First-principles calculation of diamagnetic band structure. I. Reduction to a one-dimensional Schrödinger equation

We present a new first-principles attack on the problem of Bloch electrons in a magnetic field which, being rigorous, preserves all symmetries and in particular all predictions of the magnetic translation group. In this first paper we show that the problem can be reduced for a wide class of crystal symmetries and for magnetic fields that are rational in the sense of group theory to a one-dimensional Hamiltonian. This is done in two steps: First we introduce a canonical transformation that diagonalizes the free-electron part of the "magnetic Hamiltonian." An analysis of the transformed crystal potential for simple rational fields allows a separation ansatz in terms of $k-q$ functions in one of the variables. The result is a set of Schrödinger equations in the remaining variables which have the structure of differential difference equations with periodic coefficients; they are solved in the accompanying paper.