Spectral Theory and Geometric Quantization

题目：光谱理论和几何量化p.i: Alejandro Uribe (University of Michigan- Ann Arbor)[摘要]将研究各种光谱、伪光谱和逆光谱问题。这包括：(1)半经典区域中非自伴随微分算子、Toeplitz算子和伪微分算子的谱和伪谱的估计；(2)轨道上椭圆算子的谱性质的研究，特别是建立半经典的迹公式；(3)拉普拉斯算子的反问题，其中数据是谱和初始数据的组合；(4)将kahler流形的量化推广到共轭轨道。所采用的方法将主要是微局部的，包括复相和hermite类型的h-伪微分算子和傅里叶积分算子的演算。光谱和伪光谱问题出现在各种各样的文本中，从振荡和扩散的应用问题到微分几何的问题。然而，这些领域的许多基本问题仍未得到充分理解。本提案中主题的一个共同主题是，它们可以通过波粒性的各种数学实现来研究，更具体地说，是半经典方法。这种方法在解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益的。

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.