On the Green function of the almost-Mathieu operator

The square tight-binding model in a magnetic field leads to the almost-Mathieu operator, which, for rational fields, reduces to a q x q matrix depending on the components mu, nu of the wavevector in the magnetic Brillouin zone. We calculate the corresponding Green function without explicit knowledge of eigenvalues and eigenfunctions and obtain analytical expressions for the diagonal and the first off-diagonal elements; the results which are consistent with the zero-magnetic-field case can be used to calculate several quantities of physical interest (e.g. the density of states over the entire spectrum, impurity levels in a magnetic field).