Hostname: page-component-76fb5796d-5g6vh Total loading time: 0 Render date: 2024-04-25T13:13:50.940Z Has data issue: false hasContentIssue false
  • English
  • Français

An inverse problem involving the Titchmarsh–Weyl m-function

Published online by Cambridge University Press:  14 November 2011

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

Synopsis

Let m(λ) denote the Titchmarsh–Weyl m-function associated with the differential equation

with Neumann boundary condition at 0. In the case q ∊ CM[0, ε) for some integer M and ε > 0 we prove the following result (Theorem 2.1): If for some integer ŋ ≧ 3 we know that Im {λ(ŋ + l)/2m0(λ)} = 0 on the ray λ = |λ| eiπ/ŋ, then q(v) (0) = 0 for all v in the set {0, 1, … M} for Which ≢ ŋ − 2 mod ŋ.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 110 , Issue 3-4 , 1988 , pp. 305 - 309
Copyright
Copyright © Royal Society of Edinburgh 1988

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-coefflcient and the spectral function for scalar second order differential expressions. Lecture Notes in Mathematics 949 (Berlin: Springer, 1982).Google Scholar
3Evans, W. D. and Everitt, W. N.. A return to the Hardy–Littlewood inequality. Proc. Roy. Soc. London Ser. A 380 (1982), 447–486.Google Scholar
4Evans, W. D. and Everitt, W. N.. On an inequality of Hardy–Littlewood type: I. Proc. Roy. Soc. Edinburgh Sect. A 101 (1985), 131–140.CrossRefGoogle Scholar
5Evans, W. D., Everitt, W. N., Hayman, W. K. and Ruscheweyh, S..On a class of inequalities of Hardy–Littlewood type. J. Analyse Math. 46 (1986), 118–147.CrossRefGoogle Scholar
6Harris, B. J.. The asymptotic form of the Titchmarsh–Weyl m-function. J. London Math. Soc. 92 (1984), 110–118.CrossRefGoogle Scholar
7Harris, 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.CrossRefGoogle Scholar
8Harris, B. J.. A property of the asymptotic series for a class of Titchmarsh–Weyl m-functions. Proc. Roy. Soc. Edinburgh Sect. A 102 (1986), 253–257.CrossRefGoogle Scholar
9Harris, B. J.. The asymptotic form of the Titchmarsh–Weyl m-functions for second order linear differential equations with analytic coefficient. J. Differential Equations 65 (1986), 219–234.CrossRefGoogle Scholar
10Harris, B. J.. The form of the spectral functions associated with a class of Sturm–Liouville equations with integrable coefficient. Proc. Roy. Soc. Edinburgh Sect. A 105(1987), 215–227.CrossRefGoogle Scholar
11Titchmarsh, E. C.. Eigenfunction Expansions associated with second order differential equations Vol. 1 (London: Oxford University Press, 1962).Google Scholar