磁拟微分算子的交换子准则

磁伪微分算子的Beals准则的证明：基于磁Gabor框架

A Beals criterion for magnetic pseudodifferential operators proved with magnetic Gabor frames

本文通过经典Gabor紧框架重新证明非磁Weyl伪微分算子的Beals交换子准则，并引入修改的磁Gabor紧框架，推导出磁性算子的Beals准则。

Fredholm criteria for pseudodifferential operators and induced representations of groupoid algebras

We characterize the groupoids for which an operator is Fredholm if, and only if, its principal symbol and all its boundary restrictions are invertible. A groupoid with this property is called {\em Fredholm}. Using results on the Effros-Hahn conjecture, we show that an almost amenable, Hausdorff, second countable groupoid is Fredholm. Many groupoids, and hence many pseudodifferential operators appearing in practice, fit into this framework. In particular, one can use these results to characterize the Fredholm operators on manifolds with cylindrical and poly-cylindrical ends, on manifolds that a…

A Magnetic Pseudodifferential Calculus for Operator-Valued and Equivariant Operator-Valued Symbols

In this monograph we develop magnetic pseudodifferential theory for operator-valued and equivariant operator-valued functions and distributions from first principles. These have found plentiful applications in mathematical physics, including in rigorous perturbation theory for slow-fast systems and perturbed periodic operators. Yet, a systematic treatise was hitherto missing. While many of the results can be found piecemeal in appendices and as sketches in other articles, this article does contain new results. For instance, we have established Beals-type commutator criteria for both cases, whi…

Born-Jordan Pseudodifferential Operators and the Dirac Correspondence: Beyond the Groenewold-van Hove Theorem

Quantization procedures play an essential role in microlocal analysis, time-frequency analysis and, of course, in quantum mechanics. Roughly speaking the basic idea, due to Dirac, is to associate to any symbol, or observable, $a(x,ξ)$ an operator $\mathrm{Op}(a)$, according to some axioms dictated by physical considerations. This led to the introduction of a variety of quantizations. They all agree when the symbol $a(x,ξ)=f(x)$ depends only on $x$ or $a(x,ξ)=g(ξ)$ depends only on $ξ$: \[ \mathrm{Op}(f\otimes1)u=fu,\quad\mathrm{Op}(1\otimes g)u=\mathcal{F}% ^{-1}(g\mathcal{F}u) \] where $\mathc…

Dirac方程中长程扰动的初等交换子方法

Elementary commutator method for the Dirac equation with long-range perturbations

本文提出了一种初等方法，用于研究Dirac方程在长程扰动下的广义特征函数不存在性和共振估计，为稳态散射理论提供理论基础。

Commutator criteria for strong mixing

We present new criteria, based on commutator methods, for the strong mixing property of discrete flows $\{U^N\}_{N\in\mathbb Z}$ and continuous flows $\{{\rm e}^{-itH}\}_{t\in\mathbb R}$ induced by unitary operators $U$ and self-adjoint operators $H$ in a Hilbert space $\mathcal H$. Our approach put into evidence a general definition for the topological degree of the curves $N\mapsto U^N$ and $t\mapsto{\rm e}^{-itH}$ in the unitary group of $\mathcal H$. Among other examples, our results apply to skew products of compact Lie groups, time changes of horocycle flows and adjacency operators on gr…