Spectral Theory

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Professor Simon plans to study the spectral theory of orthogonal polynomials and related systems, following up on the substantial progress in the subject within the past ten years. One intriguing problem he intends to study is the spectral theory of the orthogonal polynomials associated to the Cantor measure where there is currently strong numerical evidence that the recursion coefficients are asymptotically almost periodic and Prof. Simon hopes to prove theorems. A second problem concerns exponential decaying perturbations for elements of the exponential torus of finite gap Jacobi matrices where the need to employ techniques from the theory of Riemann surfaces presents some difficulties in extending the one band results obtained earlier by Damanik and Simon. A third problem concerns a conjectured dichotomy about the zeros of orthogonal polynomials on the unit circle where numeric evidence suggests that either every point on the circle is a limit of zeros or else all the zeros stay away from the unit circle. Finally, Professor Simon is planning to explore extensions of the results obtained in the past for orthogonal polynomials on the real line and unit circle to the context of more general measures in the complex plane.The proposed research concerns the spectral and inverse spectral theory of orthogonal polynomials. Inverse spectral theory is connected with determining the properties of an object from its measured spectral properties. Practical examples include using x-rays to determine the structure of crystalline matter and medical imaging technology such as CAT scans. The underlying mathematical theory is the seed corn for further developments in these important applied areas. The spectral theory of orthogonal polynomials is significant because it is a theory where there is a general explicit set of formulae for solving the inverse problem. During his career, Professor Simon has made important contributions to developing our scientific infrastructure with over 30 graduate student PhD supervisions and mentoring of almost 50 postdocs and other scientists. Many of these have gone on to distinguished careers of their own. He expects this to continue during the new grant period.