Problems in Mathematical Physics

California Institute Of Technology, Pasadena CA

Investigators

Abstract

PI: Barry Simon, California Institute of Technology DMS-0140592 Abstract Professor Simon will continue his research focusing on direct and inverse spectral problems for Schrodinger operators and their discrete analogs. He plans to explore the impact of sum rules for spectral theory both for Jacobi matrices (where he and Killip recently solved several long open conjectures from the Orthogonal Polynomial community using sum rules) and Schrodinger operators. Other problems he expects to look at include the structure of the isospectral manifold for problems with discrete spectrum, resonance counting in higher dimensions, and the study of mixed singular continuous and point spectrum. A fundamental problem in wide swaths of science is determining the relation between some object that can only be observed indirectly and the information you can observe indirectly. There are two sides of issue: determine what is indirectly observed for a given state of the system, called the direct problem, and trying to induce properties of the system from the indirect observations, called the inverse problem. Professor Simon will study various aspects of direct and inverse problems in quantum mechanics. While the focus will be on the equations of quantum theory, there are potential spinoffs to tomography and radar/sonar direct and inverse problems.