Problems in Mathematical Physics

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PI: Barry Simon, California Institute of TechnologyDMS-0140592AbstractProfessor Simon will continue his research focusing on direct and inversespectral problems for Schrodinger operators and their discrete analogs. Heplans to explore the impact of sum rules for spectral theory both for Jacobimatrices (where he and Killip recently solved several long open conjecturesfrom the Orthogonal Polynomial community using sum rules) and Schrodingeroperators. Other problems he expects to look at include the structure ofthe isospectral manifold for problems with discrete spectrum, resonancecounting in higher dimensions, and the study of mixed singular continuousand point spectrum.A fundamental problem in wide swaths of science is determining the relationbetween some object that can only be observed indirectly and the informationyou can observe indirectly. There are two sides of issue: determine whatis indirectly observed for a given state of the system, called the directproblem, and trying to induce properties of the system from the indirectobservations, called the inverse problem. Professor Simon will studyvarious aspects of direct and inverse problems in quantum mechanics. Whilethe focus will be on the equations of quantum theory, there are potentialspinoffs to tomography and radar/sonar direct and inverse problems.