Mathematical Sciences: Toeplitz Operators, Quantum Mechanicsand Mean Oscillation
Lewis Coburn Principal Investigator
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Professor Coburn will continue his study of the symbol calculus associated with the Berezin - Toeplitz quantization. One specific setting is complex n-space with Gaussian measure, and with moderate growth and oscillation conditions on the singularities of the symbols. Coburn will seek to characterize bounded Hankel operators in this setting in terms of mean oscillation of symbols. The corresponding boundedness problem for the Hardy spaces of the ball and polydisc will also be attacked. Coburn expects the techniques developed in working on these problems to be useful in the study of derivations on various naturally occurring algebras. The impetus for this general area of mathematical research comes from quantum mechanics. In the classical, pre-quantum overview of the world, the observables of a physical system are numbers, or, if one wants to keep track of the dependency of these numbers on some other quantities, functions. This is common sense, but at the microscopic level it doesn't quite work. One needs to replace functions by operators, linear transformations of a space that is usually infinite-dimensional. This replacement is what mathematical physicists call quantization. The general idea is that the operators behave almost like the functions from which they arise, except for correction terms which turn out to contain important information. In this line of endeavor, physical insight informs mathematics as often as the reverse.