Spectral Theory and Universality

William Marsh Rice University, Houston TX

Investigators

Abstract

The project concerns questions in spectral theory, which is the mathematical theory that corresponds to physical notions such as energy levels of quantum systems and vibration frequencies of mechanical systems. One of the topics of this project is universality, which is the appearance of certain universal (or common) local statistical behaviors for different systems. The research is also motivated by integrability, which is the existence of many conserved quantities in certain nonlinear systems used to describe their behavior over time. The project focuses on Schrödinger operators, central to quantum mechanics, and related systems in one dimension; the mathematical methods developed have the potential to illuminate other mathematical models and physical applications such as electron conductivity in disordered materials and signal transmission using solitons. Several graduate students and a postdoc will be trained, and conference organization as well as writing of a monograph on reflectionless operators is planned. One focus of this project is the application of de Branges canonical systems to universality limits for orthogonal polynomials and related systems. A new approach co-authored by the PI reformulates the question in terms of a limit of canonical systems and uses that to obtain a very general criterion for bulk universality; substantial extensions of this approach to other universality phenomena will be studied. Another focus of the project will be the development of inverse spectral theory for reflectionless Schrödinger operators for Dirichlet-regular Widom spectra without the “direct Cauchy theorem” property; the structure of the isospectral torus in this regime will have immediate consequences for the Korteweg-de Vries equation with almost periodic initial data. Finally, of interest will be new research directions made possible by the recent development of Stahl-Totik regularity for Schrödinger operators, such as the study of sum rules for Schrödinger operators with arbitrary essential spectra. 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.