Spectral Theory and Geometric Quantization

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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.