  Strong connection between the single-particle and collective excitations
stands out as one of the features of Bose-Einstein condensates (BECs). We
discuss theoretically these excitations of BECs focusing on the exact
properties of the one-body and two-body Green's functions developed by Gavoret
and Nozi\`eres. We also investigate these excitations by using the many-body
approximation theory at nonzero temperatures. First, we revisited the earlier
study presented by Gavoret and Nozi\`eres, involving the subsequent results
given by Nepomnyashchii and Nepomnyashchii, in terms of the matrix formalism
representation. This formalism is an extension of the Nambu representation for
the single-particle Green's function of BECs to discuss the density and current
response functions efficiently. We describe the exact low-energy properties of
the correlation functions and the vertex functions, and discuss the
correspondence of the spectra between the single-particle and density
excitations in the low-energy and low-momentum limits at $T=0$. After deriving
the exact low-energy structures of the one-body and two-body Green's functions,
we develop a many-body approximation theory of BECs using the matrix formalism
for describing the single-particle Green's function and the density response
function at nonzero temperatures. We show how the peaks of the single-particle
spectral function and the density response function behave with an increasing
temperature. Many-body effect on the single-particle spectral function and the
density response function is included within a random phase approximation,
where satellite structures emerge because of beyond-mean-field effects.
Criticisms are also made on recent theories casting doubt upon the conventional
wisdom of the BEC: the equivalence of the dispersion relations between the
single-particle and collective excitations in the low-energy and low-momentum
regime.
