  In this paper, a real-space analytical expression for the free Green's
function (propagator) of bilayer graphene is derived based on the
effective-mass approximation. Green's function displays highly spatial
anisotropy with three-fold rotational symmetry. The calculated local density of
states (LDOS) of a perfect bilayer graphene produces the main features of the
observed scanning tunneling microscopy (STM) images of graphite at low bias
voltage. Some predicted features of the LDOS can be verified by STM
measurements. In addition, we also calculate the LDOS of bilayer graphene with
vacancies by using the multiple-scattering theory (scatterings are localized
around the vacancy of bilayer graphene). We observe that the interference
patterns are determined mainly by the intrinsic properties of the propagator
and the symmetry of the vacancies.
