Weyl–Titchmarsh Function of an Abstract Boundary Value Problem, Operator Colligations, and Linear Systems with Boundary Control

Abstract.

The paper defines the Weyl–Titchmarsh function for an abstract boundary value problem and shows that it coincides with the transfer function of some explicitly described linear boundary control system. On the ground of obtained results we explore interplay among boundary value problems, operator colligations, and the linear systems theory that suggests an approach to the study of boundary value problems based on the open systems theory founded in works of M. S. Livšic. Examples of boundary value problems for partial differential equations and calculations of their Weyl–Titchmarsh functions are offered as illustration. In particular, we give an independent derivation of the Weyl–Titchmarsh function for the three dimensional Schrödinger operator introduced by W.O. Amrein and D.B. Pearson. Relationships to the Schrödinger operator with singular potential supported by the unit sphere are clarified and other possible applications of the developed approach in mathematical physics are noted.