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.