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2015 Survey Article: Self-adjoint ordinary differential operators and their spectrum
Anton Zettl, Jiong Sun
Rocky Mountain J. Math. 45(3): 763-886 (2015). DOI: 10.1216/RMJ-2015-45-3-763

Abstract

We survey the theory of ordinary self-adjoint differential operators in Hilbert space and their spectrum. Such an operator is generated by a symmetric differential expression and a boundary condition. We discuss the very general modern theory of these symmetric expressions which enlarges the class of these expressions by many dimensions and eliminates the smoothness assumptions required in the classical case as given, e.g., in the celebrated books by Coddington and Levinson and Dunford and Schwartz. The boundary conditions are characterized in terms of square-integrable solutions for a real value of the spectral parameter, and this characterization is used to obtain information about the spectrum. Many of these characterizations are quite recent and widely scattered in the literature, some are new. A comprehensive review of the deficiency index (which determines the number of independent boundary conditions required in the singular case) is also given for an expression $M$ and for its powers. Using the modern theory mentioned above, these powers can be constructed without any smoothness conditions on the coefficients.

Citation

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Anton Zettl. Jiong Sun. "Survey Article: Self-adjoint ordinary differential operators and their spectrum." Rocky Mountain J. Math. 45 (3) 763 - 886, 2015. https://doi.org/10.1216/RMJ-2015-45-3-763

Information

Published: 2015
First available in Project Euclid: 21 August 2015

zbMATH: 1369.47057
MathSciNet: MR3385967
Digital Object Identifier: 10.1216/RMJ-2015-45-3-763

Subjects:
Primary: 05C38, 15A15
Secondary: 05A15, 15A18‎

Rights: Copyright © 2015 Rocky Mountain Mathematics Consortium

JOURNAL ARTICLE
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Vol.45 • No. 3 • 2015
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