Landau Quantization of a 2D Antidot Lattice

We derive the Schr\"{o}dinger eigen-energy dispersion relation for a two dimensional sheet of electrons in a one dimensional periodic lattice of quantum antidot potential barriers, with a perpendicular quantizing magnetic field. This system is in the nature of a Kr\"{o}nig-Penney model with a high magnetic field present and we construct the appropriate Green's function which we use to formulate the dispersion relation for the energy spectrum.