Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-27T07:30:21.914Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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