Analysis of Pseudo-Differential Operators

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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.