The proposed research will focus on the theory and application of singular potentials and their relationship to quantum field theory. Particular emphasis will be given to those systems exhibiting asymptotic ultraviolet conformal invariance. The conformal anomaly will be analyzed in quantum mechanics problems that have applications in molecular physics such as the anisotropic inverse square potential. Path integral methods will be used to study this problem as well as a whole family of conformally invariant interactions. A second project will be to compare two methods of analyzing conformally invariant problems, one being a renormalization group analysis and the other being the method of self-adjoint extensions. The third project is to understand whether black hole thermodynamics can be understood in terms of the near-horizon conformal symmetry of a scalar field. The fourth project is to understand the role of singular potentials in dimensional reduction schemes where the low energy physics is confined to a brane but gravity propagates in one extra dimension.
