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The asymptotic form of the spectral function in Sturm–Liouville problems with a large potential like −xc (c ≦ 2)

Published online by Cambridge University Press:  14 November 2011

M. S. P. Eastham
M. S. P. Eastham*
Affiliation:
Department of Computer Science, University of Wales, Cardiff, U.K.
*
*Author's correspondence address: 4 Howard Road, Bridport, Dorset DT6 4SH, U.K.
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Extract

A new formula is given for the spectral function in singular Sturm–Liouville problems where the potential is like – (const.)xc (0 < c ≦ 2). The formula is used to answer an open question concerning the asymptotic form of the spectral function.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 128 , Issue 1 , 1998 , pp. 37 - 45
Copyright
Copyright © Royal Society of Edinburgh 1998

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References

1Atkinson, 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 (Berlin: Springer, 1982).Google Scholar
2Bennewitz, C.. Spectral asymptotics for Sturm–Liouville equations. Proc. London Math. Soc. (3) 59 (1989), 294–338.CrossRefGoogle Scholar
3Cassell, J. S.. An extension of the Liouville–Green asymptotic formula for oscillatory second-order differential equations. Proc. Roy. Soc. Edinburgh Sect. A 100 (1985), 181–90.CrossRefGoogle Scholar
4Coddington, E. A. and Levinson, N.. Theory of Ordinary Differential Equations (New York: McGraw-Hill, 1955).Google Scholar
5Eastham, M. S. P.. The Asymptotic Solution of Linear Differential Systems, London Math. Soc. Monographs 4 (Oxford: Clarendon Press, 1989).Google Scholar
6Eastham, M. S. P.. The asymptotic nature of spectral functions in Sturm–Liouville problems with continuous spectrum. J. Math. Anal. Appl. 213 (1997), 573–82.CrossRefGoogle Scholar
7Eastham, M. S. P. and Kalf, H.. Schrödinger-type Operators with Continuous Spectra, Research Notes in Mathematics 65 (London: Pitman, 1982).Google Scholar
8Fulton, C. T. and Pruess, S. A.. Eigenvalue and eigenfunction asymptotics for regular Sturm–Liouville problems. J. Math. Anal. Appl. 188 (1994), 215–27.CrossRefGoogle Scholar
9Gilbert, D. J. and Pearson, D. B.. On subordinancy and analysis of the spectrum of one-dimensional Schrödinger operators. J. Math. Anal. Appl. 128 (1987), 30–56.CrossRefGoogle Scholar
10Harris, B. J.. The asymptotic form of the spectral functions associated with a class of Sturm–Liouville equations. Proc. Roy. Soc. Edinburgh Sec. A 100 (1985), 343–60.CrossRefGoogle Scholar
11Levitan, B. M.. Remark on a theorem of V. A. Marčenko. Amer. Math. Soc. Transl. (2) 101 (1973), 105–06.Google Scholar
12Levitan, B. M.. On the asymptotic behavior of the spectral function of a self-adjoint second-order differential equation. Amer. Math. Soc. Transl. (2) 101 (1973), 192–221.Google Scholar
13Marčenko, V. A.. Some questions in the theory of one-dimensional linear differential operators of the second order. Amer. Math. Soc. Trans. (2) 101 (1973), 1–104.Google Scholar
14Marcenko, V. A.. Theorems of Tauberian type in the spectral analysis of differential operators. Amer. Math. Soc. Transl. (2) 102 (1973), 145–90.Google Scholar
15Stolz, G.. Bounded solutions and absolute continuity of Sturm–Liouville operaiors. J. Math. Anal. Appl. 169(1992), 210–28.CrossRefGoogle Scholar
16Titchmarsh, E. C.. Eigenfunction Expansions, Part 1, 2nd edn (Oxford: Clarendon Press, 1962).Google Scholar
17Weyl, H.. Über gewöhnliche Differentialgleichungen mit Singularitaten und die zugehorigen Entwicklungen willkurlicher Functionen. Math. Ann. 68 (1910), 220–69.CrossRefGoogle Scholar