Spectral analysis of unitary perturbations of contractions

By the methods of the theory of extensions, for a unitary operator $U$ with a simple spectrum one constructs all possible contractions $T_\nu$, differing from $U$ by a one-dimensional one, and with the aid of the unitary dilation $\widetilde U$ of the contraction $T$ one carries out a comparative spectral analysis of the operators $U$ and $T$. Writing the contraction $T$ in the spectral representation of the operator $U$, we construct a model of a contraction which turns out to be useful in the investigation of the process of the “connection of a communication channel” of a conservative system with an exterior universe. The technique described in the paper is useful for the construction of explicitly solved problems of resonance scattering and in the investigation of the serial structure of resonators.