Peierls substitution and the Maslov operator method

Abstract

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: Open image in new window , where Open image in new window is the corresponding energy level of some auxiliary Schrodinger 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 Open image in new window with the operator Open image in new window represented as a function depending only on the operators of kinetic momenta \( \hat P_j = - i\mu \partial _{x_j } + A_j \left( x \right) \).