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Asymptotics of the Titchmarsh-Weyl m-coefficient for integrable potentials*

Published online by Cambridge University Press:  14 November 2011

Hans G. Kaper  and
Man Kam Kwong
Hans G. Kaper
Affiliation:
Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, IL 60439-4844, U.S.A.
Man Kam Kwong
Affiliation:
Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, IL 60439-4844, U.S.A.
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Extract

This article is concerned with the asymptotic behaviour of m(λ), the Titchmarsh-Weyl m-coefficient, for the singular eigenvalue equation y“ + (λ − q(x))y = 0 on [0, ∞), as λ →∞ in a sector in the upper half of the complex plane. It is assumed that the potential function q is integrable near 0. A simplified proof is given of a result of Atkinson [7], who derived the first two terms in the asymptotic expansion of m(λ), and a sharper error bound is obtained. Theproof is then generalised to derive subsequent terms in the asymptotic expansion. It is shown that the Titchmarsh-Weyl m-coefficient admits an asymptotic power series expansion if the potential function satisfies some smoothness condition. A simple method to compute the expansion coefficients is presented. The results for the first few coefficients agree with those given by Harris [9].

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 103 , Issue 3-4 , 1986 , pp. 347 - 358
Copyright
Copyright © Royal Society of Edinburgh 1986

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References

1Weyl, H.. Ueber gewöhnliche Differentialgleichungen mit Singularitäten und die zugehörigen Entwicklungen willkürlicher Funktionen. Math. Ann. 68 (1910), 220–269.Google Scholar
2Titchmarsh, E. C.. Eigenfunction expansions associated with second order differential equations, Part I, 2nd ed. (Oxford University Press, 1962).Google Scholar
3Everitt, W. N. and Bennewitz, C.. Some remarks on the Titchmarsh-Weyl m-coefficient. In Tribute to Ake Pleijel, pp. 49–108 (Dept. of Mathematics, University of Uppsala, Sweden, 1980).Google Scholar
4Hille, E.. Lectures on ordinary differential equations (London: Addison-Wesley, 1969).Google Scholar
5Everitt, W. N.. On a property of the m-coefficient of a second-order linear differential equation. J. Londonth. Soc. 4 (1972), 443–457.CrossRefGoogle Scholar
6Everitt, W. N. and Halvorsen, S. G.. On the asymptotic form of the Titchmarsh-Weyl m-coefficient. Applicable Anal. 8 (1978), 153–169.Google Scholar
7Atkinson, F. V.. On the location of the Weyl circles. Proc. Roy. Soc. Edinburgh Sect. A 88 (1981), 345–356.CrossRefGoogle Scholar
8Atkinson, F. V. and Fulton, C. T.. The Titchmarsh-Weyl m-coefficient for non-integrablepotentials (preprint, 1985).Google Scholar
9Harris, B. J.. On the Titchmarsh-Weyl m-function. Proc. Roy. Soc. Edinburgh Sect. A 95 (1983), 223–237.CrossRefGoogle Scholar
10Harris, B. J.. The asymptotic form of the Titchmarsh-Weyl m-function. J. London. Math. Soc. (2) 30 (1984), 110–118.CrossRefGoogle Scholar
11Harris, B. J.. The asymptotic form of the Titchmarsh-Weyl m-function associated with a second order differential equation with locally integrable coefficient, Proc. Roy. Soc. Edinburgh Sect. A 102 (1986), 243–252.Google Scholar
12Piccinini, L. C., Stampacchia, G. and Vidossich, G.. Ordinary Differential Equations in Rn (New York: Springer, 1984).Google Scholar