Peierls' substitution via minimal coupling and magnetic pseudo-differential calculus

We revisit the celebrated Peierls-Onsager substitution for weak magnetic fields with no spatial decay conditions. We assume that the non-magnetic Gamma*-Periodic Hamiltonian has an isolated spectral band whose Riesz projection has a range which admits a basis generated by N exponentially localized composite Wannier functions. Then we show that the effective magnetic band Hamiltonian is unitarily equivalent to a Hofstadter-like magnetic matrix living in inverted right perpendicular l(2)(Gamma)(N) inverted left perpendicular In addition, if the magnetic field perturbation is slowly variable in space, then the perturbed spectral island is close (in the Hausdorff distance) to the spectrum of a Weyl quantized minimally coupled symbol. This symbol only depends on xi and is if Gamma*-Periodic N = 1, the symbol equals the Bloch eigenvalue itself. In particular, this rigorously formulates a result from 1951 by J. M. Luttinger.