Hostname: page-component-848d4c4894-v5vhk Total loading time: 0 Render date: 2024-06-30T10:10:03.407Z Has data issue: false hasContentIssue false
  • English
  • Français

On bounds for Titchmarsh–Weyl m-coefficients and for spectral functions for second-order differential operators

Published online by Cambridge University Press:  14 November 2011

F. V. Atkinson
F. V. Atkinson
Affiliation:
Department of Mathematics, University of Toronto, Toronto, Ontario, Canada M5S 1A1
Get access Rights & Permissions [Opens in a new window]

Synopsis

Order-of-magnitude results are extended to the case of general second-order term, with coefficient not necessarily of fixed sign, with general positive weight-function. The bounds are used to establish the expression for the Titchmarsh–Weyl function m(λ) as a Nevanlinna function in terms of the spectral function.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 97 , 1984 , pp. 1 - 7
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

1Atkinson, F. V.. On the location of the Weyl circles. Proc. Roy. Soc. Edinburgh Sect. A 88 (1981), 345–356.CrossRefGoogle Scholar
2Atkinson, F. V.. On the asymptotic behaviour of the Titchmarsh–Weyl m-coefficient and the spectral function for scalar second-order differential expressions. Lecture Notes in Mathematics 964, 1-27 (Heidelberg: Springer, 1982).Google Scholar
3Bennewitz, C. and Everitt, W. N.. Some remarks on the Titchmarsh–Weyl m-coefficient. In Tribute to Åke Pleijel (Proc. Conf. on Differential Equations, Uppsala, 1979) 49–108 (Uppsala, Sweden).Google Scholar
4Everitt, W. N.. On a property of the m-coefficient of a second–order linear differential equation. J. London Math. Soc. 4 (1972), 443–457.CrossRefGoogle Scholar
5Everitt, W. N. and Halvorsen, S. G.. On the asymptotic form of the Titchmarsh–Weyl m-coefficient. Applicable Anal. 8 (1978), 153–169.CrossRefGoogle Scholar
6Everitt, W. N. and Zettl, A.. On a class of integral inequalities. J. London Math. Soc. 17 (1978), 291–303.CrossRefGoogle Scholar
7Halvorsen, S. G.. Asymptotics of the Titchmarsh–Weyl m-coefficient. In Proc. Conference on Differential Equations, Birmingham, Alabama, 1983 (ed. Knowles, I. W. and Lewis, R. T.) (Amsterdam: North Holland, to appear).Google Scholar
8Hille, E.. Green's transforms and singular boundary-value problems. J. Math. Pures Appl. (9) 42 (1963), 331–349.Google Scholar
9Hille, E.. Lectures on Ordinary Differential Equations (London: Addison-Wesley, 1969).Google Scholar
10Kac, I. S.. Power-asymptotic estimates for spectral functions of generalized boundary-value problems of second order. Dokl. Akad. Nauk SSSR 203 (1972), 752–755; also Amer. Math. Soc. Transl. (2) 13 (1972), 453-457.Google Scholar
11Kac, I. S.. A generalization of the asymptotic formula of V. A. Marčenko for the spectral functions of a second-order boundary-value problem. Izv. Akad. Nauk SSSR, Ser. Mat. 37 (1973), 422–436.Google Scholar