  Successive quantum transitions in an intermediate regime are shown to exist
between the superfluid and Mott insulating states for interacting bosonic atoms
in one dimension with a trapping potential. These transitions, which are caused
by the interplay of the trapping potential with the competition between the
kinetic energy and the interaction, reveal novel many-body effects as reflected
by low-lying excitation behavior, unconventional long-range correlations and an
even-odd alternating squeezing process of superfluid bosons into the Mott
insulating state. These features, most likely being generic for all dimensions
when a trapping potential is involved, are relevant for both experimental
observations and physical interpretation of the Mott insulator transition.
