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Green's matrix and the formula of Titchmarsh-Kodaira for singular left-definite canonical eigenvalue problems

Published online by Cambridge University Press:  14 November 2011

Bernd Schultze
Bernd Schultze
Affiliation:
Department of Mathematics, University of Essen
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Synopsis

The theory of singular left-definite canonical eigenvalue problems treated by Nieβen and Schneider in is generalized to arbitrary λ∈(ℂ\ℝ∪{0}. In this enlarged theory the Green's matrix of the problem is evaluated and a natural analogue of the Titchmarsh-Kodaira formula is proved. This formula permits the explicit computation of the spectral matrix playing the main role in the expansion theorems of this theory.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 83 , Issue 1-2 , 1979 , pp. 147 - 183
Copyright
Copyright © Royal Society of Edinburgh 1979

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References

1Everitt, W. N.. Some remarks on a differential expression with an indefinite weight function. Spectral theory and asymptotics of differential equations (Amsterdam: North-Holland, 1974).Google Scholar
2Kodaira, K.. On ordinary differential equations of any even order and the corresponding eigenfunction expansions. Amer. J. Math. 72 (1950), 502–544.CrossRefGoogle Scholar
3Nieβen, H. D. und Schneider, A.. Integraltransformationen zu singulären S-hermiteschen Rand-Eigenwertproblemen. Manuscripta Math. 5 (1971), 133–145.Google Scholar
4Nieβen, H. D.. Singuläre S-hermitesche Rand-Eigenwertprobleme. Manuscripta Math. 3 (1970), 35–68.Google Scholar
5Nieβen, H. D.. Zum verallgemeinerten zweiten Weylschen Satz. Arch. Math. 22 (1971), 648–656.Google Scholar
6Nieβen, H. D.. Greensche Matrix und die Formel von Titchmarsh-Kodaira für singuläre S-hermitesche Eigenwertprobleme. J. Reine Angew. Math. 261 (1972), 164–193.Google Scholar
7Schäfke, F. W. und Schneider, A.. S-hermitesche Rand-Eigenwertprobleme I. Math. Ann. 162 (1965), 9–26.CrossRefGoogle Scholar
8Schäfke, F. W. und Schneider, A.. S-hermitesche Rand-Eigenwertprobleme II. Math. Ann. 165 (1966), 236–260.CrossRefGoogle Scholar
9Schäfke, F. W. und Schneider, A.. S-hermitesche Rand-Eigenwertprobleme III. Math. Ann. 177 (1968), 67–94.CrossRefGoogle Scholar
10Schneider, A. und Nieβen, H. D.. Linksdefinite singuläre kanonische Eigenwertprobleme I. J. Reine Angew. Math. 281 (1976), 13–52.Google Scholar
11Schneider, A. und Nieβen, H. D.. Linksdefinite signuläre kanonische Eigenwertprobleme II. J. Reine Angew. Math. 289 (1977), 62–84.Google Scholar
12Schneider, A.. Untersuchungen über singulare reelle S-hermitesche Diflerentialgleichungssysteme im Normalfall. Math. Z. 107 (1968), 271–296.Google Scholar
13Schneider, A.. Weitere Untersuchungen über singulare reelle S-hermitesche Diflerentialgleichungssysteme im Normalfall. Math. Z. 109 (1969), 153–168.CrossRefGoogle Scholar
14Schneider, A.. Die Greensche Matrix S-hermitescher Rand-Eigenwertprobleme im Normalfall. Math. Ann. 180 (1969), 307–312.CrossRefGoogle Scholar
15Schneider, A.. Zum Entwicklungssatz bei rellen singulären Differentialgleichungssystemen. Arch. Math. 21 (1970), 192–197.Google Scholar
16Weyl, H.. Über gewöhnliche lineare Differentialgleichungen mit singulären Stellen und ihre Eigenfunktionen. Nachr. Kgl. Gesellschaft Wiss. Gottingen; Math.-Phys. Kl. 5 (1910), 442–167.Google Scholar