Splitting of Landau levels of a 2D electron due to electron-phonon interactions

We show that in a very strong magnetic field $B$ electron-phonon interaction gives rise to a splitting of Landau levels of a 2D electron into a series of infinitely degenerate sublevels. We provide both qualitative and quantitative description of this phenomenon. The cases of interaction with acoustic and polar optical phonons are considered. The energy distance between nearest sublevels in both cases tends to zero as $B^{-1/2}$ at large $B$.