An alternative theory for time dependent correlations: recurrence relation and the exponential expansion

We discuss the general theory of time correlation functions and their spectra as given by Mori and Zwanzig, and present an alternative formulation based on recurrence relations of new basis vectors in the Hilbert space of the analyzed property. We also show that, in a very general way, spectra can be decomposed in series of Lorentzian functions. Consequently, the correlation functions can always, on a theoretical basis, be represented by a series of exponentials. Examples of application of the theory are given.