A system coupling and Donoghue classes of Herglotz-Nevanlinna functions

TL;DR

This paper introduces inverse Donoghue classes for Herglotz-Nevanlinna functions, explores their properties, and studies how coupling L-systems affects their impedance functions, leading to the concept of system attractors.

Contribution

It defines inverse generalized Donoghue classes, establishes their connection with standard classes, and analyzes the effects of coupling on impedance functions, including the concept of system attractors.

Findings

Impedance functions belong to inverse Donoghue classes under certain conditions.

Coupling of L-systems results in impedance functions in the product class .

The concept of system attractor is introduced through infinite coupling models.

Abstract

We study the impedance functions of conservative L-systems with the unbounded main operators. In addition to the generalized Donoghue class $\sM_\kappa$ of Herglotz-Nevanlinna functions considered by the authors earlier, we introduce "inverse" generalized Donoghue classes $\sM_\kappa^{-1}$ 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 $\sM_\kappa^{-1}$ presented. In addition, we establish a connection between "geometrical" properties of two L-systems whose impedance functions belong to the classes $\sM_\kappa$ and $\sM_\kappa^{-1}$, 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 $\sM_{\kappa_1}$($\sM_{\kappa_1}^{-1}$) and $\sM_{\kappa_2}$($\sM_{\kappa_2}^{-1}$), then the impedance function of the coupling falls into the class $\sM_{\kappa_1\kappa_2}$. Consequently, we obtain that if an L-system whose impedance function belongs to the standard Donoghue class $\sM=\sM_0$ is coupled with any other L-system, the impedance function of the coupling belongs to $\sM$ (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.