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

Horia D. Cornean, Bernard Helffer, Radu Purice

TL;DR

The paper extends Bloch-Floquet theory to periodic magnetic pseudodifferential operators with periodic vector potentials, providing explicit fibre-wise distribution kernels on the torus and as matrices on $\ell^2(\Gamma_*)$. It shows that the fibre operators are toroidal pseudodifferential operators and constructs a symmetric toroidal Weyl calculus, connecting the fibre decomposition to a principal-bundle viewpoint over the torus. A key technical achievement is the explicit derivation of fibre kernels and the demonstration that, for zero-mean flux, magnetic quantization coincides with standard Weyl quantization, enabling a unified torus-calculus approach. These results furnish a robust framework to analyze spectral properties, domain decomposition, and regularity of fibre operators, with potential applications to multi-field magnetic scenarios and torus-based quantum systems.

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.

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

TL;DR

The paper extends Bloch-Floquet theory to periodic magnetic pseudodifferential operators with periodic vector potentials, providing explicit fibre-wise distribution kernels on the torus and as matrices on . It shows that the fibre operators are toroidal pseudodifferential operators and constructs a symmetric toroidal Weyl calculus, connecting the fibre decomposition to a principal-bundle viewpoint over the torus. A key technical achievement is the explicit derivation of fibre kernels and the demonstration that, for zero-mean flux, magnetic quantization coincides with standard Weyl quantization, enabling a unified torus-calculus approach. These results furnish a robust framework to analyze spectral properties, domain decomposition, and regularity of fibre operators, with potential applications to multi-field magnetic scenarios and torus-based quantum systems.

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 -dimensional torus, and also when they are seen as infinite matrices acting on a discrete space via a discrete Fourier transform. Moreover, using these distribution kernels we prove that the fibre operators are toroidal pseudo-differential operators.
Paper Structure (31 sections, 14 theorems, 145 equations)

This paper contains 31 sections, 14 theorems, 145 equations.

Table of Contents

  1. Introduction.
  2. Some notations
  3. The framework
  4. The pair of $d$-dimensional tori
  5. The magnetic field.
  6. The Fourier transform.
  7. The Hörmander type symbols.
  8. A brief summary of our main results
  9. The Bloch-Floquet decomposition of periodic magnetic pseudo-differential operators.
  10. Periodic magnetic pseudo-differential operators
  11. Two unitary transformations
  12. The Bloch-Floquet (BF) transformation.
  13. The Bloch-Floquet-Zak (BFZ) transformation.
  14. The Bloch-transformation of $\mathfrak{Op}(F)$
  15. The fibre operators
  16. ...and 16 more sections

Key Result

Theorem 2.2

Suppose given a symbol $F\in{S}^p_1(\Xi)$ for some $p\in\mathbb{R}$ and a regular magnetic field $B=dA$ with $A\in\mathlarger{\mathlarger{\mathbf{\Lambda}}}_{\text{\tt bd}}^1(\mathcal{X})$. Then $\mathfrak{Op}^A(F)$ also has a "standard" Weyl symbol $F^A_W\in{S}^p_1(\Xi)$ such that $\mathfrak{Op}^A( and thus

Theorems & Definitions (40)

  • Definition 1.1
  • Definition 2.1
  • Theorem 2.2
  • proof
  • Remark 2.4
  • Proposition 2.5
  • proof
  • Proposition 2.7
  • proof
  • Remark 2.9
  • ...and 30 more