Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-19T08:38:17.942Z Has data issue: false hasContentIssue false
  • English
  • Français

The deficiency indices of powers of second order expressions with large leading coefficient

Published online by Cambridge University Press:  14 November 2011

Thomas T. Read
Thomas T. Read
Affiliation:
Department of Mathematics, Western Washington University, Bellingham, Washington 98225, U.S.A
Get access Rights & Permissions [Opens in a new window]

Synopsis

The deficiency index of each power of the differential expression M[y] = w−1(−(py′)′ + qy), defined on [a, ∞), is calculated exactly in terms of the behaviour of a simple function of p and w for a large class of expressions satisfying a hypothesis which requires that p be large compared with w and q. In general, not all powers of M are limit-point.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 101 , Issue 3-4 , 1985 , pp. 227 - 235
Copyright
Copyright © Royal Society of Edinburgh 1985

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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
2Coppel, W. A.. Disconjugacy. Lecture Notes in Mathematics 220 (Berlin: Springer, 1970).Google Scholar
3Evans, 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
4Everitt, W. N. and Giertz, M.. On some properties of the powers of a formally self-adjoint differential expression. Proc. London Math. Soc. 24 (1972), 149–170.CrossRefGoogle Scholar
5Hartman, P.. Ordinary Differential Equations (New York: John Wiley, 1964).Google Scholar
6Kauffman, R. M.. Polynomials and the limit-point condition. Trans. Amer. Math. Soc. 201 (1975), 347–366.CrossRefGoogle Scholar
7Kauffman, R. M.. On the limit-n classification of powers of positive coefficient ordinary differential expressions. Proc. Roy. Soc. Edinburgh Sect. A 97 (1984), 131–144.CrossRefGoogle Scholar
8Kauffman, R. M., personal communication.Google Scholar
9Kauffman, R. M., Read, R. T. and Zettl, A.. The Deficiency Index Problem for Powers of Ordinary Differential Expressions. Lecture Notes in Mathematics 621 (Berlin: Springer, 1977).Google Scholar
10Read, T. T.. On the limit point condition for polynomials in a second order differential expression. J. London Math. Soc. 10 (1975), 357–366.CrossRefGoogle Scholar
11Read, T. T.. Limit-point criteria for polynomials in a non-oscillatory expression. Proc. Roy. Soc. Edinburgh Sect. A 76 (1976), 13–29.CrossRefGoogle Scholar
12Read, T. T.. Factorization and discrete spectra for second order differential expressions. J. Differential Equations 35 (1980), 388–406.CrossRefGoogle Scholar
13Wintner, A.. Asymptotic integrations of the adiabatic oscillator in its hyperbolic range. Duke Math. J. 15 (1948), 55–67.CrossRefGoogle Scholar