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Asymptotic expansions of generalized Nevanlinna functions Q are investigated by means of a factorization model involving a part of the generalized zeros and poles of nonpositive type of the function Q. The main results in this paper arise from the explicit construction of maximal Jordan chains in the root subspace R∞(SF ) of the so-called generalized Friedrichs extension. A classification of maximal Jordan chains is introduced and studied in analytical terms by establishing the connections to the appropriate asymptotic expansions. This approach results in various analytic characterizations of the spectral properties of selfadjoint relations in a Pontryagin space and, conversely, translates spectral theoretical properties into analytic properties of the associated Weyl functions.

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A generalized Nevanlinna function Q(z) with one negative square has precisely one generalized zero of nonpositive type in the closed extended upper halfplane. The fractional linear transformation defined by Qτ (z) = (Q(z) − τ )/(1 + τ Q(z)), τ ∈ R ∪ {∞}, is a generalized Nevanlinna function with one negative square. Its generalized zero of nonpositive type α(τ ) as a function of τ is being studied. In particular, it is shown that it is continuous and its behavior in the points where the function extends through the real line is investigated.

In this note we consider regular Sturm -Liouville equations with a floating singularity of a special type: the coefficient of the second order derivative contains the eigenvalue parameter. We determine the form of the boundary conditions which make the problem selfadjoint after linearizing. In general the boundary conditions for the linearized system give rise to boundary conditions which involve the eigenvalue parameter in the original, non -linearized, problem. The boundary conditions give rise to a 2 x 2 matrix function, the so-called Titchmarsh-Weyl coefficient. The characteristic properties of this function are studied. The formal aspects of the theory of this class of equations turn out to be quite parallel to those for the usual situation when there is no floating singularity.