  The dynamical correlated properties of one-dimensional (1D) Bose gases
provide profound understanding of novel physics emergent from collective
excitations, for instance, the breakdown of off-diagonal long-range order, and
the establishment of Tomonaga-Luttinger liquid (TLL) theory. However, due to
the non-perturbative nature of 1D many-body systems, the exact evaluation of
correlation functions is notoriously difficult. Here by means of form factor
approach based on algebraic Bethe ansatz and numerics, we present a thorough
study on the one-body dynamical correlation function (1BDCF) of the
Lieb-Liniger model at finite temperature. The influence of thermal fluctuation
and dynamical interaction on the behavior of 1BDCF has been demonstrated and
analyzed from various perspectives, including the spectral distribution, the
line-shape of 1BDCF, and the static correlations etc. The static correlation
properties, i.e. the momentum distribution and one body density matrix are
shown in good agreement with the TLL prediction.
