Numerical techniques spectral densities

Memory effects play an important role for the description of dynamical effects in open quantum systems. As mentioned above, Meier and Tannor [32] developed a time-nonlocal scheme employing the numerical decomposition of the spectral density. The TL approach as discussed above as well as the approaches by Yan and coworkers [33-35] use similar techniques. Few systems exist for which exact solutions are available and can serve as test beds for the various theories. Among them is the damped harmonic oscillator for which a path-integral solution exists [1], In the simple model of an initially excited... 

In postulating the stationarity of the stochastic process, very strong assumptions regarding the structure of the process are made. Once these assumptions are dropped, the process can become nonstationary in many different ways. In the framework of the spectral analysis of nonstationary processes, Priestley (see, e.g., Priestley 1999) introduced the evolutionary power spectral density (EPDS) function. The EPSD function has essentially the same type of physical interpretation of the PSD function of stationary processes. The main difference is that whereas the PSD function describes the power-frequency distribution for the whole stationary process, the EPSD function is time dependent and describes the local power-frequency distribution at each instant time. The theory of EPSD function is the only one which preserves this physical interpretation for the nonstationary processes. Moreover, since the spectrum may be estimated by fairly simple numerical techniques, which do not require any specific assumption of the structure of the process, this model, based on the EPSD function, is nowadays the most adopted model for the analysis of structures subjected to nonstationary processes as the seismic motion due to earthquakes. 

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