Incommensurate fractal spectrum of a two-dimensional non-Bravais lattice under a magnetic field

We study the spectrum of an electron gas subjected to a two-dimensional non-Bravais lattice potential and a perpendicular magnetic field simultaneously. We derive the difference equations, which are not periodic in the x direction for any value of the gauge potential in Landau gauge (0,Hx,0). The spectrum has a fractal structure and is incommensurate along the magnitude of the magnetic field.