This dissertation analyzes the Wigner-Poisson system in the presence
of an external Coulomb potential. In the first part, the Weyl
transform is defined and used to derive an exact quantum mechanical
equation for the Weyl transform of the density function ρw (the Wigner
function) known as the Wigner equation. This equation holds for any
Hamiltonian which is a function of the position and momentum
operators. The Wigner-Poisson system is then formally derived by
imposing various assumptions on the structure of the Hamiltonian. This
system describes the behaviour of an effective one-particle
distribution in the presence of a large ensemble of particles.
Furthermore, it allows the particles to either attract or repel each
other as well as attract or repel as a whole from a fixed Coulomb
source located at the origin. The second part details the question of
existence and uniqueness for the Wigner-Poisson system. It is shown
that provided the initial Wigner function is sufficiently regular
[special characters omitted] and is a valid Wigner distribution, then
the Wigner-Poisson system has a unique global mild solution [special
characters omitted]. This result is independent of both the nature of
the external Coulomb potential as well as the interparticle
interaction.The proof of this result is accomplished by first
transforming the Wigner-Poisson system into a countably infinite set
of Schrödinger equations which results in what is referred to as the
Schrödinger Poisson system. Using standard semigroup theory arguments,
existence and uniqueness of the Schrödinger-Poisson system is
established. The properties of the Wigner-Poisson system are then
obtained by reversing the transformation step. Regularity results for
both the Schrödinger-Poisson and the Wigner-Poisson systems are
compared to the case with no external Coulomb potential. In addition,
the known regularity results are extended when there is no external
field. The results illustrate that the introduction of an external
Coulomb potential slightly reduces the regularity of the solution.
This confirms a conjecture of Brezzi and Markowich. The third part
analyzes the asymptotic behaviour of the Wigner-Poisson system. If the
configurational energy Εₐ,ᵦ(t) is positive for all times then by
considering the Schrödinger-Poisson system, solutions will decay in
the sense of Lᵖ for 2 < p < 6. This generalizes a result of Illner,
Lange and Zweifel. Moreover, If the total energy is negative then the
solutions will not decay in the sense of Lᵖ for any 2 < p ≤ ∞. This
generalizes a result of Chadam and Glassey. Decay estimates for both
the Schrödinger-Poisson and the Wigner-Poisson systems are compared to
the case with no external Coulomb field. As with the regularity
results, the introduction of an external Coulomb field degrades the
reported decay rates of the solution. / Graduate
| Identifer | oai:union.ndltd.org:uvic.ca/oai:dspace.library.uvic.ca:1828/8461 |
| Date | 25 August 2017 |
| Creators | Bohun, Christopher Sean |
| Contributors | Illner, Reinhard |
| Source Sets | University of Victoria |
| Language | English, English |
| Detected Language | English |
| Type | Thesis |
| Format | application/pdf |
| Rights | Available to the World Wide Web |
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