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On the strong limit-point and Dirichlet properties of second order differential expressions

Published online by Cambridge University Press:  14 November 2011

D. Race
D. Race
Affiliation:
Mathematics Department, University of Surrey, Guildford, Surrey GU2 5XH
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Synopsis

Second order differential expressions of the form w−1(−(pf′)′ + qf) are considered at a singular end-point. Some of the known relationships between properties such as Dirichlet, weak Dirichlet and strong limit-point, are extended to incorporate an arbitrary, positive weight function and complexvalued coefficients.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 101 , Issue 3-4 , 1985 , pp. 283 - 296
Copyright
Copyright © Royal Society of Edinburgh 1985

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References

1Amos, R. J.. On a Dirichlet and limit-circle criterion for second-order ordinary differential expressions. Quaestiones Math. 3 (1978), 53–65.CrossRefGoogle Scholar
2Atkinson, F. V.. Limit-n criteria of integral type. Proc. Roy. Soc. Edinburgh Sect. A 73 (1974/1975), 167–198.CrossRefGoogle Scholar
3Evans, W. D.. On the spectra of Schrödinger operators with a complex potential. Math. Ann. 255 (1981), 57–76.CrossRefGoogle Scholar
4Evans, W. D. and Everitt, W. N.. A return to the Hardy-Littlewood integral inequality. Proc. Roy. Soc. London Ser. A 380 (1982), 447–486.Google Scholar
5Everitt, W. N.. On the strong limit-point condition of second-order differential expressions. International Conference on Differential Equations, 287–307 (New York: Academic Press, 1975).Google Scholar
6Everitt, W. N.. A note on the Dirichlet condition for second-order differential expressions. Canad. J. Math. 28 (1976), 312–320.CrossRefGoogle Scholar
7Everitt, W. N.. A general integral inequality associated with certain ordinary differential operators. Quaestiones Math. 2 (1978), 479–494.CrossRefGoogle Scholar
8Everitt, W. N., Giertz, M. and Weidmann, J.. Some remarks on a separation and limit-point criterion of second-order, ordinary differential expressions. Math. Ann. 200 (1973), 335–346.CrossRefGoogle Scholar
9Everitt, W. N. and Wray, S. D.. On quadratic integral inequalities associated with second-order symmetric differential expressions. Lecture Notes in Mathematics 1032, 170–223 (Berlin: Springer, 1983).Google Scholar
10Hinton, D. B.. Strong limit-point and Dirichlet criteria for ordinary differential expressions of order 2n. Proc. Roy. Soc. Edinburgh Sect. A 76 (1977), 301–310.CrossRefGoogle Scholar
11Kalf, H.. Remarks on some Dirichlet-type results for semibounded Sturm-Liouville operators. Math. Ann. 210 (1974), 197–205.CrossRefGoogle Scholar
12Kauffman, R. M., Read, T. T. and Zettl, A.. The deficiency index problem for powers of ordinary differential expressions. Lecture Notes in Mathematics 621 (Berlin: Springer, 1977).Google Scholar
13Knowles, I. W.. Dissipative Sturm-Liouville operators. Proc. Roy. Soc. Edinburgh Sect. A 88 (1981), 329–343.CrossRefGoogle Scholar
14Kwong, M. K.. Note on the strong limit point condition of second order differential expressions. Quart. J. Math. Oxford 28 (1977), 201–208.CrossRefGoogle Scholar
15Kwong, M. K.. Conditional Dirichlet property of second order differential expressions. Quart. J. Math. Oxford 28 (1977), 329–338.Google Scholar
16Naimark, M. A.. Linear differential operators, part II (New York: Ungar, 1968).Google Scholar
17Race, D.. The theory of J-selfadjoint extensions of J-symmetric operators. J. Differential Equations 57 (1985), 258–274.CrossRefGoogle Scholar
18Read, T. T.. On the spectral theory of some nonsymmetric second order differential operators. Lecture Notes on Mathematics 1032, 407–426 (Berlin: Springer, 1983).Google Scholar