Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-26T06:51:25.860Z Has data issue: false hasContentIssue false
  • English
  • Français

A property of the asymptotic series for a class of Titchmarsh–Weyl m-functions

Published online by Cambridge University Press:  14 November 2011

B. J. Harris
B. J. Harris
Affiliation:
Department of Mathematical Sciences, Northern Illinois University, DeKalb, Illinois 60115-2888, U.S.A.
Get access Rights & Permissions [Opens in a new window]

Synopsis

In an earlier paper [6] we showed that if q ϵ CN[0, ε) for some ε > 0, then the Titchmarsh–Weyl m(λ) function associated with the second order linear differential equation

has the asymptotic expansion

as |A| →∞ in a sector of the form 0 < δ < arg λ < π – δ.

We show that if the real valued function q admits the expansion

in a neighbourhood of 0, then

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 102 , Issue 3-4 , 1986 , pp. 253 - 257
Copyright
Copyright © Royal Society of Edinburgh 1986

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

1Atkinson, F. V.. On the asymptotic behaviour of the Titchmarsh–Weyl m-coefficients and the spectral functions for scalar second order differential expressions. Lecture Notes in Mathematics 949. (Berlin: Springer, 1982).Google Scholar
2Bennewitz, C. and Everitt, W. A.. Some remarks on the Titchmarsh–Weyl m-coefficient. In Tribute to Åke Pleijel (Uppsala: Department of Mathematics, University of Uppsala, 1980).Google Scholar
3Erdélyi., A.Asymptotic expansions (New York: Dover, 1956).Google Scholar
4Harris, B. J.. The asymptotic form of the Titchmarsh-Weyl m-function. J. London Math. Soc. (2) 30 (1984), 110–118.CrossRefGoogle Scholar
5Harris, B. J.. The asymptotic form of the spectral functions associated with a class of Sturm–Liouville equations. Proc. Roy. Soc. Edinburgh Sect. A 100 (1985), 343–360.Google Scholar
6Harris, B. J.. The asymptotic form of the Titchmarsh-Weyl m-function for second order linear differential equations with analytic coefficient. J. Differential Equations (to appear).Google Scholar
7Titchmarsh, E. C.. Eigenfunction expansions associated with second order differential equations Vol. 1. Oxford: Oxford University Press, 2nd edition, 1962).Google Scholar
8Weyl, H.. Uber gewöhnliche differentialgleichungen mit singularitäten und die zugehörigen entwicklungen willkurlicher funktionen. Math. Ann. 68 (1910), 220–269.Google Scholar