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The number of integrable-square solutions of products of differential expressions*

Published online by Cambridge University Press:  14 February 2012

W. N. Everitt  and
A. Zettl
W. N. Everitt
Affiliation:
Department of Mathematics, University of Dundee
A. Zettl
Affiliation:
Northern Illinois University, De Kalb, Illinois, USA
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Synopsis

Given differential expressions τ1; τ2, …, τn— not necessarily symmetric—which are regular on [0,∞), we investigate the relationship between the number of linearly independent L2(0,∞) solutions of the equations τjy = 0 and of the product equation (τ1τ2 … τn)y = 0. Our results extend those recently obtained in [15, 16, 17] for the special case τJ = τ for j = 1, …, n and τ is symmetric. In particular they include the classification results of Everitt and Giertz [4,5,6] for this special case when τ is a real second-order symmetric expression.

Information

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 76 , Issue 3 , 1977 , pp. 215 - 226
Copyright
Copyright © Royal Society of Edinburgh 1977

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References

1Chaudhuri, J. and Everitt, W. N.. On the square of a formally self-adjoint differential expression. J. London Math. Soc. 1 (1969), 661–673.CrossRefGoogle Scholar
2Dunford, N. and Schwartz, J. T.. Linear operators II (New York: Interscience, 1963).Google Scholar
3Everitt, W. N.. Integrable-square solutions of ordinary differential equations. Quart. J. Math. Oxford Ser. 10 (1959), 145–155.CrossRefGoogle Scholar
4Everitt, W. N. and Giertz, M.. Some properties of the domains of certain differential operators. Proc. London Math. Soc. 23 (1971), 301–324.CrossRefGoogle Scholar
5Everitt, W. N. and Giertz, M.. One some properties of the powers of a formally self-adjoint differential expression. Proc. London Math. Soc. 24 (1972), 149–170.CrossRefGoogle Scholar
6Everitt, W. N. and Giertz, M.. On some properties of the domains of powers of certain differential operators. Proc. London Math. Soc. 24 (1972), 756–768.CrossRefGoogle Scholar
7Glazman, I. M.. On the theory of singular differential operators. Uspehi Mat. Nauk 5, 6 (40) (1950), 102–135 (Russian); Amer. Math. Soc. Transl. 96 (1953).Google Scholar
8Goldberg, S.. Unbounded linear operators (New York: McGraw Hill, 1966).Google Scholar
9Halperin, I.. Closures and adjoints of linear differential operators. Ann. of Math. 38 (1937), 880–919.CrossRefGoogle Scholar
10Kato, T.. Perturbation theory for linear operators (New York: Springer, 1966).Google Scholar
11Kauffman, R. M.. Polynomials and the limit-point condition. Trans. Amer. Math. Soc. 201 (1975), 347–366.CrossRefGoogle Scholar
12Kogan, V. I. and Rofe-Beketov, F. S.. On the question of deficiency indices of differential operators with complex coefficients. Mat. Fiz. Funk. Anal. 2 (1971), 45–60 (Russian); Proc. Roy. Soc. Edinburgh Sect. A 72 (1975), 281–298 (English).Google Scholar
13McLeod, J. B.. The number of integrable-square solutions of ordinary differential equations. Quart. J. Math. Oxford Ser. 17 (1966), 285–290.CrossRefGoogle Scholar
14Neumark, M. A.. Lineare Differentialoperatoren (Berlin: Akademie-Verlag, 1960).CrossRefGoogle Scholar
15Zettl, A.. Deficiency indices of polynomials in symmetric differential expressions. Lecture Notes in Mathematics 415, 293–301 (Berlin: Springer, 1974).Google Scholar
16Zettl, A.. Deficiency indices of polynomials in symmetric differential expressions II. Proc. Roy. Soc. Edinburgh Sect. A 73 (1975), 301–306.CrossRefGoogle Scholar
17Zettl, A.. The limit point and limit circle cases for polynomials in a differential operator. Proc. Roy. Soc. Edinburgh Sect. A 72 (1975), 219–224.CrossRefGoogle Scholar
18Karlsson, B.. Generalization of a theorem of Everitt. J. London Math. Soc. 9 (1974), 131–141.CrossRefGoogle Scholar
19Walker, P. W.. A vector-matrix formulation for formally symmetric ordinary differential equations with applications to solutions of integrable square. J. London Math. Soc. 9 (1974), 151–159.CrossRefGoogle Scholar
20Walker, P. W.. A note on differential equations with all solutions of integrable-square. Pacific J. Math. 56 (1975), 285–289.CrossRefGoogle Scholar