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Kungsman, Jimmy. "Resonances of Dirac Operators." Doctoral thesis, Uppsala universitet, Matematiska institutionen, 2014. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-223841.
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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.
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Bär, Christian. "Das Spektrum von Dirac-Operatoren." Bonn : [s.n.], 1991. http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&doc_number=003506032&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA.
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Kim, Yonne Mi. "Unique continuation theorems for the Dirac operator and the Laplace operator." Thesis, Massachusetts Institute of Technology, 1989. http://hdl.handle.net/1721.1/14469.
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Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1989.Title as it appeared in M.I.T. Graduate List, Feb. 1989: Carleman inequalities and strong unique continuation.
Includes bibliographical references (leaf 59).
by Yonne Mi Kim.
Ph.D.
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Thumstädter, Torsten. "Parameteruntersuchungen an Dirac-Modellen." [S.l. : s.n.], 2003. http://www.bsz-bw.de/cgi-bin/xvms.cgi?SWB10633955.
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Dumais, Guy. "Killing spinors and spectral properties of the Dirac operator." Thesis, McGill University, 1994. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=55442.
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A survey of the spectral properties of the classical Dirac operator on a Riemannian spin manifold is made. Killing spinors, which are special eigenfunctions of the Dirac operator, are studied and necessary conditions for their existence are given. Killing spinors on $ IR sp{n}$, $S sp{n}$ and $H sp{n}$ are also computed explicitly. Finally the transformation law for Dirac operator under conformal change of the metric is computed and a lower bound for the eigenvalues is given.
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Andersson, Linnéa. "Linear-scaling recursive expansion of the Fermi-Dirac operator." Thesis, Uppsala universitet, Tillämpad matematik och statistik, 2019. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-382829.
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Stadtmüller, Christoph Martin. "Horizontal Dirac Operators in CR Geometry." Doctoral thesis, Humboldt-Universität zu Berlin, 2017. http://dx.doi.org/10.18452/18130.
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In dieser Dissertation beschäftigen wir uns mit angepassten Zusammenhängen und ihren (horizontalen) Dirac-Operatoren auf strikt pseudokonvexen CR-Mannigfaltigkeiten. Einen Zusammenhang nennen wir dann angepasst, wenn er die relevanten Daten parallelisiert. Wir beschreiben den Raum der angepassten Zusammenhänge, indem wir ihre Torsionstensoren studieren, von denen gewisse Teile durch die Geometrie der Mannigfaltigkeit festgelegt sind, während andere frei wählbar sind. Als Anwendung betrachten wir die Eigenschaften der Dirac-Operatoren, die zu diesen Zusammenhängen gehören.
Weiter betrachten wir horizontale Dirac-Operatoren, die nur in Richtung des horizontalen Bündels H ableiten. Diese Operatoren sind besser an die Sub-Riemannsche Struktur einer CR-Mannigfaltigkeit angepasst als die vollen Dirac-Operatoren. Wir diskutieren, wann diese Operatoren formal selbstadjungiert sind und beweisen eine Weitzenböck-Typ-Formel. Wir konzentrieren uns dann auf den horizontalen Dirac-Operator zum Tanaka-Webster-Zusammenhang. Dieser ändert sich konform kovariant, wenn wir die Kontaktform konform ändern. Für diesen Operator betrachten wir weiterhin zwei Beispiele: Wir betrachten S^1-Bündel über Kähler-Mannigfaltigkeiten, insbesondere berechnen wir für Sphären einen Teil des Spektrums. Außerdem betrachten wir kompakte Quotienten der Heisenberggruppe und berechnen hier in den Dimensionen 3 und 5 das volle Spektrum.
