Inverse Stieltjes-like functions and inverse problems for systems with Schrödinger operator

Abstract

A class of scalar inverse Stieltjes-like functions is realized as linearfractional 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 inverse Stieltjes function of this class can be realized in the unique way so that the main operator A possesses a special semi-boundedness property. We derive formulas that restore the system uniquely and allow to find the exact value of a non-real boundary parameter h of the operator 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 α from the integral representation of the realizable function.