In a semi-classical regime, we study a periodic magnetic Schrödinger operator in ℝ2. This is inspired by recent experiments on artificial magnetism with ultra cold atoms in optical lattices, and by the new interest for the operator on the hexagonal lattice describing the behavior of an electron in a graphene sheet. We first review some results for the square (Harper), triangular and hexagonal lattices. Then, we study the case when the periodicity is given by the kagome lattice considered by Hou. Following the techniques introduced by Helffer–Sjöstrand and Carlsson, we reduce this problem to the study of a discrete operator on ℓ2(ℤ2;ℂ3) and a pseudo-differential operator on L2(ℝ;ℂ3), which keep the symmetries of the kagome lattice. We estimate the coefficients of these operators in the case of a weak constant magnetic field. Plotting the spectrum for rational values of the magnetic flux divided by 2πh where h is the semi-classical parameter, we obtain a picture similar to Hofstadter's butterfly. We study the properties of this picture and prove the symmetries of the spectrum and the existence of flat bands, which do not occur in the case of the three previous models.
