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We study realizations generated by the original Weyl–Titchmarsh functions m ∞ (z) and m α (z) . It is shown that the Herglotz–Nevanlinna functions (- m ∞ (z)) and (1 / m ∞ (z)) can be realized as the impedance functions of the corresponding Shrödinger L-systems sharing the same main dissipative operator. These L-systems are presented explicitly and related to Dirichlet and Neumann boundary problems. Similar results but related to the mixed boundary problems are derived for the Herglotz–Nevanlinna functions (- m α (z)) and (1 / m α (z)) . We also obtain some additional properties of these realizations in the case when the minimal symmetric Shrödinger operator is non-negative. In addition to that we state and prove the uniqueness realization criteria for Shrödinger L-systems with equal boundary parameters. A condition for two Shrödinger L-systems to share the same main operator is established as well. Examples that illustrate the obtained results are presented in the end of the paper. [ABSTRACT FROM AUTHOR]

We study linear perturbations of Donoghue classes of scalar Herglotz–Nevanlinna functions by a real parameter Q and their representations as impedance of conservative L-systems. Perturbation classes M Q , M κ Q , M κ - 1 , Q are introduced and for each class the realization theorem is stated and proved. We use a new approach that leads to explicit new formulas describing the von Neumann parameter of the main operator of a realizing L-system and the unimodular one corresponding to a self-adjoint extension of the symmetric part of the main operator. The dynamics of the presented formulas as functions of Q is obtained. As a result, we substantially enhance the existing realization theorem for scalar Herglotz–Nevanlinna functions. In addition, we solve the inverse problem (with uniqueness condition) of recovering the perturbed L-system knowing the perturbation parameter Q and the corresponding non-perturbed L-system. Resolvent formulas describing the resolvents of main operators of perturbed L-systems are presented. A concept of a unimodular transformation as well as conditions of transformability of one perturbed L-system into another one are discussed. Examples that illustrate the obtained results are presented. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]