Die horizontalen Dirac-Operatoren sind nicht mehr elliptisch, sondern „elliptisch in Richtung von H“. Mithilfe des Heisenbergkalküls stellen wir fest, dass die horizontalen Dirac-Operatoren nicht hypoelliptisch sind. Im Fall des Tanaka-Webster-Zusammenhangs lässt sich aber zeigen, dass der zugehörige Operator auf gewissen Teilen des Spinorbündels hypoelliptisch ist. Dies genügt, um zu beweisen, dass er (nun auf dem gesamten Spinorbündel) ein reines Punktspektrum hat und die Eigenräume, bis auf den Kern, endlich-dimensional sind und aus glatten Eigenspinoren bestehen.In the present thesis, we study adapted connections and their (horizontal) Dirac operators on strictly pseudoconvex CR manifolds. An adapted connection is one that parallelises the relevant data. We describe the space of adapted connections through their torsion tensors, certain parts of which are determined by the geometry of the manifold, while others may be freely chosen. As an application, we study the properties of the Dirac operators induced by these connections. We further consider horizontal Dirac operators, which only derive in the direction of the horizontal bundle H. These operators are more adapted to the essentially sub-Riemannian structure of a CR manifold than the full Dirac operators. We discuss the question of their self-adjointness and prove a Weitzenböck type formula for these operators. Focusing on the horizontal Dirac operator associated with the Tanaka-Webster connection, we show that this operator changes in a covariant way if we change the contact form conformally. Moreover, for this operator we discuss two examples: On S^1-bundles over Kähler manifolds, we can compute part of the spectrum and for compact quotients of the Heisenberg group, we determine the whole spectrum in dimensions three and five. The horizontal Dirac operators are not elliptic, but rather "elliptic in some directions". We review the Heisenberg Calculus for such operators and find that in general, the horizontal Dirac operators are not hypoelliptic. However, in the case of the Tanaka-Webster connection, the associated horizontal Dirac operator is hypoelliptic on certain parts of the spinor bundle and this is enough to prove that its spectrum consists only of eigenvalues and except for the kernel, the corresponding eigenspaces are finite-dimensional spaces of smooth sections.
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Richert, Manfred. "Streutheorie für Diracsche Aussenraumaufgaben." Bonn : [Math.-Naturwiss. Fak. der Univ.], 1992. http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&doc_number=005421124&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA.
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Hachem, Ghias. "Théorie spectrale de l'opérateur de Dirac avec un potentiel électromagnétique à croissance linéaire à l'infini." Paris 13, 1988. http://www.theses.fr/1988PA132008.
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L'objet de cette thèse est la théorie spectrale de l'opérateur de Dirac associé à un champ électrique extérieur. Notre approche est celle de la théorie de la diffusion. Dans un premier temps on étudie l'opérateur non perturbe dont le potentiel est une fonction linéaire d'une variable (champ électrique constant). On construit alors les fonctions propres généralisées de cet opérateur, pour cela on étudie une équation différentielle du second ordre dépendant d'un paramètre. On donne ensuite des estimations pour les fonctions propres généralisées et le théorème d'absorption limite. Dans la deuxième partie on étudie les perturbations de cet opérateur de base par des potentiels de courte portée, on donne une description du spectre de ces opérateurs, on obtient la représentation spectrale de ces opérateurs ainsi que des estimations montrant la décroissance dans le temps des états de diffusion.
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Jakubassa-Amundsen, Doris. "Spectral Theory of the Atomic Dirac Operator in the No-Pair Formalism." Diss., lmu, 2004. http://nbn-resolving.de/urn:nbn:de:bvb:19-23824.
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Tạ, Ngọc Trí. "Results on the number of zero modes of the Weyl-Dirac operator." Thesis, Lancaster University, 2009. http://eprints.lancs.ac.uk/30804/.
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For a given magnetic potential A one can define the Weyl-Dirac operator σ·(-i∇-A) on R³. An L² eigenfunction of σ·(-i∇-A) corresponding to 0 is called a zero mode. In this thesis we will be concerned with the zero mode problem for the Weyl-Dirac operator and some related problems. The main results are: (i) upper bounds for the number of zero modes of the Weyl-Dirac operator in three dimensions when scaling a given magnetic field. A similar version for the Dirac operator in two dimensions is also obtained. There are also related results to estimate the number of zero modes of the massless Dirac operator, and the dimension of the eigenspaces at threshold energies for the Dirac operator with positive mass. (ii) construction of Dirac operators on the unit ball S² of R³ as well as the determination of their spectrum in case of "constant" magnetic fields. We also show another proof for the Aharonov-Casher theorem for S² based on results about spectral properties of Dirac operators that we have obtained. (iii) a formula giving the number of zero modes of the Weyl-Dirac operator for a special magnetic field, which is the result of pullbacks from the "constant" volume form of S². We also obtain a lower bound for the number of zero modes for the Weyl-Dirac operator corresponding to certain scaled magnetic fields; the magnetic fields are parallel to fibres of the Hopf fibration (pulled-back to R³ using inverse stereographic projection).
