Boundary Behavior of Analytic Functions

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The research will focus on several problems concerningthe boundary behavior of analytic functions and theirapplications in functional analysis and mathematicalphysics. In analytic function theory the PI plans tostudy the interaction between singular and non-singularcomponents of Cauchy integrals near the boundary of acomplex domain. The results will then be applied tostudy the resolvent functions of linear operators inperturbation problems. This area has connections withmathematical models of solid state physics, whichprovides most of the motivation. Another part of theproject is the study of the connections between theBeurling-Malliavin theory, inverse spectral problemsfor partial differential equations and properties ofkernels of Toepliz operators.Analytic Function Theory is one of the classical yetrapidly developing parts of modern mathematics. Thepresent stage of its development features manypromising new applications in various parts ofmathematics and physics. One of the cornerstones ofAnalytic Function Theory is the integral formulanamed after the famous French mathematician Cauchy.The importance of this formula is due to the factthat most analytic functions in complex domains canbe defined through Cauchy integrals. The PI plansto study various properties of Cauchy integrals nearthe boundaries of their domains. The results willthen be applied in several areas of mathematics aswell as in mathematical physics. The applicationsin mathematical physics will concern, among otherthings, mathematical models of wave propagation.The general question that will be considered is howto recover full information about a disorderedenvironment from the spectrum of the wave operator andpartial information on the potential field.