Professor Simon will study various problems in the spectral theory of orthogonal polynomials. He intends to study the zero structure of the polynomials associated to self-similar measures like the Cantor measure, thereby linking to recent research in dynamical systems and harmonic analysis. He also intends to look beyond the classical spectral theory regimes of the real line and unit circle. This research piggybacks on recent progress by Professor Simon and coworkers, extending it in new directions.
Orthogonal polynomials have impacts on a wide swath of science and engineering, including filter theory in electrical engineering, problems in statistics, and recently, problems in quantum computing. The kinds of basic research that Professor Simon does, has long term spinoff in these more applied areas. Professor Simon has directly supervised over thirty graduate students and mentored about that number of postdocs. His papers and books have had considerable impact, as seen by the 8,645 citations in MathSciNet (over roughly the past ten years). He is one of a handful of mathematicians with over 5,000 citations.
involves the understanding of the connection between the basic parameters of a mathematical or physical object and its spectral properties. Examples include the use of computer tomography or sonar. The spectral theory of orthogonal polynomials is a perfect laboratory for understanding spectral theory because in general the inverse problem is hard or has not been solved in general whereas for the orthogonal polynomials the inverse problem is simple and allows the explication of deep connections in this model that may lead to understanding in other arenas. During the recently completed grant, a major theme of Prof. Simon's research has been study the spectral theory of orthogonal polynomials on the real line when the essential spectrum is a finite band set. This joint work with Jacob Christensen (now of Lund Sweden) and Maxim Zinchenko (now at U. New Mexico) is mathematically fascinating because of the occurrence of almost periodic functions. We succeeded in a complete description of the so called Szego class for this setting. For this class Rupert Frank (now at Caltech) and Prof. Simon succeeded in proving the Nevai conjecture. Another theme of this grant research has been the understanding of a recent breakthrough of Remling on the connection of absolutely continuous spectrum and what are known as reflectionless potentials. Motivated by this breakthrough, Prof. Simon with postdocs Jonathan Breuer (now at Hebrew University) and Eric Ryckman (now at Goldman Sachs) returned to a thirty year old conjecture of Davies and Simon and proved that two rather different definitions of reflectionless - one based on dynamics and one on spectral theory - are equivalent. Breuer and Simon also used the ideas of reflectionless objects to return to and increase our understanding of a problem of interest to classical analysts for over 150 years - namely the subject of natural boundaries for functions defined by power series.
