Asymptotic analysis integral time

Perform time step integration Perform the asymptotic analysis Store C t) coefficients END looj) on A END loop on t END loop on J Calculate final (juantitics Print outputs... 

As another application of this method of asymptotic integration, we shall consider the problem of the Fourier coefficients pfU(P t) i 1 the limit of long times. As mentioned above, we do not wish to give here a detailed proof of the transport equation for pk] p] t) (see, for instance, Ref. 31). The main result of this analysis is, however, very simple in the limit of long times (t —> oo), the correlations are entirely determined by the velocity distribution function p< p t). One has ... 

At the core of the analysis and methods that are discussed in this Chapter is the consistent consideration of the fact that the form of each resonance wavefunction is = fl I o+Xas (Eq. (4.1) of text), if necessary, the extension to multi-dimensional forms is obvious. Depending on the formalism, the coefficient a and the asymptotic part, Xas, are functions of either the energy (real or complex) or the time. The many-body square-integrable, %, represents the localized part of the decaying (unstable) state, i.e., the unstable wavepacket which is assumed to be prepared at f = 0. its energy, Eo, is real and embedded inside the continuous spectrum, it is a minimum of the average value of the corresponding state-specific effective Hamiltonian that keeps all particles bound. 

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