: On selfadjoint dilation of the dissipative extension of a direct sum differential operator
zbMATH Open document ID: 1275.47095
author: Bilender P. Allahverdiev
author: Ekin Ugurlu
published in: Banach Journal of Mathematical Analysis
publication date: 2013-03-20T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.bjma/1363784231
review text: The authors consider the differential expression \[ \ell(y):= \frac{1}{w(x)}\left[-y''+\frac{\nu^2-\frac{1}{4}}{x^2}y+q(x)y\right],\quad x\in\Omega:=\Omega_1\cup\Omega_2, \] where \(\Omega_1:=(0; c)\), \(\Omega_2 := (c;\infty)\), \(0 < c < \infty\) and \(0 \leq \nu < 1\). Throughout the paper, it is assumed that the points \(0, c\) and \(\infty\) are singular for the differential expression \(\ell\). Further, it is assumed that \(w\) and \(q\) are real-valued, Lebesgue measurable functions on \(\Omega\), locally integrable functions on \(\Omega_1\) and \(\Omega_2\), and \(w(x) > 0\) for almost all \(x\in\Omega\). The authors describe all maximal dissipative, maximal accretive and selfadjoint extensions of the minimal symmetric operators \(L_0\) corresponding to the differential expression \(\ell\). Further using the equivalence of the Lax-Phillips scattering function and the Sz.-Nagy-Foiaş characteristic function, they show that all eigenfunctions and associated functions of the maximal dissipative extension of \(L_0\) are complete in \(L_{w}^{2}(\Omega)\).
reviewed by: Michael Perelmuter
Mathematics Subject Classification ID: 47E05
Mathematics Subject Classification ID: 47B44
Mathematics Subject Classification ID: 47A20
Mathematics Subject Classification ID: 34B20
Mathematics Subject Classification ID: 34B24
Mathematics Subject Classification ID: 34B37
zbMATH DE Number: 6147467
zbMATH Keywords: direct sum differential operators
zbMATH Keywords: scattering matrix
zbMATH Keywords: characteristic function
zbMATH Keywords: spectral analysis
MaRDI profile type: Publication
OpenAlex ID: W2137986834
cites work: Spectral analysis of dissipative Schrödinger operators
cites work: Dissipative Sturm-Liouville operators with nonseparated boundary conditions
cites work: Sturm–Liouville eigenproblems with an interior pole
cites work: Sturm-Liouville differential operators in direct sum spaces
cites work: Q3679511
cites work: Q3699450
cites work: Q5519463
cites work: Q3683061
cites work: LIMIT-CIRCLE CRITERIA FOR SECOND-ORDER DIFFERENTIAL EXPRESSIONS
cites work: Q4862628
cites work: Q5579836
cites work: Q5624013
cites work: Spectral analysis of a nonselfadjoint Schrödinger problem with eigenparameter in the boundary condition
cites work: Q4302209
cites work: Q5631423
cites work: Q5539945
cites work: Q3279399
DOI: 10.15352/BJMA/1363784231
description: scientific article; zbMATH DE number 6147467