Stieltjes like Functions and Inverse Problems for Systems with Schrödinger Operator

A class of scalar Stieltjes like functions is realized as linear-fractional transformations of transfer functions of conservative systems based on a Schrödinger operator Th in L2[a,+∞) 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 A of a system is an accretive (∗)-extension of a Schrödinger operator Th. 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 Th as well as a real parameter μ 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 μ in terms of the changing free term γ from the integral representation of the realizable function. It turns out 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 μ are described by hyperbolas.