Bound states of the one-dimensional relativistic polaron in one-loop approximation

Quasirelativistic model of the narrow-gap semiconductor is considered. Interaction of Dirac electron fields with the nondispersive phonon, field in a $\delta$-function type impurity potential is taken into account. Classical solutions corresponding to the discrete spectrum are found. Fermion vacuum polarisation effect by phonon field is analized in one-loop approximation.