Quantum Mechanics with Non-Abelian Fields and Potentials.

The first part of this thesis explores solutions to the Schrodinger equation for systems subject to classical Yang-Mills fields. Under a weak set of assumptions on the potentials, we prove the existence of a family of operators, called the Schrodinger evolution, which map vectors in Hilbert space to solutions of the Schrodinger equation. By strengthening our assumptions it is possible to show that these evolution operators are integral operators. The collection of their kernels is commonly called the propagator in the physics literature. Through a constructive technique, an explicit formula for the propagator is found. The second part of this dissertation derives a class of sum rules, commonly known as Levinson's theorem, for a single particle system. These rules relate the number of bound states to the energy integral of the trace of the time delay operator. In particular we will incorporate into these rules detailed information about the spin structure of the system.