  In this paper, we study non-interacting bosons in a disordered
one-dimensional optical lattice in a harmonic potential. We consider the case
of deterministic disorder produced by an Aubry-Andr\'{e} potential. Using exact
diagonalization, we investigate both the zero temperature and the finite
temperature properties. We investigate the localization properties by using an
entanglement measure. We find that the extreme sensitivity of the localization
properties to the number of lattice sites in finite size closed chains
disappear in open chains. This feature continues to be present in the presence
of a harmonic confining potential. The disorder is found to strongly reduce the
Bose-Einstein condensation temperature and the condensate fraction in open
chains. The low temperature thermal depletion rate of the condensate fraction
increases considerably with increasing disorder strength. We also find that the
critical disorder strength required for localization increases with increasing
strength of the harmonic potential. Further, we find that the low temperature
condensate fraction undergoes a sharp drop to 0.5 in the localization
transition region. The temperature dependence of the specific heat is found to
be only marginally affected by the disorder.
