Landau-level subband structure of electrons on a square lattice

For crystal electrons in two dimensions in magnetic field, each Landau level is split into subbands by a periodic potential. The dependence of the energy spectrum on magnetic field exhibits a recursive structure. For the case of a square lattice, this structure was discovered empirically by Hofstadter [Phys. Rev. B 14, 2239 (1976)]. In this paper we present a proof of the statements used by Hofstadter to characterize this recursive structure. In addition an interpretation is provided for the quantum number that, along with the Hall conductivity, labels the possible gaps within the Landau level.