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The Simplified approach to the Bose gas was introduced by Lieb in 1963 tostudy the ground state of systems of interacting Bosons. In a series of recentpapers, it has been shown that the Simplified approach exceeds earlierexpectations, and gives asymptotically accurate predictions at both low andhigh density. In the intermediate density regime, the qualitative predictionsof the Simplified approach have also been found to agree very well with QuantumMonte Carlo computations. Until now, the Simplified approach had only beenformulated for translation invariant systems, thus excluding externalpotentials, and non-periodic boundary conditions. In this paper, we extend theformulation of the Simplified approach to a wide class of systems withouttranslation invariance. This also allows us to study observables in translationinvariant systems whose computation requires the symmetry to be broken. Such anobservable is the momentum distribution, which counts the number of particlesin excited states of the Laplacian. In this paper, we show how to compute themomentum distribution in the Simplified approach, and show that, for the SimpleEquation, our prediction matches up with Bogolyubov's prediction at lowdensities, for momenta extending up to the inverse healing length.

The Simplified approach to the Bose gas without translation invariance