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 method. 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.
Joint work with Horia Cornean and Viorel Iftimie.
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