  We investigate the number fluctuations of spatially split many-boson systems
employing a theorem about the maximally and minimally attainable variances of
an observable. The number fluctuations of many-boson systems are given for
different numbers of lattice sites and both mean-field and many-body wave
functions. It is shown which states maximize the particle number fluctuations,
both in lattices and double-wells. The fragmentation of the states is
discussed, and it is shown that the number fluctuations of some fragmented
states are identical to those of fully condensed states.
