Stieltjes like functions and inverse problems for systems with Schr\"odinger operator

TL;DR

This paper characterizes a class of Stieltjes-like functions as transfer functions of systems based on Schr"odinger operators with non-selfadjoint boundary conditions, providing formulas for system reconstruction and parameter determination.

Contribution

It introduces a unique realization framework for Stieltjes-like functions via Schr"odinger operators and derives explicit formulas for system parameters and their dynamics.

Findings

Parameters h and μ follow geometric trajectories (circles and hyperbolas) as functions of the free term γ.

The main operator is an accretive extension of the Schr"odinger operator.

Explicit formulas enable exact parameter recovery in the system.

Abstract

A class of scalar Stieltjes like functions is realized as linear-fractional transformations of transfer functions of conservative systems based on a Schr\"odinger operator T_h in $L_2[a,+\infty)$ with a non-selfadjoint boundary condition. In particular it is shown that any Stieltjes function of this class can be realized in the unique way so that the main operator $\bA$ of a system is an accretive (*)-extension of a Schr\"odinger operator T_h. We derive formulas that restore the system uniquely and allow to find the exact value of a non-real parameter h in the definition of T_h as well as a real parameter $\mu$ that appears in the construction of the elements of the realizing system. An elaborate investigation of these formulas shows the dynamics of the restored parameters h and $\mu$ in terms of the changing free term $\gamma$ from the integral representation of the realizable function. It turns our that the parametric equations for the restored parameter h represent different circles whose centers and radii are determined by the realizable function. Similarly, the behavior of the restored parameter $\mu$ are described by hyperbolas.