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On the location of eigenvalues of second order linear differential operators*

Published online by Cambridge University Press:  14 November 2011

Ian Knowles
Ian Knowles
Affiliation:
Department of Mathematics, University of the Witwatersrand and Department of Mathematics, University of Dundee
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Synopsis

This paper is concerned with finding upper bounds on the set of eigenvalues of self-adjoint differential operators generated in the Hilbert space L2[0, ∞) by the differential expression

on [0,∞), together with a real homogeneous boundary condition at t = 0.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 80 , Issue 1-2 , 1978 , pp. 15 - 22
Copyright
Copyright © Royal Society of Edinburgh 1978

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References

1Akhiezer, N. I. and Glazman, I. M.Theory of linear operators in Hilbert space I (New York: Ungar, 1961).Google Scholar
2Borg, G.On the point spectra of y″ + (λ − q(x))y = 0. Amer. J. Math. 73 (1951), 122–126.CrossRefGoogle Scholar
3Chaudhuri, J. and Everitt, W. N.On the spectrum of ordinary, second-order differential operators. Proc. Roy. Soc. Edinburgh Sect. A. 68 (1969), 95–119.Google Scholar
4Dunford, N. and Schwartz, J. T.Linear operators II (New York: Interscience, 1963).Google Scholar
5Eastham, M. S. P. On the absence of square integrable solutions of the Sturm-Liouville equation. Proc. Conf. Ordinary and Partial Differential Equations, Dundee 1976. Lecture Notes in Mathematics 564, 72, 77 (Berlin: Springer-Verlag, 1976).Google Scholar
6Eastham, M. S. P. Sturm-Liouville equations and purely continuous spectra (Preprint).Google Scholar
7Hartman, P.The number of L 2-solutions of x″ + q(t)x = 0. Amer. J. Math. 73 (1951), 635–646.CrossRefGoogle Scholar
8Knowles, I.On the number of L 2-solutions of second-order linear differential equations, Proc. Roy. Soc. Edinburgh Sect. A. 80 (1978) 1–13.CrossRefGoogle Scholar
9Naimark, M. A.Linear differential operators II (New York: Ungar, 1968).Google Scholar
10von Neumann, J. and Wigner, E.Über merkwürtige discrete Eigenwert. Z. Phys. 30 (1929), 465–467.Google Scholar
11Wallach, S.On the location of spectra of differential equations. Amer. J. Math. 70 (1948), 833–841.CrossRefGoogle Scholar
12Walter, J.Absolute continuity of the essential spectrum of −d 2/dt 2 + q(t) without the monotony of q. Math. Z. 129 (1972), 83–94.CrossRefGoogle Scholar
13Weidmann, J.Zur spektraltheorie von Sturm-Liouville operatoren. Math. Z. 98 (1967), 268–302.CrossRefGoogle Scholar