  We revisit the celebrated Peierls-Onsager substitution employing the magnetic
pseudo-differential calculus for weak magnetic fields with no spatial decay
conditions, when the non-magnetic symbols have a certain spatial periodicity.
We show in great generality that the symbol of the magnetic band Hamiltonian
admits a convergent expansion. Moreover, if the non-magnetic band Hamiltonian
admits a localized composite Wannier basis, we show that the magnetic band
Hamiltonian is unitarily equivalent to a Hofstadter-like magnetic matrix. In
addition, if the magnetic field perturbation is slowly variable, then the
spectrum of this matrix is close to the spectrum of a Weyl quantized, minimally
coupled symbol.
