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 $${\mathfrak {M}}_\kappa $$ of Herglotz–Nevanlinna functions considered by the authors earlier, we introduce “inverse” generalized Donoghue classes $${\mathfrak {M}}_\kappa ^{-1}$$ of functions satisfying a different normalization condition on the generating measure, with a criterion for the impedance function $$V_\Theta (z)$$ of an L-system $$\Theta $$ to belong to the class $${\mathfrak {M}}_\kappa ^{-1}$$ presented. In addition, we establish a connection between “geometrical” properties of two L-systems whose impedance functions belong to the classes $${\mathfrak {M}}_\kappa $$ and $${\mathfrak {M}}_\kappa ^{-1}$$ , 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 $${\mathfrak {M}}_{\kappa _1}$$ ( $${\mathfrak {M}}_{\kappa _1}^{-1}$$ ) and $${\mathfrak {M}}_{\kappa _2}$$ ( $${\mathfrak {M}}_{\kappa _2}^{-1}$$ ), then the impedance function of the coupling falls into the class $${\mathfrak {M}}_{\kappa _1\kappa _2}$$ . Consequently, we obtain that if an L-system whose impedance function belongs to the standard Donoghue class $${\mathfrak {M}}={\mathfrak {M}}_0$$ is coupled with any other L-system, the impedance function of the coupling belongs to $${\mathfrak {M}}$$ (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.