  We investigate the behavior of a Bose-Einstein Condensate (BEC) under the
influence of a central barrier as the particle number trends towards the
thermodynamic limit. In order to perform these studies, we present a novel
method which is tractable in the large-$N$ limit. This method employs what may
be considered to be a generalized Wannier basis, which successfully
incorporates features of previous theoretical and computational assays to the
splitting problem, including mean field effects, and has access to the
dimensionality, trap parameters, and particle numbers relevant to recent
experiments. At any barrier height we are able to discern between a two-mode
state and a state which is described sufficiently by mean field theory and,
further, give a criterion and technique for matching the two-mode theory to the
zero-barrier state. We compare the basis used in this model to the de-localized
basis functions underlying alternate models used in recent theoretical work on
the double-well splitting problem and show that only the generalized Wannier
basis displays the level crossing and emergence of two complex order parameters
with overall $U(1) \oplus U(1)$ symmetry as expected from a large-$N$ analogue
of the Superfluid to Mott insulator transition. Using this model, we identify a
universal structure, independent of $N$, in this phase transition. We also
present an analytic and model-independent description of this universal
structure and discuss its consequences for realizing true two-mode physics with
a BEC which trends towards the thermodynamic limit.
