We give quantitative estimates on the asymptotics of the linearized Boltzmann collision operator and its associated equation from angular cutoff to non cutoff. On one hand, the results disclose the link between the hyperbolic property resulting from the Grads cutoff assumption and the smoothing property due to the long-range interaction. On the other hand, with the help of the localization techniques in the phase space, we observe some new phenomenon in the asymptotic limit process. As a consequence, we give the affirmative answer to the question that there is no jump for the property that the collision operator with cutoff does not have the spectrum gap but the operator without cutoff does have for the moderate soft potentials.
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