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

Abstract

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.

Boundary Value Problems, Weyl Functions, and Differential Operators

“…Under certain circumstances the Weyl function in the limit-point case (of a usual Sturm-Liouville operator) belongs to the Kac class; see for instance [420]. For further results in this direction, see [384,385,386,387]; in these cases there is a distinguished selfadjoint extension, namely the generalized Friedrichs extension; see the notes on Chapter 5. Sturm-Liouville equations with vector-valued coefficients are beyond the scope of this text; see [345,346] and for a recent contribution [330].…”