The fibre operators in the Bloch-Floquet decomposition of periodic magnetic pseudo-differential operators

We study the structure of the fibre operators corresponding to periodic magnetic pseudo-differential operators having periodic magnetic potentials. We obtain explicit formulas for their distribution kernel, both when these fibres are seen as operators on the $d$-dimensional torus, and also when they are seen as infinite matrices acting on a discrete ${\mathrm{\ell }}^2$ space via a discrete Fourier transform. Moreover, using these distribution kernels we prove that the fibre operators are toroidal pseudo-differential operators.