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On the location of spectral concentration for Sturm–Liouville problems with rapidly decaying potential

Part of: Boundary value problems

Published online by Cambridge University Press:  26 February 2010

M. S. P. Eastham
M. S. P. Eastham
Affiliation:
Department of Computer Science, Cardiff University of Wales, P.O. Box 916, Cardiff CF2 3XF
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Extract

The spectral function ρα(μ) (−∞<μ<∞) associated with the Sturm–Liouville equation

and a boundary condition

is a non-decreasing function of μ which is defined in terms of the Titchmarsh–Weyl function mα(λ) for (1.1) and (1.2). Thus, taking into account a standardization of the sign attached to mα(λ), we have

[4, Chapter 9, Theorem 3.1; 9, Section 2.3; 21, Sections 3.3 and 6.7].

MSC classification

Secondary: 34B24: Sturm-Liouville theory
Type
Research Article
Information
Mathematika , Volume 45 , Issue 1 , June 1998 , pp. 25 - 36
Copyright
Copyright © University College London 1998

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References

1.Brändas, E., Rittby, M. and Elander, N.. Titchmarsh–Weyl theory and its relations to scattering theory: Spectral densities and cross sections; Theory and applications. J. Math. Phys., 26 (1985), 2648–2658.CrossRefGoogle Scholar
2.Brown, B. M., Eastham, M. S. P. and McCormack, D. K. R.. Spectral concentration and rapidly decaying potentials. J. Comp. Appl. Math., 81 (1997), 333–348.CrossRefGoogle Scholar
3.Brown, B. M., Eastham, M. S. P. and McCormack, D. K. R.. Spectral concentration and perturbed discrete spectra. J. Comp. Appl. Math. 86 (1997), 415–425.CrossRefGoogle Scholar
4.Coddington, E. A. and Levinson, N.. Theory of ordinary differential equations (McGraw-Hill, New York, 1955).Google Scholar
5.Eastham, M. S. P.. The spectral theory of periodic differential equations (Scottish Academic Press, Edinburgh, 1973).Google Scholar
6.Eastham, M.S. P.. The asymptotic nature of spectral functions in Sturm-Liouville problems with continuous spectrum. J. Math. Anal. Appl. 213 (1997), 573–582.CrossRefGoogle Scholar
7.Eastham, M. S. P.. The asymptotic form of the spectral function in Sturm-Liouville problems with a large potential like -xc (c≤2). Proc. Roy. Soc. Edinburgh, 128A (1998), 37–45.CrossRefGoogle Scholar
8.Eastham, M. S. P., Fulton, C. T. and Pruess, S. A.. Using the SLEDGE package on Sturm- Liouville problems having nonempty essential spectra. ACM Trans. Math. Software, 22 (1996), 423–446.CrossRefGoogle Scholar
9.Eastham, M. S. P. and Kalf, H.. Schrödinger-type operators with continuous spectra. Research Notes in Mathematics 65 (Pitman, London, 1982).Google Scholar
10.Eastham, M. S. P. and McLeod, J. B.. The existence of eigenvalues embedded in the continuous spectrum of ordinary differential operators. Proc. Roy. Soc. Edinburgh, 79A (1977), 25–34.CrossRefGoogle Scholar
11.Froese, R.. Asymptotic distribution of resonances in one dimension. J. Dijf. Equations, 137 (1997), 251–272.CrossRefGoogle Scholar
12.Gilbert, D. J. and Pearson, D. B.. On subordinacy and analysis of the spectrum of one-dimensional Schrodinger operators. J. Math. Anal. Appl., 128 (1987), 30–56.CrossRefGoogle Scholar
13.Hartman, P.. Ordinary differential equations, 2nd ed. (Birkhauser, 1982).Google Scholar
14.Hehenberger, M.. Weyl's theory applied to predissociation by rotation III. Jeffreys' approximation. J. Chem. Phys., 67 (1977), 1710–1711.CrossRefGoogle Scholar
15.Hislop, P. D. and Sigal, I. M.. Introduction to spectral theory (Springer, New York, 1996).CrossRefGoogle Scholar
16.Hochstadt, H.. Asymptotic estimates for the Sturm-Liouville spectrum. Comm. Pure Appl Math., 14(1961), 749–764.CrossRefGoogle Scholar
17.Marietta, M.. The detection of quantum-mechanical resonances using the Pruess method. Technical Note ACM 90-14 (Royal Military College of Science, 1990).Google Scholar
18.Pruess, S. A. and Fulton, C. T.. Mathematical software for Sturm-Liouville problems. ACM Trans. Math. Software, 19 (1993), 360–376.CrossRefGoogle Scholar
19.Pruess, S. A., Fulton, C. T. and Xie, Y.. Performance of the Sturm-Liouville software package SLEDGE. Colorado School of Mines Department of Math, and Comp. Sci. Tech. Rep. MCS-91-19, 1991.Google Scholar
20.Stolz, G.. Bounded solutions and absolute continuity of Sturm-Liouville operators. J. Math. Anal. Appl., 169 (1992), 210–228.CrossRefGoogle Scholar
21.Titchmarsh, E. C.. Eigenfunction expansions, Part I, 2nd ed. (Clarendon Press, Oxford, 1962).Google Scholar
22.Weyl, H.. Über gewöhnlicher Differentialgleichungen mit Singularitaten und die zugehörigen Entwicklungen willkürlicher Funktionen. Math. Ann., 68 (1910), 220–269.CrossRefGoogle Scholar