A Class of Nevanlinna Functions Related to Singular Sturm-Liouville Problems

Seppo Hassi, H.S.V. de Snoo

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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.
    Original languageEnglish
    Pages (from-to)2885-2893
    Number of pages9
    JournalProceedings of the american mathematical society
    Volume134
    Issue number10
    Publication statusPublished - 2006

    Keywords

    • Titchmarsh-Weyl coefficient
    • Nevanlinna function
    • Kac class
    • integral representation
    • Sturm-Liouville problem

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