Dual Lanczos vector space. III. Generalized polynomials and polynomial operators in the investigation of spectral and temporal properties of globally linear dynamical systems

Abstract

Dual Lanczos transformation theory is further developed by incorporating generalized polynomials and polynomial operators into its mathematical infrastructure. With the aid of these quantities, we obtain a number of new and useful results for understanding and determining the spectral and temporal properties of globally linear dynamical systems. Among these results are powerful practical tools for (i) investigating generalized Langevin equations and memory function hierarchies; (ii) making mathematically rigorous statements about expected temporal character, including the possible existence of separations in time scales and memory effects; and (iii) extracting and utilizing dynamically embedded information to determine the time evolution or spectral densities of dynamical variables and their ensemble averages, autocorrelation functions, and cross correlation functions. The reported results apply to any time evolution and spectral density problem associated with a globally linear dynamical system, including problems that are inaccessible by matrix diagonalization techniques and those that may not be given the usual continued fraction representation.