Complex Methods in Spectral and Scattering Problems

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Applications of complex function theory in spectral and scattering problems for differential equations present examples of the undisputed relevance of pure mathematics in the areas of theoretical physics and engineering. One of the focal points will be the non-linear Fourier transform (NLFT), closely related to scattering transform for the Dirac system of differential equations. The Dirac system is the quantum electrodynamical law governing spin ½ particles and should be thought of as a relativistic generalization of the Schrödinger equation. Better understanding of NLFT can lead to progress in several areas of mathematical analysis and non-linear differential equations. Questions of convergence and maximal estimates for NLFT will be studied. These are considered by the experts to be among the main problems of non-linear harmonic analysis. The second part will concern extensions and generalizations of classical results of spectral theory, which can be obtained via a new approach developed recently based on the use of so-called Toeplitz operators. Some related questions will be investigated jointly with graduate students. The materials of the project will be used in minicourses and a special topics course aiming at junior researchers and graduate students.A large part of this project concerns problems of convergence of the scattering data for a Dirac system of differential equations. Scattering is commonly viewed as a non-linear version of the classical Fourier transform, which connects this project to the maximal estimates for NLFT. These connections lead to natural questions of establishing versions of the classical results of Fourier analysis in the non-linear settings of scattering. They have been appearing in various forms for most of the last century and remain an object of active research today. As an example, one can look at the non-linear version of Parseval's identity, which can be traced as far back as the work of Verblunski in the 1930s, and a non-linear analog of Hausdorff-Young inequality, which appears in more recent work of Christ and Kiselev. Several of such questions will be studied, as well as ones in the areas of inverse spectral theory and completeness in various spaces of analytic functions.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.