Mathematical Sciences: Microlocal Approaches to Spectral Problems II

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Professor Uribe will continue his research on problems in global analysis. In particular he will study the spectral asymptotics of elliptic operators and their relationships with symplectic geometry. One main problem is to relate the asymptotic properties of the Schrodinger operator to the classical mechanics of the system represented by the operator. A second problem is to develop the theory of the Fourier integral operator in the setting of quantized Kahler manifolds. The motivation for this research comes from mathematical physics and inverse spectral problems. Inverse spectral theory tries to understand an operator by studying how the operator separates input into discrete parts. The Schrodinger operator in particular plays a central role in the modeling of many physical systems and a better understanding of the spectral properties of this operator would have broad application.