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The limit-point, limit-circle nature of rapidly oscillating potentials

Published online by Cambridge University Press:  14 February 2012

F. V. Atkinson ,
M. S. P. Eastham  and
J. B. McLeod
F. V. Atkinson
Affiliation:
Department of Mathematics, University of Toronto
M. S. P. Eastham
Affiliation:
Department of Mathematics, Chelsea College, University of London
J. B. McLeod
Affiliation:
Wadham College, University of Oxford
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Synopsis

The Weyl limit-point, limit-circle nature of the equation y″(x)–q(x)y(x) = 0(0≦ x< ∞) is analysed When q(x) has the form q(x) = xαp(xβ), where α and β are positive constants and p(t) is a continuous periodic function of t.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 76 , Issue 3 , 1977 , pp. 183 - 196
Copyright
Copyright © Royal Society of Edinburgh 1977

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References

1Atkinson, F. V.. On asymptotically periodic linear systems. J. Math. Anal. Appl. 24 (1968), 646–653.Google Scholar
2Atkinson, F. V., Limit-n criteria of integral type. Proc. Roy. Soc. Edinburgh Sect. A 73 (1975), 167–198.Google Scholar
3Brinck, I.. Self-adjointness and spectra of Strum-Liouville operators. Math. Scand. 7 (1959), 219–239.Google Scholar
4Coddington, E. A. and Levinson, N.. Theory of ordinary differential equations (New York: McGraw-Hill, 1955).Google Scholar
5Eastham, M. S. P.. On a limit–point method of Hartman. Bull. London Math. Soc. 4 (1972), 340–344.CrossRefGoogle Scholar
6Eastham, M. S. P.. Limit–circle differential expressions of the second-order with an oscillating coefficient. Quart. J. Math. Oxford Ser. 24 (1973), 257–263.CrossRefGoogle Scholar
7Eastham, M. S. P.. Second- and fourth-order differential equations with oscillatory coefficients and not of limit-point type. Proceedings of the Scheveningen Conference on the Spectral Theory and Asymptotics of Differential Equations (Ed. de Jager, E. M.). (Amsterdam: North-Holland, 1974).Google Scholar
8Eastham, M. S. P.. The spectral theory of periodic differential equations (Edinburgh: Scottish Academic Press, 1973).Google Scholar
9Eastham, M. S. P.. The limit-3 case of self-adjoint differential expressions of the fourth order with oscillating coefficients. J. London Math. Soc. 8 (1974), 427–437.CrossRefGoogle Scholar
10Hartman, P.. The number of L 2-solutions of x”+q(t)x =0. Amer. J. Math. 73 (1951), 635–645.Google Scholar
11Ismagilov, R. S.. Conditions for self–adjointness of differential operators of higher order. Soviet Math. 3 (1962), 279–283.Google Scholar
12Ismagilov, R. S.. On the self-adjointness of the Sturm-Liouville operator. Uspehi Mat. Nauk 18 (1963), 161–166.Google Scholar
13Knowles, I.. Note on a limit–point criterion. Proc. Amer. Math. Soc. 41 (1973), 117–119.CrossRefGoogle Scholar
14McLeod, J. B.. On the spectrum of wildly oscillating functions. J. London Math. Soc. 39 (1964), 623–634.Google Scholar
15McLeod, J. B.. On the spectrum of wildly oscillating functions. II. J. London Math. Soc. 40 (1965), 655–661.CrossRefGoogle Scholar
16McLeod, J. B.. On the spectrum of wildly oscillating functions. III. J. London Math. Soc. 40 (1965), 662–666.CrossRefGoogle Scholar
17Mahony, J. J.. Asymptotic results for the solutions of a certain differential equation. J. Austral. Math. Soc. 13 (1972), 147–158.Google Scholar
18Sears, D. B.. On the solutions of a linear second-order differential equation which are of integrable square. J. London Math. Soc. 24 (1949), 207–215.Google Scholar
19Walter, W.. Differential and integral inequalities (Berlin: Springer, 1970).CrossRefGoogle Scholar