Analysis of Pseudo-Differential Operators

本研究是对海森堡群上的次拉普拉斯关于中心的逆傅立叶变换所产生的扭曲拉普拉斯的研究的推广。扭转拉普拉斯函数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格林函数。扭曲拉普拉斯的一个特殊特征是，它的每个本征值都有无限的重数，因此揭示了很少关于能级结构的信息。Gramchev，Pilipovic，Rodino和Me最近发现扭转拉普拉斯算子与其转置算子的乘积具有特征值1，2，…，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重数有关。扭转双拉普拉斯函数的热核和格林函数是最近由我和我的一位博士生构造的。还计算了热核和格林函数的迹和行列式。这个建议的目的之一是微调这些现有的结果，构造扭双拉普拉斯幂的热核和格林函数，并获得扭双拉普拉斯幂的计数函数的渐近展开式。相关问题有Zeta函数、正则化迹和Zeta函数、热核的正则行列式和Green函数。扭曲的比拉普拉斯的幂是与海森堡群有关的伪微分算子，海森堡群具有一维中心，可以被认为是时间。最近，我和我以前的博士生一起研究的另一个方向是关于一类具有多维中心(时间)的Heisenberg群上的伪微分算子，以及所谓的H型群，它也具有多维中心(时间)。本研究的另一个目的是在这些具有多维中心的Heisenberg群上构造扭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用数论和组合学的方法来理解具有多维中心的Heisenberg群上扭双拉普拉斯算子的本征值的多重性，找出多维中心在几何和数学物理上的意义，解释和澄清对具有多维中心的Heisenberg群上的伪微分算子研究的最深层次在于非对易量子化。

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.