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We point out that by the ``smoothness means fast decay'' principle in Fourieranalysis, it is possible to infer the smoothness (or nonsmoothness) of theautocorrelation function from a mere glimpse of the initial state.Specifically, for a generic system with smooth eigenstates, the smoother aninitial state is, the faster its decomposition coefficients with respect to theeigenstates of the system decay, and in turn the smoother the autocorrelationfunction is. The idea is illustrated with three increasingly smooth functionsin an infinite square well. By using the Mellin transform, we also find thatthe nonsmoothness or singularity of the initial state affects the short-timebehavior of its autocorrelation function. In particular, a sufficientlynonsmooth or singular initial state could decay in a nonquadratic power law,with the exponent continuously tunable. We now understand the periodic cusps ofthe autocorrelation function in the quench dynamics of a Bloch state, which wasobserved previously [Zhang and Yang, EPL 114, 60001 (2016)].

Inferring the smoothness of the autocorrelation function from that of the initial state