This thesis consists of a summary of four papers dealing with resonances of Dirac operators on Euclidean 3-space. In Paper I we show that the Complex Absorbing Potential (CAP) method is valid in the semiclassical limit for resonances sufficiently close to the real line if the potential is smooth and compactly supported. In Paper II  we continue the investigations initiated in Paper I but here we study clouds of resonances close to the real line and show that in some sense the CAP method remains valid also for multiple resonances. In Paper III we study perturbations of Dirac operators with smooth decaying scalar potentials  and show that these possess many resonances near certain points related to the maximum and the minimum of the potential. In Paper IV we show a trace formula of Poisson type for Dirac operators having compactly supported potentials which is related to resonances. The techniques mainly stem from complex function theory and scattering theory.

Thesis (Doctoral)--Universität Bonn, 1990. / Includes bibliographical references.

In the thesis we study particular sequences of invariant differ- ential operators of first and second order which live on homogeneous spaces of a particular type of parabolic geometries. We show that they form a reso- lution of the kernel of the first operator and that they descend to resolutions of overdetermined, constant coefficient, first order systems of PDE's called the k-Dirac operators. This gives uniform description of resolutions of the k-Dirac operator studied in Clifford analysis. We give formula for second order operators which appear in the resolutions. 1

Orientador: Marcos Benevenuto Jardim / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-08T16:20:15Z (GMT). No. of bitstreams: 1 Leao_RafaeldeFreitas_D.pdf: 1758484 bytes, checksum: a1d5ed8e2a4224e43550ff157cc3a680 (MD5) Previous issue date: 2007 / Resumo: Nesta tese estudamos basicamente como o acoplamento por uma conexão arbitraria influencia o comportamento do espectro do operador de Dirac, real e complexo. Atraves dos resultados classicos da literatura, e destes resultados vemos que, de modo geral, estruturas geometricas influenciam o espectro do operador de Dirac, acoplado ou não. Embora exista uma grande literatura a respeito de estruturas geometricas e o operador de Dirac, sobretudo para o operador não acoplado, existem alguns casos, possivelmente bastante interessantes, que não foram considerados. Com o recente desenvolvimento de geometria complexa generalizada, podemos nos perguntar sobre a possibilidade de definirmos operadores de Dirac neste contexto e se isto traz resultados novos ou entendimento sobre resultados ja conhecidos. Por se tratar de uma area recente existem varios problemas envolvidos na tentativa de estudarmos operadores de Dirac sobre variedades com estruturas complexas generalizadas. O proprio conceito de conexão para este tipo de geometria ainda não e muito claro, uma vez que não assumimos a priori uma metrica na variedade base não podemos considerar a conexão Levi-Civita, ficando a pergunta que se neste contexto existe alguma conexão natural analoga a conexão de Levi-Civita. Outra questão importante e com relação ao fibrado de spinores. No caso de variedades riemannianas a maneira mais usual de construirmos fibrados de Dirac e atraves de uma estrutura Spin na variedade base. Porem este tipo de estrutura tambem e definida em termos de uma metrica ficando a pergunta de como poderíamos construir fibrados de Dirac de maneira natural sobre uma variedade complexa generalizada. Caso seja possível respondermos estas questões podemos falar em operadores de Dirac sobre variedades complexas generalizadas. Podendo, a partir dai, investigar formulas do tipo Weitzenbock e o comportamento do espectro do operador de Dirac. Alem disso podemos nos perguntar se este tipo de operador e de fato um objeto totalmente novo ou se o mesmo se relaciona com operadores conhecidos da variedade base. Outro situação pouco explorada na literatura e a de operadores de Dirac sobre variedades algebricas imersas em CPn. Na literatura existem artigos, [5, 16], que exploram sobretudo estruturas Spin e spinores. Mas não existe tentativas de usar explicitamente que certas variedades podem ser consideradas como variedades algebricas imersas em CPn para tentar obter estimativas mais finas para o espectro do operador de Dirac, como por exemplo e feito para subvariedades Lagrangianas em [8]. Para considerarmos este problema devemos entender como considerar explicitamente que estamos lidando com variedades algebricas imersas em CPn. É possível que existam duas formas de fazermos isto. A primeira e aparentemente mais direta e considerar a imersão em si, na linha do que foi feito com subvariedades Lagrangianas em [8], e estudar propriedades da mesma. Para fazermos isto é possiível que tenhamos que restringir a classe de variedades em questão. A segunda forma, que parece ser um pouco mais delicada, é tentar escrever o operador de Dirac de forma a levar em consideração a estrutura algebrica da variedade. Pode ser possível que escrevendo o operador de Dirac na linguagem algebrica obtenhamos informações que nos permitirão encontrar estimativas para o espectro do mesmo / Abstract: Not informed. / Doutorado / Geometria / Doutor em Matemática

In this PhD thesis we investigate the space R^inv(M): the space of riemannian metrics on a spin manifold M whose associated Dirac operator is invertible. In particular we are interest in the bond between the topology of R^inv(M) and the topology of the underlying manifold M. We conjecture that the homotopy type of R^inv(M) is invariant under spin cobordism.

