Spectral Theory and Geometric Quantization

建议DMS-0401064标题：光谱理论和几何量化P.I.：Alejandro Uribe(密歇根大学安娜堡分校)将研究各种光谱、伪谱和逆谱问题。这些问题包括：(1)非自伴微分算子、Toeplitz算子和伪微分算子在半经典情形下的谱和伪谱的估计；(2)研究椭圆算子的谱性质，特别是建立半经典迹公式；(3)拉普拉斯逆问题，其中的数据是谱和初始数据的组合；以及(4)把Kahler型流形的量子化推广到辛或对称流形。所采用的方法将主要是微局部的，包括复位相和Hermite类型的h-伪微分算子和傅立叶积分算子的计算。谱和伪谱问题出现在从振荡和扩散的应用问题到微分几何问题的巨大分歧中。然而，在这些领域有许多根本性的问题仍然鲜为人知。这一建议中的一个共同主题是，它们可以通过波粒二象性的各种数学实现来研究，更具体地说，是半经典方法。这种方法在分析对象和几何对象之间建立了联系，这对于这里提出的问题应该是非常有成效的。

Proposal DMS-0401064Title: Spectral theory and geometric quantizationP.I.: Alejandro Uribe (University of Michigan- Ann Arbor)ABSTRACTA variety of spectral, pseudospectral and inverse spectral problemswill be studied. These include: (1) The estimation of the spectra andpseudospectra of non self-adjoint differential, Toeplitz andpseudodifferential operators in the semiclassical regime, (2) The studyof the spectral properties of elliptic operators on orbifolds, inparticular establishing a semiclassical trace formula, (3) Inverseproblems for the Laplacian, where the data are a combination ofspectral and initial data, and (4) Extending the quantization of Kahlermanifolds to symplectic orbifolds. The methods employed will beprimarily microlocal, including the calculus of h-pseudodifferentialoperators and Fourier integral operators both of complex phase and ofHermite types.Spectral and pseudospectral problems arise in enormously diversecontexts, from applied problems in oscillations and diffusions toproblems in differential geometry. Yet there are many fundamentalquestions in these areas that remain poorly understood. A common themeof the topics in this proposal is that they are amenable to studythrough the various mathematical realizations of the wave-particleduality, more specifically semi-classical methods. This approachestablishes a connection between analytical and geometrical objects,which should be very fruitful for the problems proposed here.