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

数学物理学报 ›› 2026, Vol. 46 ›› Issue (2): 770-818.

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

熊枭^*(), 余秋石(), 张鑫宇()

哈尔滨工业大学数学研究院哈尔滨 150001

收稿日期:2026-01-15 修回日期:2026-01-27 出版日期:2026-04-26 发布日期:2026-04-27
通讯作者: 熊枭 E-mail:xxiong@hit.edu.cn;757082626@qq.com;804586989@qq.com
作者简介:余秋石, Email：757082626@qq.com
张鑫宇, Email：804586989@qq.com
基金资助:
国家自然科学基金(12371138);国家自然科学基金(W2441002)

Singular Value Asymptotics of Pseudodifferential Operators on the Quantum Torus

Xiao Xiong^*(), Qiushi Yu(), Xinyu Zhang()

Institute for Advanced Study in Mathematics of HIT, Harbin 150001

Received:2026-01-15 Revised:2026-01-27 Online:2026-04-26 Published:2026-04-27
Contact: Xiao Xiong E-mail:xxiong@hit.edu.cn;757082626@qq.com;804586989@qq.com
Supported by:
NSFC(12371138);NSFC(W2441002)

摘要/Abstract

摘要：

该文给出了量子环上拟微分算子的谱渐近极限公式, 即对量子环上阶数为 $-m < 0$ 的古典拟微分算子 $T \in \mathrm{C}\Psi^{-m}(C^\infty(\mathbb{T}_\theta^d))$, 其谱渐近极限 $\lim_{t \to \infty} t^{\frac{m}{d}} \mu(t, T) $ 为其主象征 $\sigma(T)_{-m}$ 在单位球面上的 $ L_{\frac d m}$ 积分. 这回答了 McDonald 和 Ponge 的文章 [Adv Math, 2023, 412: 108815] 中提出的一个猜想. 作为推论, 该文给出量子环上拟微分算子的 Weyl 律, 包括 Laplace-Beltrami 算子的 Weyl 律.

关键词: 量子环, 拟微分算子, 谱渐近极限, Weyl 律

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

中图分类号:

O177.2

熊枭, 余秋石, 张鑫宇. 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[J]. 数学物理学报, 2026, 46(2): 770-818.

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.

