Perturbations of Donoghue classes and inverse problems for L-systems

Abstract

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 $\mathfrak M^Q$, $\mathfrak M^Q_\kappa$, $\mathfrak M^{-1,Q}_\kappa$ 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.