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Oscillation criteria and the discreteness of the spectrum of self-adjoint, even order, differential operators

Published online by Cambridge University Press:  14 November 2011

Ondřej Došlý
Ondřej Došlý
Affiliation:
Department of Mathematics, Masaryk University, Janáčkovo nám. 2a, 662 95 Brno, Czechoslovakia
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Synopsis

This paper deals with the oscillation properties of self-adjoint differential equations

The oscillation criteria are derived, which allows a unified approach to the investigation of (*) near a finite or infinite singularity. These criteria are used to study spectral properties of singular differential operators associated with (*).

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 119 , Issue 3-4 , 1991 , pp. 219 - 232
Copyright
Copyright © Royal Society of Edinburgh 1991

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References

1Ahlbrandt, C. D.. Equivalent boundary value problems for self-adjoint differential systems. J. Differential Equations 9 (1971), 420–435.CrossRefGoogle Scholar
2Ahlbrandt, C. D., Hinton, D. B. and Lewis, R. T.. The effect of variable change on oscillation and disconjugacy criteria with application to spectral theory and asymptotic theory. J. Math. Anal. Appl. 81 (1981), 234–277.CrossRefGoogle Scholar
3Ahlbrandt, C. D., Hinton, D. B. and Lewis, R. T.. Necessary and sufficient conditions for the discreteness of the spectrum of certain singular differential operators. Canad. J. Math. 33 (1981), 229–246.CrossRefGoogle Scholar
4Coppel, W. A.. Disconjugacy. Lecture Notes in Mathematics 220 (Berlin: Springer, 1971).CrossRefGoogle Scholar
5Došlý, O.. Oscillation criteria for self-adjoint linear differential equations (to appear).Google Scholar
6Došlý, O.. Transformations of linear Hamiltonian systems preserving oscillation behaviour. Arch Math, (submitted).Google Scholar
7Došlý, O. and Fiedler, F.. A remark on Nehari-type oscillation criteria for self-adjoint linear differential equations (to appear).Google Scholar
8Evans, W. D., Kwong, M. K. and Zettl, A., Lower bounds for the spectrum of ordinary differential operators. J. Differential Equations 48 (1983), 123–155.CrossRefGoogle Scholar
9Fiedler, F.. Oscillation criteria for a special class of 2n-order ordinary differential equations. Math. Nachr. 131 (1987), 205–218.CrossRefGoogle Scholar
10Fiedler, F.. About certain fourth-order ordinary differential operators—oscillation and discreteness of their spectrum. Math. Nachr. 142 (1989), 235–250.CrossRefGoogle Scholar
11Glazman, I. M.. Direct Methods of Qualitative Spectral Analysis of Singular Differential Operators (Jerusalem: Davey, 1965).Google Scholar
12Hinton, D. B. and Lewis, R. T.. Discrete spectra criteria for singular differential operators with middle terms. Math. Proc. Cambridge Philos. Soc. 77 (1975), 337–347.CrossRefGoogle Scholar
13Lewis, R. T.. The discreteness of the spectrum of self-adjoint, even order, differential operators. Proc. Amer. Math. Soc. 42 (1974), 480–482.CrossRefGoogle Scholar