Extraction and utilization of dynamically embedded information and dual Lanczos transformations in the investigation of spectral and temporal properties of globally linear dynamical systems

Abstract

The concept of extraction and utilization of dynamically embedded information in dual Lanczos transformation theory is put on a sound theoretical foundation. Also, our earlier work on dual Lanczos transformations is cast in a different form that may be readily applied to problems with an initial representation characterized by either discrete or continuous indices. This provides a more lucid and flexible interpretation of their significance, giving the theory a simpler and more coherent formal structure. We show that dual Lanczos transformations represent a particular mechanism for carrying out the process of extraction and utilization of dynamically embedded information and correspond to the transformation of the global dynamics of a system into the closed subdynamics of a dynamically invariant subspace. Although this subdynamics is not necessarily equivalent to the global dynamics, it is shown that knowledge of the subdynamics is sufficient for determining certain spectral and temporal properties of a system depending on the bias built into the dynamically invariant subspace. We report a closed set of equations for carrying out dual Lanczos transformations by means of symbolic manipulation on computers in the treatment of local and nonlocal classical models in the classical phase space representation.