Impurity states in the one-dimensional Bose gas

The detailed study of the low-energy spectrum for a mobile impurity in the one-dimensional bosonic enviroment is performed. Particularly we have considered only two analytically accessible limits, namely, the case of an impurity immersed in a dilute Bose gas, where one can use many-body perturbation techniques for low-dimensional bosonic systemsm and the case of the Tonks-Girardeau gas, for which the usual fermionic diagrammatic expansion up to the second order is applied.