The Peierls-Onsager substitution in the framework of magnetic $\Psi$DO
Our objectif is to show that the magnetic pseudodifferential
calculus and its integral kernel form give a precise mathematical framework to
obtain a gauge covariant formulation of the Peierls-Onsager substitution
We concentrate mainly on the isolated spectral band situation. Once we can
isolate a spectral island for a periodic Hamiltonian, we can define
symbols of the associated band operators (Hamiltonian, projection, etc.) and
the corresponding magnetic operators while being singular perturbations of the
free ones, can be approximated by the magnetic quantization of the symbols of
the free ones.
When a Wannier basis is supposed to exist for the given spectral island, then
a generalization of the result of Gh. Nenciu is obtained.