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Roos, Saskia [Verfasser]. "The Dirac operator under collapse with bounded curvature and diameter / Saskia Roos." Bonn : Universitäts- und Landesbibliothek Bonn, 2018. http://d-nb.info/1170777902/34.
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Fischmann, Matthias. "Conformally covariant differential operators acting on spinor bundles and related conformal covariants." Doctoral thesis, Humboldt-Universität zu Berlin, Mathematisch-Naturwissenschaftliche Fakultät II, 2013. http://dx.doi.org/10.18452/16703.
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Konforme Potenzen des Dirac Operators einer semi Riemannschen Spin-Mannigfaltigkeit werden untersucht. Wir präsentieren einen neuen Beweis, basierend auf dem Traktor Kalkül, für die Existenz von konformen ungeraden Potenzen des Dirac Operators auf semi Riemannschen Spin-Mannigfaltigkeiten. Desweiteren konstruieren wir eine neue Familie von konform kovarianten linearen Differentialoperatoren auf dem standard spin Traktor Bündel. Weiterhin verallgemeinern wir den Existenzbeweis für konforme ungerade Potenzen des Dirac Operators auf semi Riemannsche Spin-Mannigfaltigkeiten. Da die Existenzbeweise konstruktive sind, erhalten wir explizite Formeln für die konforme dritte und fünfte Potenz des Dirac Operators. Basierend auf den expliziten Formeln zeigen wir, dass die konforme dritte und fünfte Potenz des Dirac Operators formal selbstadjungiert (anti selbstadjungiert) bezüglich des L2-Skalarproduktes auf dem Spinorbündel ist. Abschliessend präsentieren wir neue Strukturen der konformen ersten, dritten und fünften Potenz des Dirac Operators: Es existieren lineare Differentialoperatoren auf dem Spinorbündel der Ordnung kleiner gleich eins, so dass die konforme erste, dritte und fünfte Potenz des Dirac Operators ein Polynom in jenen Operatoren ist.Conformal powers of the Dirac operator on semi Riemannian spin manifolds are investigated. We give a new proof of the existence of conformal odd powers of the Dirac operator on semi Riemannian spin manifolds using the tractor machinery. We will also present a new family of conformally covariant linear differential operators on the standard spin tractor bundle. Furthermore, we generalize the known existence proof of conformal power of the Dirac operator on Riemannian spin manifolds to semi Riemannian spin manifolds. Both proofs concering the existence of conformal odd powers of the Dirac operator are constructive, hence we also derive an explicit formula for a conformal third- and fifth power of the Dirac operator. Due to explicit formulas, we show that the conformal third- and fifth power of the Dirac operator is formally self-adjoint (anti self-adjoint), with respect to the L2-scalar product on the spinor bundle. Finally, we present a new structure of the conformal first-, third- and fifth power of the Dirac operator: There exist linear differential operators on the spinor bundle of order less or equal one, such that the conformal first-, third- and fifth power of the Dirac operator is a polynomial in these operators.
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Kusterer, Daniel-Jens. "Quark properties, topology and confinement from Lattice Gauge Theory." [S.l. : s.n.], 2004. http://www.bsz-bw.de/cgi-bin/xvms.cgi?SWB11514075.
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Downes, Robert James. "Cosserat elasticity, spectral theory of first order systems, and the massless Dirac operator." Thesis, University College London (University of London), 2014. http://discovery.ucl.ac.uk/1417503/.
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This thesis is concerned with the study of the massless Dirac operator in dimension three and is, in part, based upon [12, 22, 21, 26, 25]. An introduction is given in Chapter 1. In Chapter 2 we study a special version of Cosserat elasticity, with deformations induced by rotations only, and no displacements. We prove that for a particular choice of elastic moduli and in the stationary setting (harmonic dependence on time) our mathematical model reduces to the massless Dirac equation. Chapter 3 contains a description of the progress recently made in the spectral theory of first order systems, with a particular focus on dimension three presented in Chapter 4. We prove in Chapter 5 that the second asymptotic coefficient of the counting function of a first order system has the geometric meaning of the massless Dirac action. Finally, in Chapter 6 we examine the spectral asymmetry of the massless Dirac operator. We work on a 3-torus equipped, initially, with a Euclidean metric and consider the behaviour of the spectrum under a perturbation of the metric. We derive an explicit asymptotic formula for the eigenvalue closest to zero.
