#### Location

Cupples I Room 218

#### Start Date

7-21-2016 4:00 PM

#### End Date

21-7-2016 4:20 PM

#### Description

We study the impedance functions of conservative L-systems with unbounded main operators. In addition to the generalized Donoghue class $\mathfrak M_\kappa$ of Herglotz-Nevanlinna functions considered earlier, we introduce ``inverse" generalized Donoghue classes $\mathfrak M_\kappa^{-1}$ of functions satisfying a different normalization condition on the generating measure. 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. After that 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. The talk is based on joint work with K.~A.~Makarov and E. Tsekanovski\u i (see the reference below). \begin{itemize} \item[{[1]}] S. Belyi, K. A.~ Makarov, and E.~Tsekanovski\u i: \textit{A system coupling and Donoghue classes of Herglotz-Nevanlinna functions}, Complex Analysis and Operator Theory, (2016), 10 (4), pp. 835-880. \end{itemize}

A system coupling and Donoghue classes of Herglotz-Nevanlinna functions.

Cupples I Room 218

We study the impedance functions of conservative L-systems with unbounded main operators. In addition to the generalized Donoghue class $\mathfrak M_\kappa$ of Herglotz-Nevanlinna functions considered earlier, we introduce ``inverse" generalized Donoghue classes $\mathfrak M_\kappa^{-1}$ of functions satisfying a different normalization condition on the generating measure. 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. After that 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. The talk is based on joint work with K.~A.~Makarov and E. Tsekanovski\u i (see the reference below). \begin{itemize} \item[{[1]}] S. Belyi, K. A.~ Makarov, and E.~Tsekanovski\u i: \textit{A system coupling and Donoghue classes of Herglotz-Nevanlinna functions}, Complex Analysis and Operator Theory, (2016), 10 (4), pp. 835-880. \end{itemize}