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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A class of Nevanlinna functions related to singular Sturm-Liouville problems


Authors: Seppo Hassi, Manfred Möller and Henk de Snoo
Journal: Proc. Amer. Math. Soc. 134 (2006), 2885-2893
MSC (2000): Primary 30D15; Secondary 34B20
Published electronically: April 7, 2006
MathSciNet review: 2231612
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Abstract: The class of Nevanlinna functions consists of functions which are holomorphic off the real axis, which are symmetric with respect to the real axis, and whose imaginary part is nonnegative in the upper halfplane. The Kac subclass of Nevanlinna functions is defined by an integrability condition on the imaginary part. In this note a further subclass of these Kac functions is introduced. It involves an integrability condition on the modulus of the Nevanlinna functions (instead of the imaginary part). The characteristic properties of this class are investigated. The definition of the new class is motivated by the fact that the Titchmarsh-Weyl coefficients of various classes of Sturm-Liouville problems (under mild conditions on the coefficients) actually belong to this class.


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Additional Information

Seppo Hassi
Affiliation: Department of Mathematics and Statistics, University of Vaasa, P.O. Box 700, 65101 Vaasa, Finland
Email: sha@uwasa.fi

Manfred Möller
Affiliation: Department of Mathematics, University of the Witwatersrand, Wits, 2050, South Africa
Email: manfred@maths.wits.ac.za

Henk de Snoo
Affiliation: Department of Mathematics and Computing Science, University of Groningen, P.O. Box 800, 9700 AV Groningen, Nederland
Email: desnoo@math.rug.nl

DOI: http://dx.doi.org/10.1090/S0002-9939-06-08295-5
PII: S 0002-9939(06)08295-5
Keywords: Nevanlinna function, Kac class, integral representation, Sturm-Liouville problem, Titchmarsh-Weyl coefficient
Received by editor(s): August 3, 2004
Received by editor(s) in revised form: April 12, 2005
Published electronically: April 7, 2006
Additional Notes: The authors gratefully acknowledge the support of the NRF of South Africa under GUN 2053746, of the Research Institute for Technology of the University of Vaasa, and of the Dutch Association for Mathematical Physics under MF04/62a.
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2006 American Mathematical Society