This project deals with Jacobi matrices and more general difference operators with some absolutely continuous spectrum and related dynamical properties of such operators. The dynamical systems that are induced by the shift map are relevant when one is interested in the absolutely continuous spectrum, and then maps implemented by Toda flows deserve special attention because they preserve the structural properties of such sytems. The project will pursue this approach and give a prominent role to Toda maps in the study of the absolutely continuous spectrum.
The mathematical questions that will be studied in this project are related to several classical areas of pure mathematics; however, they derive their main motivation from the theory of quantum mechanics. More specifically, they are directly related to questions about the long time behavior of a quantum particle (say, an electron). The rigorous mathematical investigation of such issues often leads to a clearer and more secure understanding of basic mechanisms and subtle effects in the quantum world.
