Spectral Theory and Geometric Quantization

Regents Of The University Of Michigan - Ann Arbor, Ann Arbor MI

Investigators

Abstract

Proposal DMS-0401064 Title: Spectral theory and geometric quantization P.I.: Alejandro Uribe (University of Michigan- Ann Arbor) ABSTRACT A variety of spectral, pseudospectral and inverse spectral problems will be studied. These include: (1) The estimation of the spectra and pseudospectra of non self-adjoint differential, Toeplitz and pseudodifferential operators in the semiclassical regime, (2) The study of the spectral properties of elliptic operators on orbifolds, in particular establishing a semiclassical trace formula, (3) Inverse problems for the Laplacian, where the data are a combination of spectral and initial data, and (4) Extending the quantization of Kahler manifolds to symplectic orbifolds. The methods employed will be primarily microlocal, including the calculus of h-pseudodifferential operators and Fourier integral operators both of complex phase and of Hermite types. Spectral and pseudospectral problems arise in enormously diverse contexts, from applied problems in oscillations and diffusions to problems in differential geometry. Yet there are many fundamental questions in these areas that remain poorly understood. A common theme of the topics in this proposal is that they are amenable to study through the various mathematical realizations of the wave-particle duality, more specifically semi-classical methods. This approach establishes a connection between analytical and geometrical objects, which should be very fruitful for the problems proposed here.