  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 Grad's 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.
