Spectral Theory and Scaling Limits

The investigator studies problems in spectral theory, the mathematical theory that pertains to physical notions such as energy levels of quantum systems and vibration frequencies of mechanical systems. The research is motivated by universality, which is the appearance of certain common local statistical behaviors for different systems, and integrability, which is the existence of many conserved quantities in certain nonlinear systems used to describe their behavior over time. The project focuses on commonly studied one-dimensional systems, such as orthogonal polynomials and Schrödinger operators. One focus of this project is the study of spectral properties of large finite truncations of an infinite system, on a microscopic scale. These problems have immediate interpretations in terms of random matrix theory and one-dimensional quantum mechanics; moreover, mathematical methods developed on these systems have the potential to illuminate other mathematical models and physical applications. The project provides research training opportunities for graduate students and postdoctoral scholars. Asymptotic local zero distributions of orthogonal polynomials, as the degree goes to infinity, are described by scaling limits of Christoffel-Darboux kernels. Analogously, local eigenvalue distributions of truncations of some operators are described by scaling limits of corresponding reproducing kernels. A modern approach based on de Branges canonical systems gives optimal results for scaling limits in the fixed measure setting; one goal of this project is to develop this approach in the varying measure setting. Another goal is to describe new scaling behaviors beyond the standard universality classes, where instead of a scaling limit there may be a limit cycle or a one-sided characterization of allowed behavior; this is motivated by almost periodic operators in critical/supercritical regimes and ergodic operators with zero measure spectrum. The investigator also studies related questions in direct spectral theory, to develop techniques for fine control of spectral measures around a point in the spectrum, and sum rules with respect to essential spectra with infinitely many gaps, building upon recent applications of potential theory to differential operators. 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.