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Bär, Christian, and Nicolas Ginoux. "Classical and quantum fields on Lorentzian manifolds." Universität Potsdam, 2012. http://opus.kobv.de/ubp/volltexte/2012/5997/.
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We construct bosonic and fermionic locally covariant quantum fields theories on curved backgrounds for large classes of fields. We investigate the quantum field and n-point functions induced by suitable states.
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Smith, Douglas Andrew. "Structure of the QCD vacuum and low-lying eigenmodes of the Wilson-Dirac operator." Thesis, University of Edinburgh, 1997. http://hdl.handle.net/1842/11413.
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This thesis details a study of the vacuum structure of QCD using the tool of lattice gauge theory. Chapter 1 gives an introduction to path integrals, semi-classical approximations to path integrals, instantons, topological charge and instanton phenomenology. Chapter 2 introduces lattice gauge theory and the problems of studying topological charge on the lattice. The cooling method and its pitfalls are discussed and details are given of a study undertaken of under-relaxed cooling. In Chapter 3 the algorithms that were developed to study the instantons on the cooled configurations are discussed. Chapter 4 gives the results for the structure of the vacuum: size distributions, spatial distributions, correlations between charges, and scaling of distributions with the lattice spacing. Chapter 5 discusses an exploratory study of the low-lying eigenmodes of the Wilson-Dirac operator. The zero-modes of both the unimproved and improved operators on cold and heated instantons are calculated and the lattice artefacts investigated. Chapter 6 contains my conclusions and suggestions for future work.
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Leão, Rafael de Freitas 1979. "Auto-valores do operador de Dirac e do laplaciano de Dobeault." [s.n.], 2007. http://repositorio.unicamp.br/jspui/handle/REPOSIP/306021.
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Orientador: Marcos Benevenuto JardimTese (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
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Le, Thu Hoai. "Hyperholomorphic structures and corresponding explicit orthogonal function systems in 3D and 4D." Doctoral thesis, Technische Universitaet Bergakademie Freiberg Universitaetsbibliothek "Georgius Agricola", 2014. http://nbn-resolving.de/urn:nbn:de:bsz:105-qucosa-150508.
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Die Reichhaltigkeit und breite Anwendbarkeit der Theorie der holomorphen Funktionen in der komplexen Ebene ist stark motivierend eine ähnliche Theorie für höhere Dimensionen zu entwickeln. Viele Forscher waren und sind in diese Aufgaben involviert, insbesondere in der Entwicklung der Quaternionenanalysis. In den letzten Jahren wurde die Quaternionenanalysis bereits erfolgreich auf eine Vielzahl von Problemen der mathematischen Physik angewandt. Das Ziel der Dissertation besteht darin, holomorphe Strukturen in höheren Dimensionen zu studieren. Zunächst wird ein neues Holomorphiekonzept vorgelegt, was auf der Theorie rechtsinvertierbarer Operatoren basiert und nicht auf Verallgemeinerungen des Cauchy-Riemann-Systems wie üblich. Dieser Begriff umfasst die meisten der gut bekannten holomorphen Strukturen in höheren Dimensionen. Unter anderem sind die üblichen Modelle für reelle und komplexe quaternionenwertige Funktionen sowie Clifford-algebra-wertige Funktionen enthalten. Außerdem werden holomorphe Funktionen mittels einer geeignete Formel vom Taylor-Typ durch spezielle Funktionen lokal approximiert. Um globale Approximationen für holomorphe Funktionen zu erhalten, werden im zweiten Teil der Arbeit verschiedene Systeme holomorpher Basisfunktionen in drei und vier Dimensionen mittels geeigneter Fourier-Entwicklungen explizit konstruiert. Das Konzept der Holomorphie ist verbunden mit der Lösung verallgemeinerter Cauchy-Riemann Systeme, deren Funktionswerte reellen Quaternionen bzw. reduzierte Quaternionen sind. In expliziter Form werden orthogonale holomorphe Funktionensysteme konstruiert, die Lösungen des Riesz-Systems bzw. des Moisil-Teodorescu Systems über zylindrischen Gebieten im R3, sowie Lösungen des Riesz-Systems in Kugeln des R4 sind. Um konkrete Anwendungen auf Randwertprobleme realisieren zu