A System Coupling and Donoghue Classes of Herglotz–Nevanlinna Functions

Abstract

We study the impedance functions of conservative L-systems with the unbounded main operators. In addition to the generalized Donoghue class Mκ of Herglotz-Nevanlinna functions considered by the authors earlier, we introduce "inverse" generalized Donoghue classes M −1 κ of functions satisfying a different normalization condition on the generating measure, with a criterion for the impedance function V Θ (z) of an L-system Θ to belong to the class M −1 κ presented. In addition, we establish a connection between "geometrical" properties of two L-systems whose impedance functions belong to the classes Mκ and M −1 κ , respectively. In the second part of the paper we introduce a coupling of two L-system and show that if the impedance functions of two L-systems belong to the generalized Donoghue classes Mκ 1 (M −1 κ 1 ) and Mκ 2 (M −1 κ 2 ), then the impedance function of the coupling falls into the class Mκ 1 κ 2 . Consequently, we obtain that if an L-system whose impedance function belongs to the standard Donoghue class M = M 0 is coupled with any other L-system, the impedance function of the coupling belongs to M (the absorbtion property). Observing the result of coupling of n L-systems as n goes to infinity, we put forward the concept of a limit coupling which leads to the notion of the system attractor, two models of which (in the position and momentum representations) are presented. All major results are illustrated by various examples.

Perturbations of Donoghue Classes and Inverse Problems for L-Systems

“…Proof. The proof is completely similar to the proof of Theorem 10 except for the fact that it relies on [10,Theorem 5.6]. Again, we note that as it was shown in the proof of [10, Theorem 5.6], the realizing L-system Θ κ can be chosen as a minimal model L-system Θ 2 of the form (224) described in details in Appendix B.…”

“…Proof. The proof structure here completely resembles the one of Theorem 32 except for the fact that it also relies on the realization theorem for the class M −1 κ0 found in [10,Theorem 5.6].…”

“…Similarly (see [10]), a function M ∈ N belongs to the generalized Donoghue class M −1 κ if in the representation (24)…”

“…Relation (11) of Hypothesis 3 implies that g + − g − ∈ Dom(B) and g + − κg − ∈ Dom(T κ B ). As it was shown in [10,Section 4] (222)…”

“…is called the impedance function of Θ. This article is a part of an ongoing project studying the connections between various subclasses of Herglotz-Nevanlinna functions and conservative realizations of L-systems (see [2], [3], [4], [9], [10], [13], [19], [20]). The class of all Herglotz-Nevanlinna functions in a finite-dimensional Hilbert space E, that can be realized as impedance functions of an L-system, was described in [7], [4,Definition 6.4.1].…”