The full momentum dependence of spectrum of a point-like impurity immersed in a dilute onedimensional 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.

We exactly calculate the full temperature dependence of Casimir-like forces appearing between two and three static impurities loaded in the ideal Bose gas below the Bose–Einstein condensation transition point. Assuming the short-ranged character of the boson-impurity interaction, the calculation procedure presented here can be easily extended on a Bose system with an arbitrary number of impurities immersed.

The detailed mean-field treatment of the Bose polaron problem in two and three dimensions is presented. Particularly, assuming that impurity is immersed in the dilute Bose gas and interacts with bosons via the hard-sphere two-body potential, we calculate the low-momentum parameters of its spectrum, namely, the binding energy and the effective mass. The limits of applicability of the mean-field approach to a problem of mobile impurity in Bose-Einstein condensates are discussed by comparing our results to the Monte Carlo simulations data.

We proposed the non-perturbative scheme for calculation of the impurity spectrum in the Bose system at zero temperature. The method is based on the path-integral formulation and describes an impurity as a zero-density ideal Fermi gas interacting with Bose system for which the action is written in terms of density fluctuations. On the example of the 3 He atom immersed in the liquid helium-4 a good consistency with experimental data and results of Monte Carlo simulations is shown.

We studied the properties of a single impurity atom immersed in a dilute Bose condensate at low temperatures. In particular, we perturbatively obtained the momentum dependence of the impurity spectrum and damping. By means of the Brillouin-Wigner perturbation theory we also calculated the self-energy both for attractive and repulsive polaron in the long-wavelength limit. The stability problem of the impurity atom in a weakly-interacting Bose gas is also examined.