  For a dilute atomic gas in a strong transverse trapping potential, one
normally expects that, in the ground state, the gas will populate only the
lowest transverse level. We show, however, that for the strongly interacting
gas under a Feshbach resonance, the ground state includes a large fraction of
atoms in excited levels of the trap, even if the gas is very dilute and the
trap is very strong. This is because the effective atom-molecule coupling is
typically enhanced to many times the trap mode spacing by an induced
confinement along the untrapped dimension(s). Thus one cannot ''freeze out''
the transverse degrees of freedom except under certain conditions.
