  The full momentum dependence of spectrum of a point-like impurity immersed in
a dilute one-dimensional Bose gas is calculated on the mean-field level. In
particular we elaborate, to the finite-momentum Bose polaron, the path-integral
approach whose semi-classical approximation leads to the conventional
mean-field treatment of the problem while quantum corrections can be easily
accounted by standard loop expansion techniques. The extracted low-energy
parameters of impurity spectrum, namely, the binding energy and the effective
mass of particle, are shown to be in qualitative agreement with the results of
quantum Monte Carlo simulations.
