Overdetermined conservative 2D Systems, Invariant in One Direction and a Generalization of Potapov's theorem

Abstract

This work is a direct continuation of the authors work arXiv:0812.3779v1. A special case of conservative overdetermined time invariant 2D systems is developed and studied. Defining transfer function of such a systems we obtain a class CI of inner functions $S(λ,t_2)$, which are identity for $λ=\infty$, satisfy certain regularity assumptions and intertwines solutions of ODEs with a spectral parameter $λ$. Using translation model, we prove that every function in the class CI can be realized as a transfer function of a certain vessel. The highlight of this theory is a generalization Potapov's theorem, which gives a very special formula for such a function in the form of multiplication of Blacke-Potapov products, corresponding to the discrete spectrum of certain system operator $A_1(t_2)$ and of multiplicative integral, corresponding to the continuous spectrum of $A_1(t_2)$. This theorem is proved under a slightly more restrictive assumptions, then the development of the whole theory. Namely, we suppose that the derivative of the transfer function is a continuous function of $t_2$ for almost all $λ$. At the last part zero/pole interpolation problem for matrix functions in CI is considered and a realization theorem of such functions appeared in arXiv:0812.3779v1 (theorem 8.1) is reproved. Hermitian case is also analyzed and the corresponding realization theorem is proved.