Analysis of Pseudo-Differential Operators

本研究是在对Heisenberg群上的子拉普拉斯算子对中心进行傅里叶反变换的基础上，对扭曲拉普拉斯算子的研究进行扩展。扭曲拉普拉斯函数的特征值和特征函数是完全已知的，可以用来显式地构造热核和格林函数。扭曲拉普拉斯算子的一个特殊特征是它的每一个特征值都有无限的多重性，因此几乎没有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与其转置的乘积，一个适当地再归一化的四阶偏微分算子，具有特征值1,2，…，每个特征值n的多重性是n的狄利克雷因子的个数d(n)。这个算子称为扭曲双拉普拉斯算子，记为M。M的特征值的计数函数的渐近展开与特征值的多重性有关。我和我的一个博士生最近构造了扭曲双拉普拉斯函数的热核和格林函数。计算了热核和格林函数的迹和行列式。本文的目标之一是对这些已有的结果进行微调，构造扭曲双拉普拉斯算子的热核和幂格林函数，并得到扭曲双拉普拉斯算子幂计数函数的渐近展开式。相关的问题是热核和格林函数的正则化轨迹和正则化行列式。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，它有一个一维的中心，可以看作是时间。最近，我和我以前的博士生研究的另一个方向是集中在一类具有多维中心（时间）的Heisenberg群和所谓的h型群上的伪微分算子，它们也具有多维中心（时间）。本研究计划的另一个目标是在这些具有多维中心的Heisenberg群上构造扭曲拉普拉斯算子和相应的双拉普拉斯算子。这些新的扭曲双拉普拉斯算子及其幂是有待研究的全新的伪微分算子。本研究的另一个目的是利用数论和组合学来理解具有多维中心的Heisenberg群上扭曲双拉普拉斯子幂的特征值的多重性，找出多维中心在几何和数学物理方面的意义。解释和阐明对多维中心Heisenberg群上伪微分算子研究的最深刻理解在于非交换量子化。

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.