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Magnetic Pseudo-differential Operators with Hörmander Symbols Dominated by Tempered Weights

Mikkel Hviid Thorn

TL;DR

The paper addresses magnetic pseudodifferential operators with symbols dominated by tempered weights, extending the matrix representation in tight Gabor frames to asymmetrical quantizations and the symbol class $S_0(M)$. It develops a Parseval tight Gabor frame tailored to magnetic contexts, derives a robust matrix representation and a corresponding symbol calculus, and establishes boundedness, compactness, and Schatten-class criteria in terms of the tempered weight $M$. A key contribution is the independence of the quantization parameter $t$ in the change-of-quantization map and the construction of a magnetic Moyal algebra that governs the product of symbols. The results broaden the applicability to magnetic Schrödinger-type operators with complex decay structures and invite applications to magnetic pseudodifferential super-operators, while also outlining limitations and avenues for future work.

Abstract

We extend the matrix representation of magnetic pseudo-differential operators in a tight Gabor frame from [arXiv:1804.05220, arXiv:2212.12229] to asymmetrical quantizations and smooth symbols dominated by a tempered weight (and not just decay/growth properties in the momentum variables). This leads to new results regarding the symbol calculus of such operators and their Schatten-class properties.

Magnetic Pseudo-differential Operators with Hörmander Symbols Dominated by Tempered Weights

TL;DR

The paper addresses magnetic pseudodifferential operators with symbols dominated by tempered weights, extending the matrix representation in tight Gabor frames to asymmetrical quantizations and the symbol class . It develops a Parseval tight Gabor frame tailored to magnetic contexts, derives a robust matrix representation and a corresponding symbol calculus, and establishes boundedness, compactness, and Schatten-class criteria in terms of the tempered weight . A key contribution is the independence of the quantization parameter in the change-of-quantization map and the construction of a magnetic Moyal algebra that governs the product of symbols. The results broaden the applicability to magnetic Schrödinger-type operators with complex decay structures and invite applications to magnetic pseudodifferential super-operators, while also outlining limitations and avenues for future work.

Abstract

We extend the matrix representation of magnetic pseudo-differential operators in a tight Gabor frame from [arXiv:1804.05220, arXiv:2212.12229] to asymmetrical quantizations and smooth symbols dominated by a tempered weight (and not just decay/growth properties in the momentum variables). This leads to new results regarding the symbol calculus of such operators and their Schatten-class properties.

Paper Structure

This paper contains 12 sections, 10 theorems, 64 equations.

Table of Contents

  1. Introduction
  2. Symbol Classes and Magnetic Pseudo-differential Operators
  3. Tight Gabor Frame and Matrix Representation
  4. Tight Gabor Frame
  5. Matrix Representation
  6. Change and Composition of Symbols
  7. Continuous Change of Quantization
  8. Magnetic Moyal Algebra
  9. Boundedness, Compactness, and Schatten-class Properties
  10. Compactness
  11. Schatten-class Properties
  12. Discussion

Key Result

Lemma 2.1

CorneanHelfferPurice2024 The family $(\mathcal{G}_{\tilde{\alpha}}^A)_{\tilde{\alpha}\in{\mathbb Z}^{2d}}$ defines a Parseval frameSee Christensen2016. in $L^2({\mathbb R}^d)$ and consequentlyWe define $\langle\cdot,\cdot\rangle_{L^2}$ to be antilinear in the first entry and linear in the second. holds for every $f\in L^2({\mathbb R}^d)$ with unconditional convergence.

Theorems & Definitions (19)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Theorem 2.4
  • Corollary 3.1
  • Corollary 3.2
  • proof
  • ...and 9 more