Perturbations of Donoghue classes and inverse problems for L-systems

TL;DR

This paper investigates how scalar Herglotz-Nevanlinna functions can be perturbed and represented as impedance of L-systems, providing explicit formulas, realization theorems, and solutions to inverse problems with applications and examples.

Contribution

It introduces new classes of perturbed Donoghue functions, develops explicit formulas for L-system parameters, and solves inverse problems with uniqueness, enhancing existing realization theorems.

Findings

Explicit formulas for von Neumann and unimodular parameters.

New realization theorems for perturbed Donoghue classes.

Solution to inverse problems with uniqueness conditions.

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.