图/表 3

参考文献 47

[1]	Baaj S. Calcul pseudodifférentiel et produits croisés de $C^$-algèbres. I.C R Acad Sci Paris, Ser 1, Math, 1988, 307*(11): 581-586
[2]	Baaj S. Calcul pseudodifférentiel et produits croisés de $C^$-algèbres. II.C R Acad Sci Paris, Ser 2, Math, 1988, 307*(12): 663-666
[3]	Birman M S, Solomyak M Z. The principal term of the spectral asymptotics for "nonsmooth" elliptic problems. Funct Anal Appl, 1970, 4: 265-275
[4]	Birman M S, Solomyak M Z. Asymptotic behavior of the spectrum of pseudodifferential operators with anisotropically homogeneous symbols. Vestn Leningr Univ, 1982, 10: 237-247
[5]	Birman M S, Solomyak M Z. Asymptotic behavior of the spectrum of pseudodifferential operators with anisotropically homogeneous symbols. II.Vestn Leningr Univ Mat Mekh Astron, 1979, 13(3): 5-10
[6]	Birman M S, Solomyak M Z. Asymptotic behavior of the spectrum of variational problems on solutions of elliptic equations. Sibirsk Mat Zh, 1979, 20(1): 3-22
[7]	Birman M S, Solomyak M Z. Spectral Theory of Selfadjoint Operators in Hilbert Space. Dordrecht:D. Reidel Publishing, 1987
[8]	Chen Z, Xu Q, Yin Z. Harmonic analysis on quantum tori. Comm Math Phys, 2013, 322(3): 755-805
[9]	Connes A. $C^$-algèbres et géométrie différentielle. C R Acad Sci Paris Sér A, 1980, 290*: 599-604
[10]	Connes A. An analogue of the Thom isomorphism for crossed products of a $C^*$-algebra by an action of $\mathbb{R}$. Adv Math, 1981, 39: 31-55
[11]	Connes A. Noncommutative differential geometry. Inst Hautes études Sci Publ Math, 1985, 62: 257-360
[12]	Connes A. The action functional in noncommutative geometry. Comm Math Phys, 1988, 117: 673-683
[13]	Connes A. Noncommutative Geometry. San Diego: Academic Press, 1994
[14]	Connes A, Tretkoff P. The Gauss-Bonnet theorem for the noncommutative two torus//Arithmetic, and Related Topics. Baltimore, MD: Johns Hopkins Univ Press, 2011: 141--158
[15]	González-Pérez A M, Junge M, Parcet J. Singular integrals in quantum Euclidean spaces. Mem Amer Math Soc, 2021, 272: Art 1334
[16]	Ha H, Lee G, Ponge R. Pseudodifferential calculus on noncommutative tori, I. Oscillating integrals. Int J Math, 2019, 30(8): Art 1950033
[17]	Ha H, Lee G, Ponge R. Pseudodifferential calculus on noncommutative tori, II. Main properties. Int J Math, 2019, 30(8): Art 1950034
[18]	Ha H, Ponge R. Laplace-Beltrami operators on noncommutative tori. J Geom Phys, 2020, 150: Art 103594
[19]	Hormander L. Pseudo-differential operators. Comm Pure Appl, 1965, 18(3): 501-517
[20]	Ikehara S. An extension of Landau's theorem in the analytic theory of numbers. J Math Phys Mass Inst Technol, 1931, 10: 1-12
[21]	Kohn J J, Nirenberg L. An algebra of pseudo-differential operators. Commun Pure Appl Math, 1965, 18: 269-305
[22]	Lee G, Ponge R. Functional calculus for elliptic operators on noncommutative tori, I.J Pseudo-Differ Oper Appl, 2020, 11: 935-1004
[23]	Lein M, Mǎntoiu M, Richard S. Magnetic pseudodifferential operators with coefficients in $C^*$-algebras. Publ Res Inst Math Sci, 2010, 46: 755-788
[24]	Lévy C, Jiménez C N, Paycha S. The canonical trace and the noncommutative residue on the noncommutative torus. Trans Amer Math Soc, 2016, 368(2): 1051-1095
[25]	Lord S, Sukochev F, Zanin D. Singular Traces: Theory and Applications. Berlin: Walter de Gruyter, 2012
[26]	Lord S, Sukochev F, Zanin D. Singular Traces, Vol. 1: Theory. Berlin: Walter de Gruyter, 2021
[27]	McDonald E, Ponge R. Dixmier trace formulas and negative eigenvalues of Schrödinger operators on curved noncommutative tori. Adv Math, 2023, 412: Art 108815
[28]	McDonald E, Sukochev F, Xiong X. Quantum differentiability on quantum tori. Commun Math Phys, 2019, 371: 1231-1260
[29]	McDonald E, Sukochev F, Zanin D. A noncommutative Tauberian theorem and Weyl asymptotics in noncommutative geometry. Lett Math Phys, 2022, 122: Art 77
[30]	Mǎntoiu M, Purice R, Richard S. Twisted crossed products and magnetic pseudodifferential operators. Theta Ser Adv Math, 2005, 5: 137-172
[31]	Pisier G, Xu Q. Noncommutative $L_p$-spaces//Johnson W B, Lindenstrauss J. Handbook of the Geometry of Banach Spaces. Amsterdam: North-Holland, 2003, 2: 1459--1517
[32]	Polishchuk A. Analogues of the exponential map associated with complex structures on noncommutative two-tori. Pacific J Math, 2006, 226: 153-178
[33]	Ponge R. Connes integration and Weyl's laws. J Noncommut Geom, 2023, 17(2): 719-767
[34]	Seeley R T. Complex powers of an elliptic operator//Amer Math Soc Proc Sympos Pure Math. Providence, RI: Amer Math Soc,1967: 288-307
[35]	Shubin M A. Pseudo-Differential Operators and Spectral Theory. Berlin: Springer-Verlag, 1987
[36]	Simon B. Trace Ideals and Their Applications. Cambridge: Cambridge Univ Press, 1979
[37]	Spera M. Sobolev theory for noncommutative tori. Rend Sem Mat Univ Padova, 1992, 86: 143-156
[38]	Stein E M. Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton: Princeton Univ Press, 1993
[39]	Sukochev F, Xiong X, Zanin D. Asymptotics of singular values for quantised derivatives on noncommutative tori. J Funct Anal, 2023, 285: Art 110021
[40]	Tao J. The theory of pseudo-differential operators on the noncommutative $n$-torus. J Phys Conf Ser, 2018, 965: 1-12
[41]	Trèves F. Topological Vector Spaces, Distributions and Kernels. New York: Academic Press, 1967
[42]	Trèves F. Introduction to Pseudodifferential and Fourier Integral Operators. New York: Plenum Press, 1980
[43]	Wiener N. Tauberian theorems. Ann Math, 1932, 33(1): 1-100
[44]	Xia R, Xiong X. Mapping properties of operator-valued pseudo-differential operators. J Funct Anal, 2019, 277: 2918-2980
[45]	Xiong X, Xu Q, Yin Z. Sobolev, Besov and Triebel-Lizorkin Spaces on Quantum Tori. Providence, RI: Mem Amer Math Soc, 2018
[46]	Xu Q. Noncommutative $L_p$-spaces and Martingale Inequalities. 2007. Unpublished manuscript
[47]	许全华, 吐尔德别克, 陈泽乾. 算子代数与非交换 $L_p$ 空间引论. 北京: 科学出版社, 2010
	Xu Q, Turdebek , Chen Z Q. Introduction to Operator Algebras and Noncommutative $L_p$ Spaces. Beijing: Science Press, 2010