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 τ-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 [ℓ2(τ)]N. 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 ξ and is τ- -periodic; if N = 1, the symbol equals the Bloch eigenvalue itself. In particular, this rigorously formulates a result from 1951 by J. M. Luttinger.