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Dissipative Sturm-Liouville operators

Published online by Cambridge University Press:  14 November 2011

Ian Knowles
Ian Knowles
Affiliation:
Department of Mathematics, University of the Witwatersrand, Johannesburg 2001, South Africa
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Synopsis

Consider the differential expression

where p and w > 0 are real-valued and q is complex-valued on I. A number of criteria are established for certain extensions of the minimal operator generated by τ in the weighted Hilbert space to be maximal dissipative.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 88 , Issue 3-4 , 1981 , pp. 329 - 343
Copyright
Copyright © Royal Society of Edinburgh 1981

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References

1Atkinson, F. V.. Limit-n criteria of integral type. Proc. Roy. Soc. Edinburgh Sect. A 73 (1974/1975), 167–198.Google Scholar
2Berberian, S. K.. Measure and Integration (New York: Macmillan, 1965).Google Scholar
3Evans, W. D.. On limit-point and Dirichlet-type results for second-order differential expressions. In Proceedings Conference on Ordinary and Partial Differential Equations, Dundee, Scotland, 1976, pp. 78–92. Lecture Notes in Mathematics 564 (Berlin: Springer, 1976).Google Scholar
4Everitt, W. N. and Knowles, I. W.. Limit-point and limit-circle criteria for Sturm-Liouville equations with intermittently negative principal coefficients, in preparation.Google Scholar
5Glazman, I. M.. An analogue of the extension theory of hermitian operators and a non-symmetric one-dimensional boundary-value problem on a half-axis (Russian). Dokl. Akad. Nauk SSSR US (1957), 214–216.Google Scholar
6Glazman, I. M.. Direct Methods of Qualitative Spectral Analysis of Singular Differential Operators (Jerusalem: IPST, 1965).Google Scholar
7Knowles, I. W.. On the boundary conditions characterising J-selfadjoint extensions of Jsymmetric operators. J. Differential Equations, to appear.Google Scholar
8Knowles, Ian and Race, David. On the point spectra of complex Sturm-Liouville operators. Proc. Roy. Soc. Edinburgh Sect. A 85 (1980), 263–289.CrossRefGoogle Scholar
9Krein, S. G.. Linear Differential Equations in Banach Space. Transl. Math. Monographs Vol. 29 (Providence, Rhode Island: Amer. Math. Soc, 1971).Google Scholar
10Lidskii, V. B.. Summability of series in terms of the principal vectors of non-selfadjoint operators. Amer. Math. Soc. Transl. (2) 40 (1964), 193–228.Google Scholar
11Naimark, M. A.. Linear Differential Operators, Part II (New York: Ungar, 1968).Google Scholar
12Race, David. On the location of the essential spectra and regularity fields of complex Sturm-Liouville operators. Proc. Roy. Soc. Edinburgh Sect. A 85 (1980), 1–14.Google Scholar