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.