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Limit-point (LP) criteria for real symmetric differential expressions of order 2n†

Published online by Cambridge University Press:  14 November 2011

H. Kurss  and
G. Meyer
H. Kurss
Affiliation:
Department of Mathematics, Adelphi University, Garden City, L.I., New York 11530, U.S.A.
G. Meyer
Affiliation:
Department of Data Processing, Laguardia Community College, L.I.C., New York 11101, U.S.A.
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Synopsis

An interval-type LP criterion for

is derived in which “positive” coefficients play a prominent role. When pn = 1 and all the other pi are zero this reduces to a result of Ismagilov (1962). Successive specializations are obtained with the growth of the pi constrained by monomials in x. Previous LP criteria of Everitt (1968) and Hinton (1972, 1974) are shown to be special cases.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 88 , Issue 3-4 , 1981 , pp. 203 - 217
Copyright
Copyright © Royal Society of Edinburgh 1981

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References

1Atkinson, F. V.. Limit-n criteria of integral type. Proc. Roy. Soc. Edinburgh Sect. A 73 (1975), 167–198.CrossRefGoogle Scholar
2Coddington, E. A. and Levinson, N.. Theory of ordinary differential equations (New York: McGraw-Hill, 1955).Google Scholar
3Devinatz, A.. The deficiency index of certain fourth-order ordinary self-adjoint differential operators. Quar t. J. Math. Oxford Ser. 23 (1972), 267–286.CrossRefGoogle Scholar
4Devinatz, A.. The deficiency index problem for ordinary self-adjoint differential operators. Bull. Amer. Math. Soc. 79 (1973), 1109–1128.CrossRefGoogle Scholar
5Devinatz, A.. On limit-2 fourth order differential operators. J. London Math. Soc. 7 (1973), 135–146.CrossRefGoogle Scholar
6Eastham, M. S. P.. The limit-2 case of fourth order differential equations. Quar t. J. Math. Oxford Ser. 22 (1971), 131–134.CrossRefGoogle Scholar
7Eastham, M. S. P.. The limit-2n case of symmetric differential operators of order 2n. Proc. London Math. Soc. 38 (1979), 272–294.Google Scholar
8Evans, W. D.. On non-integrable square solutions of a fourth order differential equation and the limit-2 classifications. J. London Math. Soc. 7 (1973), 343–355.CrossRefGoogle Scholar
9Evans, W. D. and Zettl, A.. On the deficiency indices of powers of real 2nth-order symmetric differential expressions. J. London Math. Soc. 13 (1976), 543–556.CrossRefGoogle Scholar
10Evans, W. D. and Zettl, A.. Interval limit-point criteria for differential expressions and their powers. J. London Math. Soc. 15 (1977), 119–133.CrossRefGoogle Scholar
11Everitt, W. N.. Some positive definite differential operators. J. London Math. Soc. 43 (1968), 465–473.CrossRefGoogle Scholar
12Everitt, W. N.. On the limit-point classification of fourth order differential equations. J. London Math. Soc. 44 (1969), 273–281.CrossRefGoogle Scholar
13Everitt, W. N., Hinton, D. B. and Wong, J. S. W.. On the strong limit-n classification of linear ordinary differential expressions of order 2n. Proc. London Math. Soc. 29 (1974), 351–367.CrossRefGoogle Scholar
14Friedrichs, K. O.. Über die ausgezeichnete Randbedingung in der Spektraltheorie der halbbeschränkten gewöhnlichen Differentialoperatoren zweiter Ordnung. Math. Ann. 112 (1935), 1–23.CrossRefGoogle Scholar
15Hardy, G. H. and Wright, E. M.. An introduction to the theory of numbers (Oxford: Clarendon Press, 1954).Google Scholar
16Hartman, P.. The number of L2-solutions of x″+ q(t) x = 0. Ame r. J. Math. 73 (1951), 635–645.CrossRefGoogle Scholar
17Hinton, D. B.. Limit point criteria for differential equations. Cana d. J. Math. 24 (1972), 293–305.CrossRefGoogle Scholar
18Hinton, D. B.. Limit point criteria for differential equations II. Cana d. J. Math. 26 (1974), 340–351.CrossRefGoogle Scholar
19Ismagilov, R. S.. Conditions for self-adjointness of differential operators of higher order. Dokl. Akad. Nauk. SSSR 142 (1962). 1239–1242. English Transl. in Soviet Math. 3 (1962), 279–283.Google Scholar
20Kauffman, 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
21Kalf, H. and Walter, J.. Strongly singular potentials and essential self-adjointness of singular elliptic operators in C0∞(Rn\{0}). J. Functional Analysis 10 (1972), 114–130.CrossRefGoogle Scholar
22Kurss, H.. A limit-point criterion for non-oscillatory Sturm–Liouville differential operators. Proc. Amer. Math. Soc. 18 (1967), 445–449.CrossRefGoogle Scholar
23Naimark, M. A.. Linear differential operators: Part I, II (New York: Ungar 1968).Google Scholar
24Sears, D. B.. Note on the uniqueness of the Green's functions associated with certain differential equations. Cana d. J. Math. 2 (1950), 314–325.CrossRefGoogle Scholar
25Titchmarsh, E. C.. On the uniqueness of the Green's function associated with a second-order differential equation. Cana d. J. Math. 1 (1949), 191–198.CrossRefGoogle Scholar
26Walker, P. W.. Deficiency indices of fourth order singular operators. J. Differential Equations 9 (1971), 133–140.CrossRefGoogle Scholar
27Weyl, H.. Über gewöhnliche Differentialgleichungen mit Singularitäten und die zugehörigen Entwicklungen willkurlicher Funktionen. Math. Ann. 68 (1910), 220–269.CrossRefGoogle Scholar