On the self-adjoint extensions of symmetric ordinary differential operators in direct sum spaces
type: Item
instance of: scholarly article
title: On the self-adjoint extensions of symmetric ordinary differential operators in direct sum spaces
zbMATH Open document ID: 0768.34052
DOI: 10.1016/0022-0396(92)90115-4
author: Shouzhong Fu
published in: Journal of Differential Equations
publication date: 1993-04-01T00:00:00Z
review text: Let \(I_ j = (a_ j,b_ j)\), \(j=1,2,\dots,q\) be open intervals in \(\mathbb{R}\), and denote by \(T_ 0(M_ j)\), \(T_ 1(M_ j)\) the minimal and maximal operators respectively generated by a formally symmetric \(n\)th- order differential expression \(M_ j\) in \(L^ 2(I_ j)\). Set \(M=(M_ 1,\dots,M_ q)\) and define the direct sums \(T_ 1(M)=\bigoplus^ q_{j=1} T_ 1(M_ j)\), \(T_ 0(M) = \bigoplus^ q_{j=1} T_ 0(M_ j)\) and \(H = \bigoplus^ q_{j=1} L^ 2(I_ j)\). Then \(T_ 0(M)\) is a closed symmetric operator in \(H\) with adjoint \(T_ 1(M)\) and deficiency indices \(d^ + = \sum^ q_{j=1} d^ +_ j\), \(d^ - = \sum^ q_{j=1} d^ -_ j\), where \((d^ +_ j,d^ -_ j)\) are the deficiency indices of \(T_ 0(M_ j)\). It is assumed that \(d^ +_ j = d^ -_ j\) and hence \(d^ + = d^ -\), so that \(T_ 0(M)\) has self- adjoint extensions. The author obtains a characterization of all these self-adjoint extensions in terms of solutions of \(M_ jy = \pm iy\) which are square integrable near the end-points \(a_ j\), \(b_ j\) of the intervals \(I_ j\). It is shown that there are self-adjoint extensions which can not be expressed as direct sums of operators in \(L^ 2(I_ j)\), a phenomenon already observed by \textit{W. N. Everitt} and \textit{A. Zettl} [Rocky Mt. J. Math. 16, 497-516 (1986; Zbl 0624.34020)]. Other recent contributions to this problem are [\textit{W. N. Everitt} and \textit{A. Zettl}, Proc. Lond. Math. Soc., III. Ser. 64, 524-544 (1992; Zbl 0723.34022)] and [\textit{S. E. Ibrahim}, Rocky Mt. J. Math. 22, No. 3, 877- 915 (1992)].
Mathematics Subject Classification ID: 34L05
Mathematics Subject Classification ID: 47E05
zbMATH DE Number: 147078
zbMATH Keywords: minimal and maximal operators
zbMATH Keywords: symmetric operator
zbMATH Keywords: deficiency indices
zbMATH Keywords: self-adjoint extensions
reviewed by: W. D. Evans
MaRDI profile type: Publication
full work available at URL: 0022-0396(92)90115-4
OpenAlex ID: W1989557255
cites work: On the Nature of the Spectrum of Singular Second Order Linear Differential Equations
cites work: Q3770851
cites work: Sturm-Liouville differential operators in direct sum spaces
cites work: Singular self-adjoint Sturm-Liouville problems
cites work: Singular Self-Adjoint Sturm–Liouville Problems: II. Interior Singular Points
cites work: The domains of the selfadjoint extensions of ordinary symmetric differential operators over \((-\infty, \infty)\)
cites work: On the self-adjoint extensions of symmetric ordinary differential operators with middle deficiency indices
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description: scientific article; zbMATH DE number 147078