Hostname: page-component-848d4c4894-nmvwc Total loading time: 0 Render date: 2024-06-22T09:32:04.056Z Has data issue: false hasContentIssue false
  • English
  • Français

On the limit-n classification of powers of positive coefficient ordinary differential expressions

Published online by Cambridge University Press:  14 November 2011

Robert M. Kauffman
Robert M. Kauffman
Affiliation:
Department of Mathematics, Western Washington University, Bellingham, WA 98225, U.S.A.
Get access Rights & Permissions [Opens in a new window]

Synopsis

Let , with each pi a finite sum of real multiples of real powers of x, and with D = d/dx. Suppose that pi≧0, pN > 0, and p0≧ε > 0 on the interval [1, ∞). Suppose that there exists a j such that degree pi − 2j > degree pi−2i for all i ≠ j. We compute the deficiency index d(Ln) in L2[l, ∞) of any power of L. It is known that d(L) = N. We give an exact formula for d(Ln) which, if degree pi − 2j > 0, shows that d(Ln) = nN for all n if and only if j = 0, and that the limit as n approaches infinity of d(Ln)/n is N + j. We show in addition that if degree pi − 2i ≦ 0 for all i, d(Ln) = nN for all n.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 97 , 1984 , pp. 131 - 144
Copyright
Copyright © Royal Society of Edinburgh 1984

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.Google Scholar
2Evans, 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
3Everitt, W. N. and Giertz, M.. On some properties of the powers of a formally self-adjoint differential expression. Proc. London Math. Soc. 24 (1972), 756–768.CrossRefGoogle Scholar
4Goldberg, S.. Unbounded linear operators: theory and applications (New York: McGraw-Hill, 1966).Google Scholar
5Kauffman, R. M.. Polynomials and the limit point condition. Trans. Amer. Math. Soc. 201 (1975), 347–366.Google Scholar
6Kauffman, R. M.. On the limit-n classification of ordinary differential operators with positive coefficients. Proc. London Math. Soc. 35 (1972), 496–526.Google Scholar
7Kauffman, 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
8Read, T. T.. On the limit-point condition for polynomials in a second order differential expression. J. London Math. Soc. 10 (1975), 357–366.Google Scholar