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

TL;DR

This paper analyzes the structure of fibre operators in the Bloch-Floquet decomposition of periodic magnetic pseudo-differential operators, providing explicit formulas and demonstrating their pseudo-differential nature.

Contribution

It introduces explicit formulas for the distribution kernels of fibre operators and proves they are toroidal pseudo-differential operators.

Findings

Derived explicit distribution kernel formulas for fibre operators.

Established that fibre operators are toroidal pseudo-differential operators.

Provided dual perspectives on operators as on the torus and as infinite matrices.

Abstract

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 $\ell^2$ space via a discrete Fourier transform. Moreover, using these distribution kernels we prove that the fibre operators are toroidal pseudo-differential operators.