This project will investigate a number of problems involving resonances and related phenomena in scattering theory. One problem is that of describing the distribution of resonances for potential scattering in Euclidean space and that of determining a radial potential from its associated resonances. Another problem is the study of potentials for which the associated Schroedinger operators are isoresonant. The PI will also investigate isoperimetric scattering problems and the relationship between sojourn times and the scattering matrix for a manifold with infinite cylindrical ends.
These problems arise in quantum mechanics and are of importance for physics, chemistry and engineering. Resonances are complex numbers associated with the resolvent operator of a system. They identify decaying states with the real part being the frequency and the imaginary part the rate of decay. In chemistry these states are said to be metastable. Resonances also are of importance in inverse problems, target identification and quantum waveguides such as semiconductor quantum wires. Thus this research may have applications to a variety of important topics.
