Analysis of some Orthogonal Systems and Operators: One-Dimensional and Multidimensional Problems

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Analysis of some orthogonal systems and operators: one-dimensional and multidimensional systems.Abstract of proposed researchSerguei DenissovThe goal of this project is to study a broad spectrum of orthogonal systems and the corresponding operators. There are essentially two different cases: one-dimensional systems and multidimensional systems. In the one-dimensional case, the orthogonal systems with discrete index (orthogonal polynomials on the unit circle and on the real line) and continuous index (Krein systems) will be studied. The subtle problems of dependence of operator coefficients on the spectral measure (and vice versa) will be considered. The standard methods of approximation theory will be used together withextensive application of operator theory ideas. For Krein systems, even the basic theory is not developed. Thus, the systematic study of this case is warranted. The orthogonal systems are directly related to some important operators in mathematical physics - namely the Schroedinger and Dirac operators. Different questions in the spectral analysis of these operators will be studied. The deep analysis of orthogonal systems will be used to investigate possible new results for some completely integrable systems (Toda lattice, KdV, nonlinear Schroedinger equation).Considerable effort will be made to better understand the multidimensional case In particular, the Schroedinger and Dirac operators with slowly decaying or random decaying potentials will be studied. Approximation theory suggests that the Green function and its spatial asymptotics is the right object to analyze. The spectral parameter can be taken in the resolvent set, which often makes the problem treatable. Having this asymptotics, many difficult problems in scattering theory can be solved. This technique will also be tried to understand an interesting phenomenon in mathematical physics: the delocalization in Anderson model.