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m (λ)-functions for complex Sturm-Liouville operators

Published online by Cambridge University Press:  14 November 2011

David Race
David Race
Affiliation:
Department of Mathematics, University of the Witwatersrand, 1 Jan Smuts Avenue, Johannesburg 2001, South Africa
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Synopsis

In this paper, the formally J-symmetric Sturm-Liouville operator with complex-valued coefficients is considered and a generalisation of the Weyl limit-point, limit-circle dichotomy is sought by means of m (λ )-functions. These functions are then used to give an explicit description of all the associated J-selfadjoint operators with separated boundary conditions in the limit-circle case. A formulation of the eigenvalues of these operators, and a characterisation of which extensions are non-well-posed, are also found. Finally, the limit-point case is studied, mainly by means of an example.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 86 , Issue 3-4 , 1980 , pp. 275 - 289
Copyright
Copyright © Royal Society of Edinburgh 1980

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References

1Fulton, C. T.. Parametrizations of Titchmarsh's m (λ)-functions in the limit circle case. Trans. Amer. Math. Soc. 229 (1977), 51–63.Google Scholar
2Hartman, P.. Ordinary differential equations (New York: Wiley, 1964).Google Scholar
3Knowles, I. W.. Symmetric conjugate-linear operators in Hilbert space, pre-print.Google Scholar
4Knowles, I. W. and Race, D.. On the point-spectra of complex Sturm-Liouville operators. Proc. Roy. Soc. Edinburgh Sect. A 85 (1980), 263–290.Google Scholar
5Race, D.. On the location of the essential spectra and regularity fields of complex Sturm-Liouville operators. Proc. Roy. Soc. Edinburgh Sect. A 85 (1979), 1–14.Google Scholar
6Race, D.. The spectral theory of complex Sturm-Liouville operators (Ph.D. thesis, Univ. of the Witwatersrand, Johannesburg, 1979).Google Scholar
7Sims, A. R.. Secondary conditions for linear differential operators of the second order. J. Math. Mech. 6 (1957), 247–285.Google Scholar
8Titchmarsh, E. C.. Eigenfunetion expansions associated with second-order differential equations 2nd edn, vol. I (Oxford: Clarendon 1962).Google Scholar
9Weyl, H.. Über gewöhnliche Differentialgleichungen mit Singularitäten und die zugehörigen Entwicklungen willkürlicher Funktionen. Math. Ann. 68 (1910), 220–269.CrossRefGoogle Scholar