The proposed projects will investigate several mathematical problems which lie at the interface of probability, partial differential equations and spectral theory. On the probability side, the problems are formulated in terms of the behavior of stochastic processes which extend in a natural way Brownian motion. These processes, called Levy processes after the French mathematician Paul Levy, play an important role in many areas of mathematics. They have applications in fields as diverse as life and physical sciences and economics. They have been particularly useful in the modeling of financial markets. On the partial differential equations and spectral theory side, they involved extensions of classical theories to operators which are used to model phenomena involving fractional diffusions.
Mathematically the problems are as follows. (1) Trace asymptotics for Schrodinger operators on Euclidean spaces for the generator of rotationally symmetric stable processes (fractional Laplacian) with smooth potentials. In the case of the Laplacian this produces the famous heat invariants for potentials which are used in inverse scattering and other problems in spectral theory arising in areas of mathematical physics. Questions of scattering and resonances for stable processes and connections to heat invariants for the fractional Laplacian will be explored. (2) A two-term Weyl's asymptotic law for domains on Euclidean spaces. The probabilistic techniques introduced by Mark Kac in the early 50's to prove the celebrated "Weyl's First Law" on the growth of the eigenvalues of the Laplacian (Brownian motion) in terms of the volume of the domain can be used to extend this result to the eigenvalues of many other Levy processes. These probabilistic/heat equation methods fail to give "Weyl's Second Law? which is a much deeper result and which was proved in 1980 for the Laplacian by Ivrii and shortly thereafter by Melrose. On the other hand, the analysis techniques of Ivrii and Melrose also fail primarily due to the boundary conditions needed to deal with processes with jumps. The failure of both the current probabilistic and analytic methods makes these problems extremely challenging. In order to make progress on these questions, brand new techniques will have to be developed. Such techniques will likely impact several fields in mathematics, including probability and spectral theory. These projects will involve the training of graduate students. The results will be disseminated through publications in professional journals, lectures and on the web. As in previous NSF grants awarded to the PI, serious efforts will be made to involve students and young Ph.D.'s from underrepresented groups on this research.
