L-systems with Schrödinger operator

Abstract

In this chapter we apply some of the previous results to the problems related to the Schr¨odinger operator. First we give a complete characterization of all (∗)- extensions of ordinary differential operators. Then we use it to provide a thorough description of all L-systems with a Schr¨odinger operator T _h . Moreover, we describe the class of scalar Stieltjes/(inverse Stieltjes)-like functions that can be realized as impedance functions of L-systems with a Schr¨odinger operator T _h . The formulas that restore an L-system uniquely from a given Stieltjes/(inverse Stieltjes)-like function as the impedance function of this L-system are derived. These formulas allow us to solve the inverse problem and find the exact value of the parameter h in the definition of T _h as well as a real parameter μ that appears in the construction of the elements of the L-system being realized. A detailed study of these formulas shows the dynamics of the restored parameters h and μ in terms of the changing free term in the integral representation of a realizable function. We also provide a full description of accretive, sectorial, and extremal boundary value problems for a Schr¨odinger operator T _h on the half-line in terms of the boundary parameter h.