können wird eine orthogonale Zerlegung eines Rechts-Quasi-Hilbert-Moduls komplex-quaternionischer Funktionen unter gegebenen Bedingungen studiert. Die Ergebnisse werden auf die Behandlung von Maxwell-Gleichungen mit zeitvariabler elektrischer Dielektrizitätskonstante und magnetischer Permeabilität angewandtThe richness and widely applicability of the theory of holomorphic functions in complex analysis requires to perform a similar theory in higher dimensions. It has been developed by many researchers so far, especially in quaternionic analysis. Over the last years, it has been successfully applied to a vast array of problems in mathematical physics. The aim of this thesis is to study the structure of holomorphy in higher dimensions. First, a new concept of holomorphy is introduced based on the theory of right invertible operators, and not by means of an analogue of the Cauchy-Riemann operator as usual. This notion covers most of the well-known holomorphic structures in higher dimensions including real, complex, quaternionic, Clifford analysis, among others. In addition, from our operators a local approximation of a holomorphic function is attained by the Taylor type formula. In order to obtain the global approximation for holomorphic functions, the second part of the thesis deals with the construction of different systems of basis holomorphic functions in three and four dimensions by means of Fourier analysis. The concept of holomorphy is related to the null-solutions of generalized Cauchy-Riemann systems, which take either values in the reduced quaternions or real quaternions. We obtain several explicit orthogonal holomorphic function systems: solutions to the Riesz and Moisil-Teodorescu systems over cylindrical domains in R3, and solutions to the Riesz system over spherical domains in R4. Having in mind concrete applications to boundary value problems, we investigate an orthogonal decomposition of complex-quaternionic functions over a right quasi-Hilbert module under given conditions. It is then applied to the treatment of Maxwell’s equations with electric permittivity and magnetic permeability depending on the time variable
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Thakre, Varun [Verfasser], Viktor [Akademischer Betreuer] Pidstrygach, Thomas [Akademischer Betreuer] Schick, and Max [Akademischer Betreuer] Wardetzky. "Conformal Properties of Generalized Dirac Operator / Varun Thakre. Gutachter: Thomas Schick ; Max Wardetzky. Betreuer: Viktor Pidstrygach." Göttingen : Niedersächsische Staats- und Universitätsbibliothek Göttingen, 2013. http://d-nb.info/1044416165/34.
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Khochman, Abdallah. "Résonances et diffusion pour les opérateurs de Dirac et de Schrödinger magnétique." Thesis, Bordeaux 1, 2008. http://www.theses.fr/2008BOR13689/document.
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Le sujet de cette thèse est l’étude de certaines équations de physique mathématique. Dans un premier temps, on étudie les résonances et la fonction de décalage spectral pour les opérateurs de Dirac semi-classique et de Schrödinger magnétique en dimension 3. On dé?nit les résonances comme des valeurs propres d’un opérateur non-autoadjoint obtenu par distortion complexe. Pour l’opérateur de Dirac, on majore le nombre de résonances par O(h-3) où h ? 0 est le paramètre semi-classique. Dans le cas de Schrödinger magnétique, l’opérateur de référence génère des valeurs propres de multipli- cité in?nie plongées dans le spectre continu. Dans une couronne centrée en une de ces valeurs propres et de rayons (r, 2r), on établit une borne supérieure, quand r ? 0, du nombre de résonances. Une approximation de type Breit-Wigner de la dérivée de la fonction de décalage spectral en fonction des résonances et une formule de trace locale sont obtenues pour ces deux opérateurs. De plus, on prouve une formule asymptotique de Weyl pour la fonction de décalage spectral pour l’opérateur de Dirac avec un potentiel électro-magnétique. Dans un deuxième temps, on s’intéresse à l’opérateur de Dirac semi-classique en dimension 1 avec un potentiel ayant des limites constantes mais pas nécessairement les mêmes à ±8. En utilisant la méthode BKW complexe, on construit des solutions analytiques de l’opérateur de Dirac. On étudie la théorie de la di?usion en fonction des solutions entrantes et sortantes. On obtient une asymptotique semi-classique de la matrice de di?usion dans di?