  We report on ground state properties of a one-dimensional, weakly-interacting
Bose gas constrained by an infinite multi-rods periodic structure at zero
temperature. We solve the stationary Gross-Pitaevskii equation (GPE) to obtain
the Bloch wave functions from which we give a semi-analytical solution for the
density profile, as well as for the phase of the wave function in terms of the
Jacobi elliptic functions, and the incomplete elliptic integrals of the first,
second and third kind. Then, we determine numerically the energy of the ground
state, the chemical potential and the compressibility of the condensate and
show their dependence on the potential height, as well as on the interaction
between the bosons. We show the appearance of loops in the energy band spectrum
of the system for strong enough interactions, which appear at the edges of the
first Brillouin zone for odd bands and at the center for even bands. We apply
our model to predict the energy band structure of the Bose gas in an optical
lattice with subwavelength spatial structure. To discuss the density range of
the validity of the GPE predictions, we calculate the ground state energies of
the free Bose gas using the GPE, which we compare with the Lieb-Liniger exact
energies.
