Peierls substitution and the Maslov operator method

We consider a periodic Schrodinger operator in a constant magnetic field with vector potential A(x). A version of adiabatic approximation for quantum mechanical equations with rapidly varying electric potentials and weak magnetic fields is the Peierls substitution which, in appropriate dimensionless variables, permits writing the pseudodifferential equation for the new auxiliary function: E(nu)(-i mu partial derivative chi,x)phi = E phi, where E(nu) is the corresponding energy level of some auxiliary Schrodinger operator, assumed to be nondegenerate, and mu is a small parameter. In the present paper, we use V. P. Maslov's operator method to show that, in the case of a constant magnetic field, such a reduction in any perturbation order leads to the equation E(nu)((P) over cap,mu)phi = E phi with the operator E(nu)((P) over cap,mu) represented as a function depending only on the operators of kinetic momenta (P) over cap (j) = -i mu partial derivative x(j) + Aj(x).