2008/12/20 by Andrey Melnikov, Victor Vinnikov, Melnikov, Andrey +1 · 1 citation
Mathematics · Physics and Astronomy · #Nonlinear Waves and Solitons #Quantum chaos and dynamical systems #Spectral Theory in Mathematical Physics #math.FA #math.SP #msc:26B30 #msc:34K06 #msc:45D05 #msc:47E05 #msc:47N20 #msc:47N70
paper · pdf · doi:10.48550/arxiv.0812.3970
54 pages
arxiv created 2008/12/23 · arxiv updated 2009/12/01
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,t2), which are identity for λ=∞, 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 A1(t2) and of multiplicative integral, corresponding to the continuous spectrum of A1(t2). 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 t2 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.