量子环上拟微分算子的奇异值渐近性——献给陈化教授 70 寿辰

Abstract

Abstract:

This paper establishes a spectral asymptotic formula for pseudodifferential operators on the quantum torus. Specifically, for a classical pseudodifferential operator $T \in \mathrm{C}\Psi^{-m}(C^\infty(\mathbb{T}_\theta^d))$ of order $-m < 0$ the spectral asymptotics $\lim_{t \to \infty} t^{\frac{m}{d}} \mu(t, T)$ is given by the $ L_{\frac d m}$-integral of its principal symbol $\sigma(T)_{-m}$ over the unit sphere. This result confirms a conjecture posed by McDonald and Ponge [Adv Math, 2023, 412: 108815]. As a corollary, we derive the Weyal law for pseudodifferential operators on the quantum torus, including the Weyl law for the Laplace-Beltrami operator.

Key words: quantum torus, pseudodifferential operators, spectral asymptotics, weyl law

CLC Number:

O177.2

Xiao Xiong, Qiushi Yu, Xinyu Zhang. Singular Value Asymptotics of Pseudodifferential Operators on the Quantum Torus[J].Acta mathematica scientia,Series A, 2026, 46(2): 770-818.

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Singular Value Asymptotics of Pseudodifferential Operators on the Quantum Torus

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