Point impurities remove degeneracy of the Landau levels in a two-dimensional electron gas

The density of states of a two-dimensional electron gas in a magnetic field has been studied taking into account the scattering on point impurities. It is demonstrated that allowance for the electron-impurity interaction completely removes degeneracy of the Landau levels even for a small volume density of these point defects. The density of states is calculated in a self-consistent approximation taking into account all diagrams without intersections of the impurity lines. The electron density of states $\rho$ is determined by the contribution from a cut of the one-particle Green's function rather than from a pole. In a wide range of the electron energies $\omega$ (measured from each Landau level), the value of $\rho (\omega)$ is inversely proportional to the energy $|\omega|$ and proportional to the impurity concentration. The obtained results are applicable to various two-dimensional electron systems such as inversion layers, heterostructures, and electrons on the surface of liquid helium.