Vladimir Ryzhov
A Weyl-Titchmarsh Function of an Abstract Boundary Value Problem, Operator Colligations, and Linear Systems with Boundary Control
(127K, LaTeX)
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
and suggest an approach to the study of boundary value problems
based on the open systems theory due to M.~S.~Livsic.
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 Schrodinger
operator introduced by W.O.Amrein and D.B.Pearson.
Relationships to the Schrodinger operator with
singular potential supported by the unit sphere
are clarified and other possible applications of the developed
approach in mathematical physics are noted.