Peierls substitution and the Maslov operator method.

We consider a periodic Schrödinger 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: [Figure not available: see fulltext.], where [Figure not available: see fulltext.] is the corresponding energy level of some auxiliary Schrödinger operator, assumed to be nondegenerate, and µ 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 [Figure not available: see fulltext.] with the operator [Figure not available: see fulltext.] represented as a function depending only on the operators of kinetic momenta $$ \hat P_j = - i\mu \partial _{x_j } + A_j \left( x \right) $$. [ABSTRACT FROM AUTHOR]