Spectral Theory

California Institute Of Technology, Pasadena CA

Investigators

Abstract

Professor Simon will continue his research into spectral theory, especially the spectral theory of orthogonal polynomials. This project will focus on the theory where the spectrum has a finite number of gaps. It will also study perturbations of periodic Schrodinger operators, a model of impurities in solids. Spectral theory concerns the relation of mathematical models to their "spectral characteristics" and includes such areas as computer tomography, sonar analysis, and scattering of subatomic particles. Orthogonal polynomials are the simplest paradigm; they are especially useful because the inverse problem--often difficult to even show has solutions--is so explicit. In addition, orthogonal polynomials on the unit circle (on which Professor Simon has written a book) are a central tool in electronic circuit filter design.