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In this paper we study the L-system realizations generated by the original Weyl-Titchmarsh functions mα(z) in the case when the minimal symmetric Shrödinger operator in L 2 [ℓ, +∞) is non-negative. We realize functions (−mα(z)) as impedance functions of Shrödinger L-systems and derive necessary and sufficient conditions for (−mα(z)) to fall into sectorial classes S β 1 ,β 2 of Stieltjes functions. Moreover, it is shown that the knowledge of the value m∞(−0) and parameter α allows us to describe the geometric structure of the L-system that realizes (−mα(z)). Conditions when the main and state space operators of the L-system realizing (−mα(z)) have the same or not angle of sectoriality are presented in terms of the parameter α. Example that illustrates the obtained results is presented in the end of the paper.

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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.