二维超定保守系统的单向不变量及Potapov定理的推广
Overdetermined conservative 2D Systems, Invariant in One Direction and a Generalization of Potapov's theorem
摘要：
这项工作是作者工作arXiv：0812.3779v1的直接延续。研究了保守超定二维时不变系统的一种特殊情形。定义了这类系统的传递函数，得到了一类内函数$S(λ,t_2)$的CI，它们是$λ=\infty$的恒等式，满足一定的正则性假设，并且将常微分方程的解与谱参数$λ$缠绕在一起.利用平移模型，证明了CI类中的每一个函数都可以作为某个容器的传递函数来实现。该理论的亮点是推广的Potapov定理，它给出了一个非常特殊的公式，该公式以Blacke-Potapov乘积的乘积形式表示，对应于某些系统算子$A_1(t_2)$的离散谱，以乘法积分的乘积形式表示，对应于$A_1(t_2)$的连续谱。这个定理是在一个稍微限制性更强的假设下证明的，然后发展了整个理论。也就是说，我们假设对于几乎所有的$λ$，传递函数的导数是$t_2$的连续函数。最后讨论了CI中矩阵函数的零/极点插值问题，并重新证明了arXiv：0812.3779v1中关于这类函数的一个实现定理（定理8.1）。分析了厄米特情形，并证明了相应的实现定理。
英文摘要：
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.
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