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The point-continuous spectrum of second-order, ordinary differential operators

Published online by Cambridge University Press:  14 November 2011

Christine Thurlow
Christine Thurlow
Affiliation:
121 The Grove, Ealing, London W5
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Synopsis

Given any countably infinite set of isloated points on the ℷ -axis, it is shown that there is a continuous q(x) such that these points constitute exactly the point-continuous spectrum for the equation yn″(x) + (ℷ —q(x))y(x) = 0(0≦x<∞) with some homogenous boundary condition at x = 0. This extends a result given by Eastham and McLeod for countably infinite sets of isolated points on the positive ℷ-axis.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 84 , Issue 3-4 , 1979 , pp. 197 - 211
Copyright
Copyright © Royal Society of Edinburgh 1979

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References

1Akhiezer, N. I. and Glazman, I. M., Theory of linear operators in Hilbert Space, 2 vols (New York: Ungar, 1961).Google Scholar
2Eastham, M. S. P.. Theory of ordinary differential equations (London: Van Nostrand Reinhold, 1970).Google Scholar
3Eastham, M. S. P.. Sturm–Liouville equations and purely continuous spectra. Nieuw Arch. Wisk. 25 (1977), 169–181.Google Scholar
4Eastham, M. S. P. and McLeod, J. B.. The existence of eigenvalues embedded in the continuous spectrum of ordinary differential operators. Proc. Roy. Soc. Edinburgh Sect. A 79 (1977), 25–34.CrossRefGoogle Scholar
5Levitan, B. M. and Gasymov, M. G.. Determination of a differential equation by two of its spectra. Russian Math. Surveys 19 (1964), 1–63.CrossRefGoogle Scholar
6Thurlow, C. R.. A generalisation of the inverse spectral theorem of Levitan and Gasymov. Proc. Roy. Soc. Edinburgh Sect. A 84 (1979), 185–196.CrossRefGoogle Scholar
7Thurlow, C. R.. Topics in differential equations (Oxford Univ. D. Phil, thesis, 1978).Google Scholar
8Thurlow, C. R. and Eastham, M. S. P.. The existence of eigenvalues of infinite multiplicity for the Schrodinger operator, in preparation.Google Scholar