Toeplitz Order and Spectral Problems

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This project is related to the area of the Uncertainty Principle (UP) in Harmonic Analysis. Briefly, this principle says the sets where a function and its Fourier transform are non-zero cannot be simultaneously small. Stemming from the work of Norbert Wiener in mathematics and Werner Heisenberg in physics, the area of UP still presents many mathematical challenges. Its problems have a number of applications in adjacent fields. Several classical problems of UP, posed decades ago by such prominent mathematicians as Norman Levinson, Andrei Kolmogorov and Norbert Wiener, remain open. Some of such problems are studied in this project. Modern methods of Complex and Harmonic Analysis that appeared in the last 30 years suggest new approaches to the classical challenges of UP. These problems have a number of important applications in Approximation Theory, Prediction Theory, Spectral Theory of differential operators and Mathematical Physics. Among the proposed topics of research are generalizations of the so-called Gap and Type problems in Harmonic Analysis, an extension of the well-known Gelfand-Levitan theory in the area of spectral problems for differential operators and connections between Krein's canonical systems of differential equations and asymptotics of the Riemann zeta-function. Successful completion of this step of the project will create a systematic view of the large variety of problems in the area of UP based on the new notion of Toeplitz Order discussed in the proposal. The study of Toeplitz Order is a continuation of the study of the so-called Toeplitz approach to UP developed in recent papers of Nikolai Makarov (Caltech) and the principal investigator. The next stage of the applications of the Toeplitz approach contains several classical open problems of Harmonic Analysis and Spectral Theory, including general completeness problems, spectral problems for Schroedinger and Dirac operators and a Toeplitz operator version of the so-called Krein - de Branges theory, which was designed to connect Complex and Spectral Analysis. Among other applications, the project includes a problem on Uncertainty Quantification in the settings of spectral problems for Schroedinger opearators and canonical systems of differential equations.