Mathematical Sciences: Research in Scattering Theory

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This project involves three research projects in mathematical analysis. The first concerns the change in spectrum of the Laplace operator defined on automorphic functions in the upper half plane modulo a confinite subgroup of SL(2,R). The goal is to show that for the generic case, the asymptotic distribution of eigenvalues is less than the Weyl distribution calls for. The second area focuses on the circle problem in hyperbolic space which, in its simplest form, compares the number of lattice points in a plane contained in a circle with the area of the circle. Asymptotic values for the relationship have been steadily improved although comparisons with the Euclidean case are of no value. Work is proceeding on estimates of the error term in the cocompact case. Finally, work is continuing on scattering theory for symmetric spaces of noncompact type. Although this was developed in the 1970's, some of the published work contains gaps (in addition to being quite complicated). Efforts will be made to find a simpler and more coherent treatment - and to fill the gaps. Scattering theory arises in many contexts where one is able to measure the outcome of an event in which hidden obstacles are present. The theory seeks to extract information about the obstacles in terms of the initial and final data.