Remarks on Time-Dependent Behavior of Disordered Lattice (Part II. Time Dependent Problems)
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Some considerations on the results of the recent works on time-dependent behavior of disordered lattice are presented. The nature of the exponential decay of periodic disturbance obtained by Maradudin, Weiss and Jepsen and the origin of the difference between their results and Hori's are discussed from the point of view of coarse-graining of the eigenfrequency spectrum. It is concluded that the temporal behavior depends sensitively on the manner of coarse-graining of the frequency spectrum, which in turn is intimately connected with the experimental situation, or the method of observation. It is not surprising that the different results are obtained by different methods of calculation, since each method of calculation corresponds to a characteristic manner of coarse-graining of the spectrum. The relation between the asymptotic time dependence and the singularities in the frequency spectrum is also discussed. The conclusion is that the asymptotic behavior depends extremely sensitively on the exact form of the spectral density function at the extremities of the spectrum itself or of the spectral gap, if any. Since this cannot be inferred by any perturbational methods with sufficient accuracy, the asymptotic behavior obtained by such methods is doubtful to be valid.
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