Semiclassical Theory of Fermions
A blend of non-perturbative semiclassical techniques is employed to systematically construct approximations to noninteracting many-fermion systems (coupled to some external potential mimicking the Kohn-Sham potential of density functional theory). In particular, uniform asymptotic approximations are obtained for the particle and kinetic energy density in terms of the external potential acting on the fermions and the Fermi energy. Dominant corrections to the classical limit of quantum mechanics are shown to be captured by the semiclassical approximation everywhere in configuration space. As opposed to previous treatments, no singular behavior arising from inappropriate choice of representation ever arises. Such convenient properties allow us to derive a number of universal limits for the particle density and kinetic and potential energies in the semiclassical limit. Additionally, we study the performance of the semiclassical approximations in a variety of one-dimensional potentials.
In the second part of this thesis, a Dyson-like equation is derived relating the Green's function of an isolated subsystem satisfying Dirichlet boundary conditions with that of an associated infinite coupled system. We explain the relation to the Landauer model and quantum transport. In particular an analytical form for the self-energy operator is obtained for a simple model. The developed framework is illustrated with a semiclassical calculation.