  The Simplified approach to the Bose gas was introduced by Lieb in 1963 to
study the ground state of systems of interacting Bosons. In a series of recent
papers, it has been shown that the Simplified approach exceeds earlier
expectations, and gives asymptotically accurate predictions at both low and
high density. In the intermediate density regime, the qualitative predictions
of the Simplified approach have also been found to agree very well with Quantum
Monte Carlo computations. Until now, the Simplified approach had only been
formulated for translation invariant systems, thus excluding external
potentials, and non-periodic boundary conditions. In this paper, we extend the
formulation of the Simplified approach to a wide class of systems without
translation invariance. This also allows us to study observables in translation
invariant systems whose computation requires the symmetry to be broken. Such an
observable is the momentum distribution, which counts the number of particles
in excited states of the Laplacian. In this paper, we show how to compute the
momentum distribution in the Simplified approach, and show that, for the Simple
Equation, our prediction matches up with Bogolyubov's prediction at low
densities, for momenta extending up to the inverse healing length.
