Partial Differential Equations and Complex Variables

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Proposal DMS-9801555 Abstract of the Research Project of Francois Treves F. Treves plans to carry out his research in the coming years, in the following three distinct areas: 1. Global parametrices for the time-dependent Schroedinger equation on Euclidean space equipped with a non-Euclidean Riemannian metric. This is joint work with S. Chanillo (Rutgers Univ.). The first step is to construct a parametrix when there is no potential. The next step is to do the same with a potential, whose derivatives are decaying at infinity at a suitable rate (with similar contions on the metric). They propose to use the parametrix to describe the critical set and the propagation of singularities (in Lbesgue and Sobolev spaces), with special attention to the presence of trapped geodesics. 2. Boundary values of Dolbeault cohomology classes. Treves is completing a research program begun with P. D. Cordaro (Univ. of Sao Paulo, Brazil) and S. Gindikin (Rutgers Univ.). The objectives are several: to obtain integral formulas for the solutions of second-order hyperbolic and ultra-hyperbolic partial differential equations; to define spaces of generalized functions (actually hyperfunctions) well suited for the representations of the classical groups which preserve those equations; to lay the ground for the study of ``analytic extension" to domains more complicated than wedges. 3. Analyticity of the solutions of second-order linear partial differential equations of the form ``sums of squares of (analytic, real) vector fields". Recently Treves has formulated a necessary and sufficient condition for such a property to be valid; he plans to use microlocal analysis to initiate a proof of the conjecture. The solutions of the Schroedinger equation are the fundamental elements in the ``wave description" of the states of particles in Quantum Physics. The construction attempted by Chanillo and Treves is aimed at providing explicit formulas for solving the Schroedinger equation - formulas that make it possible to describe very precisely the behaviour of the particles in the large (so as to take into account distant ``radiation") and in ``curved" space (so as to take into account special constraints on the trajectories of the particles). The work of Treves on boundary values aims at forging new mathematicatools, and again explicit formulas, that could be used in the theory of the Radon and X-ray transforms (which lie at the theoretical foundations of tomography). Finally, the differential equations of the kind ``sums of squares of vector fields" describe, through their solutions, various kinds of Brownian motion and of random processes (they have been extensively studied by probability theorists). The analyticity of their solutions is a precise mathematical formulation of their great smoothness, as well as indicating the possibility of computing some of their properties in finitely many steps.