  We study the dispersion relation of the excitations of a dilute Bose-Einstein
condensate confined in a periodic optical potential and its Bloch oscillations
in an accelerated frame. The problem is reduced to one-dimensionality through a
renormalization of the s-wave scattering length and the solution of the
Bogolubov - de Gennes equations is formulated in terms of the appropriate
Wannier functions. Some exact properties of a periodic one-dimensional
condensate are easily demonstrated: (i) the lowest band at positive energy
refers to phase modulations of the condensate and has a linear dispersion
relation near the Brillouin zone centre; (ii) the higher bands arise from the
superposition of localized excitations with definite phase relationships; and
(iii) the wavenumber-dependent current under a constant force in the
semiclassical transport regime vanishes at the zone boundaries. Early results
by J. C. Slater [Phys. Rev. 87, 807 (1952)] on a soluble problem in electron
energy bands are used to specify the conditions under which the Wannier
functions may be approximated by on-site tight-binding orbitals of harmonic-
oscillator form. In this approximation the connections between the low-lying
excitations in a lattice and those in a harmonic well are easily visualized.
Analytic results are obtained in the tight-binding scheme and are illustrated
with simple numerical calculations for the dispersion relation and
semiclassical transport in the lowest energy band, at values of the system
parameters which are relevant to experiment.
