Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-27T07:38:14.549Z Has data issue: false hasContentIssue false

On the location of the Weyl circles

Published online by Cambridge University Press:  14 November 2011

F. V. Atkinson
Affiliation:
Department of Mathematics, University of Toronto, Toronto, Canada M5S 1A1

Synopsis

The paper deals with explicit estimates concerning certain circles in the complex plane which were associated with Sturm–Liouville problems by H. Weyl. By the use of Riccati equations instead of linear integral equations, improvements are obtained for results of Everitt and Halvorsen concerning the behaviour of the Titchmarsh–Weyl m-coefficient.

Type
Research Article
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

1Birger, E. S. and Kalyabin, G. A.. The theory of Weyl limit-circles in the case of non-self-adjoint second-order differential-equation systems. Differential'nye Uravnenija 12 (1976), 15311540.Google Scholar
2Brändas, E. and Hehenberger, M.. Determination of Weyl's m-cofficient for a continuous spectrum. In Ordinary and Partial Differential Equations, pp. 316322, Proceedings of the Conference held at Dundee, Scotland, March 26–29, 1974 Lecture Notes in Mathematics 415 (Berlin: Springer, 1974).Google Scholar
3Everitt, W. N.. On a property of the m-coefficient of a second-order linear differential equation. J. London Math. Soc. 4 (1972), 443457.CrossRefGoogle Scholar
4Everitt, W. N. and Bennewitz, C.. Some remarks on the Titchmarsh-Weyl m-coefficient. In A tribute to Åke Pleijel, pp. 49108 (Uppsala: Uppsala Universitet, 1980).Google Scholar
5Everitt, W. N. and Halvorsen, S. G.. On the asymptotic form of the Titchmarsh–Weyl m-coefficient. Applicable Anal. 8 (1978), 153169.CrossRefGoogle Scholar
6Fulton, C.. Parametrizations of Titchmarsh's m(λ)-functions in the limit-circle case. Trans. Amer. Math. Soc. 229 (1977), 5163 (also Ph.D. Dissertation, Rhein-Westf. Technische Hochschule, Aachen, 1973).Google Scholar
7Race, D., m(λ)-functions for complex Sturm–Liouville operators. Proc. Roy. Soc. Edinburgh Sect. A 86 (1980), 275289.CrossRefGoogle Scholar
8Weyl, H.. Ueber gewöhnliche Differentialgleichungen mit Singularitäten und die zugehörigen Entwicklungen willkürlicher Funktionen. Math. Ann. 68 (1910), 220269.CrossRefGoogle Scholar