No description available.

No description available.

We solve a random matrix ensemble called the chiral Ginibre orthogonal ensemble, or chGinOE. This non-Hermitian ensemble has applications to modelling particular low-energy limits of two-colour quantum chromo-dynamics (QCD). In particular, the matrices model the Dirac operator for quarks in the presence of a gluon gauge field of fixed topology, with an arbitrary number of flavours of virtual quarks and a non-zero quark chemical potential. We derive the joint probability density function (JPDF) of eigenvalues for this ensemble for finite matrix size N, which we then write in a factorised form. We then present two different methods for determining the correlation functions, resulting in compact expressions involving Pfaffians containing the associated kernel. We determine the microscopic large-N limits at strong and weak non-Hermiticity (required for physical applications) for both the real and complex eigenvalue densities. Various other properties of the ensemble are also investigated, including the skew-orthogonal polynomials and the fraction of eigenvalues that are real. A number of the techniques that we develop have more general applicability within random matrix theory, some of which we also explore in this thesis.

Les modèles de champ moyen en QED apparaissent naturellement dans la modélisation du nuage électronique des atomes lourds. Cette modélisation joue un rôle croissant en physique et chimie quantique, les effets relativistes ne pouvant pas être négligés pour ces atomes. En physique quantique relativiste, le vide est un milieu polarisable, susceptible de réagir à la présence de champ électromagnétique.On se place dans le cadre du modèle variationnel de Bogoliubov-Dirac-Fock (BDF) qui est une approximation de champ moyen de la QED sans photon (en particulier, les interactions considérées sont purement électrostatiques).Il est à noter que pour donner un sens au modèle BDF, il est nécessaire d'introduire une régularisation ultra-violette. Il se produit un phénomène de renormalisation de charge due à la polarisation du vide : la charge de l'électron observée dépend de la charge « nue » de l'électron et du paramètre de régularisation. On étudie rigoureusement ce phénomène ainsi que le problème de la renormalisation de la masse. Cette dernière est en lien avec l'existence d'un état fondamental pour le système d'un électron dans le vide, en l'absence de tout champ extérieur. En revanche, on montre l'absence de minimiseurs dans le cas de deux électrons.Enfin, on exhibe des points critiques de l'énergie BDF, interprétés comme des états excités du vide. On met en évidence le positronium, système métastable d'un électron et de son antiparticule le positron, ainsi que le dipositronium, molécule métastable constituée de deux électrons et de deux positrons.Les méthodes utilisées sont variationnelles (concentration-compacité, lemme de Borwein et Preiss). / In QED, mean-field models appear in the modelling of the electron clouds of heavy atoms. This modelling plays a increasing role in physics and in quantum chemistry: relativistic effects cannot be neglected in these atoms. In relativistic quantum physics the vacuum is a polarizable medium that can react to the presence of an electromagnetic field.We consider the so-called Bogoliubov-Dirac-Fock (BDF) model, a variational model which is a mean-field approximation of no-photon QED (in particular the interactions are purely electrostatic).We point out that an ultraviolet regularisation is necessary to properly define the BDF model. The vacuum polarisation leads to a \emph{renormalisation} phenomenon, the "observed" charge of the electron depends on its "bare" charge and the regularisation parameter. We rigorously study both the problem of charge renormalisation and mass renormalisation. This last one is linked to the existence of ground state in the case of an electron in the vacuum, without any external field. In contrast, we show there is no ground state in the case of two electrons.Finally we exhibit some critical points of the BDF energy which are interpreted as vacuum excited states. In particular, there are the positronium (a metastable system constituted by an electron and its antiparticle called the positron) and the dipositronium (a metastable molecule constituted by two electrons and two positrons).The methods that we use are variational (concentration-compactness, Borwein and Preiss's Lemma).

The symplectic Dirac and the symplectic twistor operators are sym- plectic analogues of classical Dirac and twistor operators appearing in spin- Riemannian geometry. Our work concerns basic aspects of these two ope- rators. Namely, we determine the solution space of the symplectic twistor operator on the symplectic vector space of dimension 2n. It turns out that the solution space is a symplectic counterpart of the orthogonal situation. Moreover, we demonstrate on the example of 2n-dimensional tori the effect of dependence of the solution spaces of the symplectic Dirac and the symplectic twistor operators on the choice of the metaplectic structure. We construct a symplectic generalization of classical theta functions for the symplectic Dirac operator as well. We study several basic aspects of the symplectic version of Clifford analysis associated to the symplectic Dirac operator. Focusing mostly on the symplectic vector space of the real dimension 2, this amounts to the study of first order symmetry operators of the symplectic Dirac ope- rator, symplectic Clifford-Fourier transform and the reproducing kernel for the symplectic Fischer product including the construction of bases for the symplectic monogenics of the symplectic Dirac operator in real dimension 2 and their extension to symplectic spaces...