Hostname: page-component-586b7cd67f-vdxz6 Total loading time: 0 Render date: 2024-11-24T01:01:42.907Z Has data issue: false hasContentIssue false
  • English
  • Français

Integrable-square solutions of a singular ordinary differential equation

Published online by Cambridge University Press:  14 November 2011

Richard C. Gilbert
Richard C. Gilbert
Affiliation:
Mathematics Department, California State University, Fullerton, California, U.S.A.
Get access Rights & Permissions [Opens in a new window]

Synopsis

Absolutely square integrable solutions are determined for the equation = λ y where the ζn−r(x) are holomorphic in a sector of the complex plane and have asymptotic expansions as x approaches infinity. It is shown that the number of such solutions depends upon the roots of the characteristic equation and their multiplicity, and upon the sign of the derivative of the characteristic polynomial. Application is made to formally symmetric ordinary differential operators.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 89 , Issue 3-4 , 1981 , pp. 267 - 279
Copyright
Copyright © Royal Society of Edinburgh 1981

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

1Gilbert, R. C.. Asymptotic formulas for solutions of a singular linear ordinary differential equation. Proc. Roy. Soc. Edinburgh. Sect. A 81 (1978), 57–70.Google Scholar
2Gilbert, R. C.. A class of symmetric ordinary differential operators whose deficiency numbers differ by an integer. Proc. Roy. Soc. Edinburgh. Sect. A 82 (1978), 117–134.Google Scholar
3Kogan, V. I. and Rofe-Beketov, F. S.. On the question of the deficiency indices of differential operators with complex coefficients. Proc. Roy. Soc. Edinburgh. Sect. A 72 (1975), 281–298.Google Scholar
4Orlov, S. A.. On the deficiency index of linear differential operators. Dokl. Akad. Nauk SSSR 92 (1953), 483–486.Google Scholar
5Wasow, W.. Asymptotic expansions for ordinary differential equations (New York: Interscience, 1965).Google Scholar
6Weyl, H.. Uber gewohnliche Differentialgleichungen mit Singularitaten und die zugehorigen Entwicklungen wilkiirlicher Funktionen. Math. Anal. 68 (1910), 220–269.CrossRefGoogle Scholar