Analysis of Pseudo-Differential Operators

伪微分算子分析

• 批准号: RGPIN-2016-05353
• 负责人: Wong, Man
• 金额: $ 1.09万
• 依托单位: York University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2018
• 资助国家: 加拿大
• 起止时间: 2018-01-01 至 2019-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=668815
• 关键词: Analysis; Pseudo; Differential; Operators

This research is to extend the research on the twisted Laplacian that comes from taking the inverse Fourier transform of the sub-Laplacian on the Heisenberg group with respect to the center. The eigenvalues and the eigenfunctions of the twisted Laplacian are completely known and can be used to construct the heat kernel and Green function explicitly. One special feature of the twisted Laplacian is that each of its eigenvalues has infinite multiplicity, thus revealing little information about the structutre of the energy levels. It has recently been found by Gramchev, Pilipovic, Rodino and me that the product of the twisted Laplacian and its transpose, a fourth-order partial differential operator suitably re-normalized, has eigenvalues 1, 2, ..., and the multiplicity of each eigenvalue n is the number d(n) of Dirichlet divisors of n. This operator is called the twisted bi-Laplacian and denoted M. The asymptotic expansion of the counting function of the eigenvalues of M is then related to the multiplicities of the eigenvalues. The heat kernel and Green function of the twisted bi-Laplacian have recently been constructed by me and a Ph.D. student of mine. The trace and the determinant of the heat kernel and the Green function have also been computed. One of the objectives of this proposal is to fine-tune these existing results, to construct heat kernels and Green functions of powers of the twisted bi-Laplacian and to obtain asymptotic expansions of the counting functions of powers of the twisted bi-Laplacian. Related problems are the zeta function regularized traces and zeta function regularized determinants of the heat kernels and Green functions. The powers of the twisted bi-Laplacian are pseudo-differential operators related to the Heisenberg group, which has a one-dimensional center and can be thought of as time. Recently, another direction of my research with my former Ph.D. students has been focused on pseudo-differential operators on a class of Heisenberg groups with multi-dimensional centers (times) and on the so-called H-type groups, which also have multi-dimensional centers (times). Another objective of this research program is to construct the twisted Laplacians and the corresponding bi-Laplacians on these Heisenberg groups with multi-dimensional centers. These new twisted bi-Laplacians and their powers are entirely new pseudo-differential operators to be studied. The other objective of this research enterprise is to understand the multiplicities of the eigenvalues of the powers of twisted bi-Laplacians on Heisenberg groups with multi-dimensional centers with number theory and combinatorics, to find out the meanings of multi-dimensional centers in terms of geometry and mathematical physics, to explain and clarify that the deepest understanding of the study of pseudo-differential operators on Heisenberg groups with multi-dimensional centers lies in non-commutative quantization.

本文的研究是对海森堡群的次拉普拉斯算子关于中心的逆傅里叶变换所得到的扭曲拉普拉斯算子的研究的扩展。扭曲拉普拉斯算子的本征值和本征函数是完全已知的，可以用来显式地构造热核和绿色函数。扭曲拉普拉斯算子的一个特点是它的每一个本征值都有无限重数，从而几乎不能揭示能级结构的信息。最近Gramchev，Pilipovic，Rodino和我发现扭曲拉普拉斯算子和它的转置的乘积，一个适当重新归一化的四阶偏微分算子，具有特征值1，2，...，并且每个特征值n的重数是n的Dirichlet因子的个数d（n）。这个算子被称为扭曲双拉普拉斯算子，记为M。M的本征值的计数函数的渐近展开则与本征值的重数有关。我和一位博士最近构造了扭曲双拉普拉斯算子的热核和绿色函数。我的学生计算了热核和绿色函数的迹和行列式。本文的目的之一是对已有的结果进行微调，构造扭双拉普拉斯算子的热核和幂的绿色函数，并得到扭双拉普拉斯算子的幂的计数函数的渐近展开式.相关的问题是热核和绿色函数的zeta函数正则化迹和zeta函数正则化行列式。扭曲双拉普拉斯算子的幂是与海森堡群相关的伪微分算子，海森堡群有一个一维中心，可以被认为是时间。最近，我和我以前的博士研究的另一个方向。学生们一直专注于一类具有多维中心（时间）的海森堡群上的伪微分算子，以及也具有多维中心（时间）的所谓H型群。本研究的另一个目的是构造具有多维中心的海森堡群上的扭曲拉普拉斯算子和相应的双拉普拉斯算子。这些新的扭双拉普拉斯算子及其幂是全新的拟微分算子。本研究的另一个目的是用数论和组合学来理解具有多维中心的海森堡群上的扭双拉普拉斯算子的幂的特征值的重数，找出多维中心在几何和数学物理方面的意义，解释并阐明了多维中心海森堡群上伪微分算子研究的最深理解在于非对易量子化。