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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

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