  This paper is a contribution to the theory of coherent crystals. We present
arguments claiming that negative minima in the Fourier transform of a soft pair
interaction may give rise to the coexistence of diagonal and off-diagonal
long-range order at high densities, and that this coexistence may be detectable
due to a periodicity seen on the off-diagonal part of the one-body reduced
density matrix, without breaking translation invariance. As an illustration, we
study the ground state of a homogenous system of bosons in continuous space,
from the interaction retaining only the Fourier modes $v(\k)$ belonging to a
single nonzero wave number $|\k|=q$. The result is a mean-field model. We prove
that for $v(\k)>0$ the ground state is asymptotically fully Bose-condensed,
while for $v(\k)<0$ at densities exceeding a multiple of $\hbar^2
q^2/2m|v(\k)|$ it exhibits both Bose-Einstein condensation and diagonal
long-range order, and the latter can be seen on both the one- and the two-body
density matrix.
