Generalized Nevanlinna Functions: Operator Representations, Asymptotic Behavior

Abstract

This article gives an introduction and short overview on generalized Nevanlinna functions, with special focus on asymptotic behavior and its relation to the operator representation. Introduction: Classical Nevanlinna FunctionsGeneralized Nevanlinna functions (scalar, matrix-or operator-valued) are functions that are meromorphic in CnR satisfying certain symmetry and sign conditions. They appeared first in connection with self-adjoint operators and relations in Pontryagin spaces (in [46] for scalar and [47] for matrix-valued functions) and have become an important tool in extension theory, e.g., for differential operators in connection with eigenvalue dependent boundary conditions. This survey is started with a short review of old and well-known facts about classical Nevanlinna functions (also known as Herglotz-, Pick-, or R-functions), which build a basis for the indefinite generalizations.Recall that a function q that maps the upper half plane C C holomorphically into C C [R is called a Nevanlinna function, q 2 N 0 .C/. These functions, which appear in many applications, e.g., as Titchmarsh-Weyl coefficients in Sturm-Liouville problems, are very well studied objects. In particular, such a function admits an integral representation ([37,58,59], presented in the following form by Cauer [16]; see also [1]):where a 2 R, b 0, and a positive Borel measure with R R d .t/ 1Ct 2 < 1. More abstractly, for every such function there exists a Hilbert space K; OE ; , a self-adjoint linear relation (i.e., a multi-valued operator) A in K and an element v in K such that with some z 0 2 %.A/ the function q can be written as q.z/ D q.z 0 / C .z z 0 / I C .z z 0 /.A z/ 1 v; v K :E-mail: luger@math.su.se Page 1 of 24Operator Theory DOI 10.1007/978-3-0348-0692-3_35-1 © Springer Basel 2015 In particular, for b D 0 one can choose K D L 2 , where is the measure in the integral representation (1), then A is the operator of multiplication by the independent variable. If b ¤ 0 then A is not an operator, but a relation with one-dimensional multi-valued part (see, e.g., [32] for the theory of linear relations and for the details of this representation, e.g., [53]).The limit behavior of q at the real line can be deduced directly from the above representations. In particular, lim z O !˛.˛ z/q.z/, where lim z O !˛d enotes the non-tangential limit to˛2 R, always exists and is zero or positive. Here the second case is equivalent to˛being an eigenvalue of the minimal representing relation A in (2), or, equivalently, f˛g ¤ 0.In several examples, however, e.g., in connection with singular Sturm-Liouville problems (see, e.g., [31,50,52]), there appear functions that are not as described above, but belong to a so-called generalized Nevanlinna class. Such functions may have non-real poles and the limit lim z O !˛.˛ z/q.z/ does not necessarily exist. However, this exceptional behavior can appear at finitely many points only, which relies on the fact that these functions admit operator representations of the form (2) but in Pontryagin spaces....

Realizations of Meromorphic Functions of Bounded Type

“…Finally, let q be a generalized Nevanlinna function and (K, A, v) be a minimal realization. Then it is well known [37], that q is analytically continuable through an open interval (a, b) ⊂ R if and only if (a, b) ⊂ ̺(A). This characterizes the resolvent set of A completely in terms of the function theoretic properties of q.…”

On spectral deformations and singular Weyl functions for one-dimensional Dirac operators

“…However, this will not be true in general. The following theorem gives a criterion when the singular Weyl function belongs to the class N ∞ κ of generalized Nevanlinna functions with no non-real poles and the only generalized pole of nonpositive type at ∞ (for further information on generalized Nevanlinna functions we refer to [24], see also [19,Appendix B]).…”