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

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.