This award will support collaborative research in mathematical physics between Dr. Peter Hislop, University of Kentucky and Dr. Jean-Michel Combes, University of Toulon and Var, La Garde, France. The objective of the proposed research is to investigate quantum mechanics of ordered and disordered media. The quantum behavior of a charged particle moving in a periodic array of atoms is determined by the bands of allowed energy. Any particle with energy in these bands will propagate to infinity in the crystal. Physically, this is an idealized situation which corresponds to matter with infinite conductivity. These transport properties of matter, for example the conductivity, change dramatically when the material is immeresed in electric and magnetic fields or when the perfect order of the crystal is broken by impurities. Changes of this type may cause states of the system to be well-localized in energy and space. A key ingredient in the formation of these localized states is quantum tunneling through potential barriers. A particle propagating with energy E will encounter regions of space where the potential is larger than E. Classically, the particle is forbidden to traverse these regions, but quantum mechnically, the wave function will penetrate them. However, the net effect of many barriers may be to well-localize the particle. In this project, the investigators will concentrate on the spectral structure of these systems with an emphasis on the role played by quantum tunnelling. They will study the resonance states and the scattering theory of crystals in electric fields. They will also study models of disordered media in which diffusive behavior is suppressed by potential barriers. The project will benefit from the significant and complementary expertise of the two investigators in the areas of quantum resonances and scattering theory.
