Passive Linear Systems and Scattering Theory

Abstract

Passive Linear Time-Invariant Systems (PLTIS’s) theory has been developed in connection with quantum mechanics and mathematical physics (spectral, scattering and other problems), with networks, control and stochastic processes theories (synthesis, stability, prediction and other problems) and there is a considerable literature on these topics (see, for example, the books [25], [28], [12], [42], [35], [21], the papers [23], [16], [17], [11], [40], [22], [7], [8] and references in these books and papers). In the first half of the 20th century the impedance formalism was developed, in which the transfer functions (t.f.’s) of PLTIS’s were the so called resistance or impedances matrices. For Conservative Linear Time-Invariant Systems (CLTIS’s) this development was intimately connected with the Riesz-Herglotz integral representation of positive-real functions with scalar, matrix or operator values. Connected to this representation and to the resolvent and spectral theory of selfadjoint and unitary operators in Hilbert space is Cauer method of synthesis of lossless electrical n-ports, etc. At the second half of this century the scattering and transmission (or chain scattering) formalism was also developed in the PLTIS’s theory.

This research was made possible in part by Grant No.UM1-298 from US CRDF and the Unkrainian Government.