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The form of the spectral functions associated with Sturm-Liouville problems with continuous spectrum

Part of: Boundary value problems

Published online by Cambridge University Press:  26 February 2010

B. J. Harris
B. J. Harris
Affiliation:
Department of Mathematical Sciences, Northern Illinois University, DeKalb, Illinois 60115-2888, U.S.A.
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Extract

We consider the spectral function, ρα(μ), for –∞<μ<∞ associated with the Sturm-Liouville equation

and the boundary condition

We suppose that q is a real-valued member of L1[0, ∞) and λ is a real parameter.

MSC classification

Secondary: 34B24: Sturm-Liouville theory
Type
Research Article
Information
Mathematika , Volume 44 , Issue 1 , June 1997 , pp. 149 - 161
Copyright
Copyright © University College London 1997

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References

1.Atkinson, F. V.. On the asymptotic behavior of the Titchmarsh Weyl m-coejficient and the spectral function for sealer second order differential expressions. Lecture Notes in Mathematics 964 (Springer, Berlin, 1982).Google Scholar
2.Coddington, E. A. and Levinson, N.. On the nature of the spectrum of singular, second order linear differential equations. Canad. J. Math., 3 (1951), 335–338.CrossRefGoogle Scholar
3.Coddington, E. A. and Levinson, N.. Theory of Ordinary Differential Equations (McGraw Hill, New York, 1955).Google Scholar
4.Eastham, M. S. P. and Kalf, H.. Schrödinger type operators with continuous spectra. Research Notes in Mathematics (Pitman, London, 1982).Google Scholar
5.Eastham, M. S. P.. The asymptotic nature of spectral functions in Sturm Liouville problems with continuous spectra. Preprint.Google Scholar
6.Fulton, C. T. and Preuss, S. A.. Eigenvalue and eigenfunction asymptotics for regular Sturm-Liouville problems. J. Math. Analysis and Appl., 188 (1994), 215–227.Google Scholar
7.Gilbert, D. and Pearson, D. B.. On subordinary and analysis of the spectrum of one dimensional Schrodinger operators. J. Math. Analysis and Appl., 128 (1987), 30–56.CrossRefGoogle Scholar
8.Harris, B. J.. The asymptotic form of the spectral functions associated with a class of Sturm–Liouville equations. Proc. Royal Soc. Edinburgh, 100A (1985), 343–360.CrossRefGoogle Scholar
9.Harris, B. J.. The asymptotic form of the Titchmarsh-Weyl m-function for second order linear differential equations with analytic coefficients. J. Diff. Equations, 65 (1986), 219–234.CrossRefGoogle Scholar
10.Harris, B. J.. The form of the spectral functions associated with a class of Sturm–Liouville equations with integrable coefficient. Proc. Royal Soc. Edinburgh, 105A (1987), 215–227.CrossRefGoogle Scholar
11.Harris, B. J. and Talarico, S. T.. On turning points. Preprint.Google Scholar
12.Titchmarsh, E. C.. Eigenfunction Expansions, Part 1 (Clarendon Press, Oxford, 2nd editions 1962).Google Scholar