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

Technical Summary: The Fibre Operators in the Bloch-Floquet Decomposition of Periodic Magnetic Pseudo-Differential Operators

Problem Statement
The paper addresses the structural analysis of fibre operators arising from the Bloch-Floquet decomposition of periodic magnetic pseudo-differential operators. While the decomposition of periodic differential operators into fibre operators acting on the $d$-dimensional torus is well-established, the corresponding theory for pseudo-differential operators, particularly those involving magnetic potentials, remains less developed. The authors aim to explicitly characterize these fibre operators, specifically determining their distribution kernels and establishing their nature as toroidal pseudo-differential operators. A central challenge lies in handling magnetic fields where the vector potential may grow at infinity or where the magnetic flux through the unit cell is non-zero, necessitating a unified framework that connects magnetic quantization with standard Weyl calculus.

Methodology
The authors employ a rigorous functional analytic framework combining magnetic pseudo-differential calculus, harmonic analysis on abelian groups, and the theory of direct integrals of Hilbert spaces.

1. Framework and Notation: The study is set in dimension $d \geq 2$ with configuration space $X \cong \mathbb{R}^d$ and a regular lattice $\Gamma \cong \mathbb{Z}^d$. The dual space $X^*$ and dual lattice $\Gamma^*$ are defined with a specific normalization involving a $2\pi$ factor in the duality pairing. The authors utilize the Schrödinger projective representation of the phase space $X \times X^*$ to define magnetic pseudo-differential operators $Op_A(\Phi)$ via a twisted quantization involving a phase factor $\Lambda_A(x,y) = \exp(-i \int_{[x,y]} A)$.

2. Equivalence of Quantizations: A critical methodological step is the proof (Theorem 2.2) that for periodic magnetic fields with "zero mean flux" (satisfying $\int_{R_{jk}} B = 0$), the magnetic pseudo-differential operator $Op_A(F)$ is unitarily equivalent to a standard Weyl operator $Op(F^A_W)$ with a modified symbol. This allows the authors to reduce the study of magnetic periodic operators to the study of standard periodic Weyl operators, provided the magnetic field admits a periodic vector potential.

3. Bloch-Floquet Transformations: The authors utilize two unitary transformations to decompose the operators:

   • The Bloch-Floquet (BF) transformation maps $L^2(X)$ to a direct integral of spaces $F_{z^*}$ over the dual torus $Q^* = X^*/\Gamma^*$.
   • The Bloch-Floquet-Zak (BFZ) transformation maps $L^2(X)$ to a decomposable family of operators acting on $L^2(Q)$, where $Q = X/\Gamma$ is the $d$-dimensional torus.

4. Kernel Analysis: By regularizing the distribution kernels of the operators and applying Poisson summation formulas, the authors derive explicit formulas for the kernels of the fibre operators. They analyze the convergence of these kernels as the regularization parameter tends to infinity.

5. Symmetric Weyl Calculus on the Torus: In Section 3, the authors construct a "symmetric" Weyl calculus specifically for the $d$-dimensional torus. This involves defining a Weyl system on the dual pair $(Q, \Gamma^*)$ and utilizing a two-fold cover of the torus (related to the square root function) to define a symmetric unitary representation that respects adjunction properties ($Op(\Phi)^* = Op(\Phi)$).

Key Contributions and Results

• Explicit Distribution Kernels: The paper provides explicit formulas for the distribution kernels of the fibre operators in two representations:

  • As operators on the $d$-dimensional torus $Q$ (Theorem 2.22, Eq. 2.41).
  • As infinite matrices acting on the discrete space $\ell^2(\Gamma^*)$ via the inverse discrete Fourier transform (Theorem 2.22, Eq. 2.44). The matrix elements are given by $[F_Q(\tilde{Op}(F)_\xi)F_Q^{-1}]_{\alpha^*, \beta^*} = \hat{F}_{\alpha^*-\beta^*}(\xi - (\alpha^*+\beta^*)/2)$.

• Identification as Toroidal Pseudo-Differential Operators: Using the derived kernels, the authors prove that the fibre operators are indeed toroidal pseudo-differential operators in the sense of existing literature (e.g., [27, 26, 21]). Specifically, they are shown to be symmetric Weyl operators on the torus, obtained by replacing the canonical differential operators on $\mathbb{R}^d$ with invariant differential operators on the torus.

• Equivalence of Magnetic and Standard Quantization: Theorem 2.2 establishes that for periodic magnetic fields with zero mean flux, the magnetic quantization is equivalent to a standard Weyl quantization with a symbol $F^A_W$ that differs from the original symbol $F$ by a term in a lower-order symbol class ($S^{p-2}_1$). This implies that the domain of the magnetic operator $H^p_A(X)$ coincides with the standard Sobolev space $H^p(X)$.

• Symmetric Weyl Calculus Construction: The authors construct a symmetric Weyl calculus on the torus (Section 3) that is not covered by previous general results on Pontryagin dual pairs (specifically because the torus is not an "odd group"). They demonstrate that the Bloch-Floquet representations of periodic Weyl operators naturally fit into this symmetric calculus.

• Spectral Properties: The paper confirms that for elliptic symbols, the fibre operators are lower-bounded, self-adjoint operators with compact resolvents, possessing a discrete spectrum diverging to $+\infty$ with smooth eigenfunctions.

Significance and Claims
The paper claims that a precise understanding of the fibre operator structure is necessary for addressing various problems involving periodic magnetic fields, as highlighted in the authors' previous works [4]-[6]. The primary significance lies in:

1. Unification: It provides a unified treatment for periodic magnetic pseudo-differential operators, showing that under the zero-flux condition, they can be analyzed using standard Weyl calculus techniques adapted to the torus.
2. Explicit Formulas: It moves beyond abstract existence results to provide concrete, explicit formulas for the distribution kernels and matrix representations of these operators, facilitating further spectral analysis.
3. Extension of Calculus: It extends the theory of pseudo-differential operators to the torus in a "symmetric" manner, filling a gap in the literature regarding Weyl calculus on abelian Lie groups like the torus.
4. Generalization Potential: The authors note that their analysis can be extended to magnetic fields with rational flux through the unit cell and potentially to more general magnetic fields by replacing invariant differential operators with non-trivial connections, though these extensions are not fully developed in this specific note.

The work is presented as a foundational step ("this note") to clarify the structure of these operators, emphasizing that the fibre operators are exactly the pseudo-differential operators on the torus obtained via the symmetric Weyl calculus constructed in the paper.