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

Thesis (Ph. D.)--University of Oregon, 1999. / Typescript. Includes vita and abstract. Includes bibliographical references (leaves 95-97). Also available for download via the World Wide Web; free to University of Oregon users. Address:http://wwwlib.umi.com/cr/uoregon/fullcit?p9957570.

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Thesis (Ph. D.)--University of Texas at Austin, 2005. / Supervisor: Chih-Kang Ken Shih. Vita. Includes bibliographical references.

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Ce mémoire présente les résultats sur l’étude de la supraconductivité et de la réponse à des impuretés locales électrique et magnétiques des semi-métaux de Luttinger. Ces semi-métaux correspondent à des matériaux tri-dimensionnels dont la relation de dispersion électronique est caractérisée par des bandes quadratiques qui se touchent, en présence d’un fort couplage spin-orbite caractérisé par une pseudo-spin-3/2. Expérimentalement, certains semi-métaux de Luttinger supraconducteurs possèdent une température critique ne pouvant être expliquée par les théories conventionnelles (BCS) se référant principalement au mécanisme des phonons. Le volet supraconductivité de notre travail s’intéresse à la résolution numérique de l’équation d’Eliashberg, une théorie microscopique de la supraconductivité, avec interactions Coulombiennes écrantées comme mécanisme d’appariement des paires de Cooper. Nos résultats concernant la température critique montrent une dépendance linéaire avec la température de Fermi du matériau et nous constatons un accord entre température critique expérimentale et de notre modèle pour divers semi-métaux de Luttinger à base de bismuth, comme YPtBi, YPdBi, LuPtBi et LuPdBi. La réponse en densité de charge et spin à des impuretés locales électriques et magnétiques est aussi étudiée à température nulle analytiquement et à température non-nulle numériquement et est comparée aux résultats connus du gaz d’électron libre et des semi-métaux de Dirac. Contrairement à ces dernier, une réponse magnétique anisotropique est observée pour les semi-métaux de Luttinger et la susceptibilité magnétique de spin résultante se trouve être diamagnétique. Un Hamiltonien d’interaction entre deux impuretés magnétiques médié par le mécanisme RKKY, l’interaction entre des impuretés magnétiques obtenue par l’intermédiaire des électrons libres du matériau, est aussi présenté et discuté pour différents semi-métaux. Cette interaction par couplage RKKY pourrait être à l’origine de phases magnétiques exotiques, comme dans le cas du pyrochlore Pr2Ir2O7. Nous terminons en soulignant les explorations possibles concernant nos résultats, en ajoutant ou modifiant des termes brisant une symétrie dans l’Hamiltonien initial. / In this master thesis, I review my work on the superconductivity and on the inhomogeneities induced by impurities in Luttinger semimetals. Luttinger semimetals are characterized by a quadratic band-touching between electron and hole bands, at a time-reversal-invariant point of the Brillouin zone, and that describes effectively pseudo-spin 3/2 fermions. The superconductivity in some Luttinger semimetals can be peculiar due to the increase of the optical dielectric constant through interband excitations. For example, in YPtBi, the superconducting critical temperature is at odds with theoretical expectations from the BCS theory where Cooper pairs are induced by lattice vibrations, the phonons. We thus explore another mechanism of superconductivity, through the microscopic theory of Eliashberg that we solve numerically and where Cooper pairs are induced by the screened Coulomb interaction. In particular we compute the critical temperature and show that it scales linearly with the Fermi temperature of electrons, and compare our results to experimental observations. The multiple bands in Luttinger semimetals also affect the inhomogeneities in charge and in spin due to a charged or a magnetic impurity. We mainly study this phenomenon at zero temperature through analytical calculations and explore the influence of temperature numerically. We compare our results with inhomogeneities in a normal and in a Dirac electron gas. In particular, our results indicate that Luttinger semimetals tend to be diamagnetic on the contrary to normal and Dirac electron gases. We also derive the effective Hamiltonian of two magnetic impurities, where their mutual interaction is mediated by conduction electrons, also known as the RKKY mechanism. This interaction by RKKY coupling could be at the origin of exotic magnetic phases, as in the case of the pyrochlore Pr2Ir2O7. We finish by highlighting possible explorations of our results, by adding or modifying terms in the initial Hamiltonian.