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: $\mathscr E^{\nu}(-i\mu\partial_x,x)\phi=E\phi$, where $\mathscr E^{\nu}$ is the corresponding energy level of some auxiliary Schrödinger 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 $\mathscr{E}^{\nu}(\widehat P,\mu)\phi=E\phi$ with the operator $\mathscr{E}^{\nu}(\widehat P,\mu)$ represented as a function depending only on the operators of kinetic momenta $\widehat P_j=-i\mu\partial_{x_j}+A_j(x)$.