érents cas, notamment dans le cas où le paradoxe de Klein apparaît. Le calcul des valeurs propres et des résonances est aussi traité pour l’opérateur de Dirac semi-classique unidimensionnelIn this thesis, we consider equations of mathematical physics. First, we study the reso- nances and the spectral shift function for the semi-classical Dirac operator and the magnetic Schrö- dinger operator in three dimensions. We de?ne the resonances as the eigenvalues of a non-selfadjoint operator obtained by complex distortion. For the Dirac operator, we establish an upper bound O(h-3), as the semi-classical parameter h tends to 0, for the number of resonances. In the Schrödinger magne- tic case, the reference operator has in?nitely many eigenvalues of in?nite multiplicity embedded in its continuous spectrum. In a ring centered at one of this eigenvalues with radiuses (r, 2r), we establish an upper bound, as r tends to 0, of the number of the resonances. A Breit-Wigner approximation formula for the derivative of the spectral shift function related to the resonances and a local trace formula are obtained for the considered operators. Moreover, we prove a Weyl-type asymptotic of the SSF for the Dirac operator with an electro-magnetic potential. Secondly, we consider the semi-classical Dirac ope- rator on R with potential having constant limits, not necessarily the same at ±8. Using the complex WKB method, we construct analytic solutions for the Dirac operator. We study the scattering theory in terms of incoming and outgoing solutions. We obtain an asymptotic expansion, with respect to the semi-classical parameter h, of the scattering matrix in di?erent cases, in particular, in the case when the Klein paradox occurs. Quantization conditions for the resonances and for the eigenvalues of the one-dimensional Dirac operator are also obtained
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Majid, Shahn, and Andreas Cap@esi ac at. "Riemannian Geometry of Quantum Groups and Finite Groups with." ESI preprints, 2000. ftp://ftp.esi.ac.at/pub/Preprints/esi902.ps.
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Veloso, Diogo. "Seiberg-Witten theory on 4-manifolds with periodic ends." Thesis, Aix-Marseille, 2014. http://www.theses.fr/2014AIXM4781/document.
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Dans cette thèse on prouve des résultats analytiques sur la théorie cohomotopique de Seiberg-Witten pour des 4-variétes Riemanniennes Spinc(4) a bouts périodiques, (X,g,τ). Nos résultats montrent, que sur certaines conditions techniques en (X, g, τ ),, cette nouvelle version est cohérente et mène a des invariants de Seiberg-Witten.Premièrement, en utilisant le critère de Taubes pour des operateurs périodiques dans des variétes a bouts périodiques, on montre que pour une 4-varieté Riemmanienne a bouts périodiques (X, g) vérifiant certaines conditions topologiques, le Laplacian ∆+ : L2(Λ2+) → L2(Λ2+) est un opérateur de Fredholm. On prouve une décomposition de type Hodge pour des 1-formes de X, a poids positif.Ensuite on prouve, en assumant certaines conditions topologiques et courbure scalaire non-negative sur les bouts, que l'opérateur de Dirac associé a une connection périodique (ASD a l'infini) est Fredholm.Dans la deuxième partie de la thèse on démontre un isomorphisme entre le groupe de cohomologie de de Rham Hd1R(X,iR), et le groupe harmonique intervenant dans la decomposition de Hodge des 1-formes de X a poids positif. On prouve l'existence de deux séquences exactes courtes liant le groupe de jauge de l'espace de modules de Seiberg-Witten et le groupe de cohomologie H1(X, 2πiZ).Dans la troisième partie on prouve les principaux résultats: la coercitivité de l'application de Seiberg-Witten et la compacité de l'espace de moduli pour une 4-varieté a bouts périodiques (X, g, τ ), vérifiant les conditions mentionnées plus haut.Finalment, utilisant la coercivité, on montre l'existence d'un invariant cohomotopique de type Seiberg- Witten type associé a (X, g, τ )In this thesis we prove analytic results about a cohomotopical Seiberg-Witten theory for a Riemannian, Spinc(4) 4-manifold with periodic ends, (X,g,τ) . Our results show that, under certain technical assumptions on (X, g, τ ), this new version is coher- ent and leads to Seiberg-Witten type invariants for this new class of 4-manifolds.First, using Taubes criteria for end-periodic operators on manifolds with periodic ends, we show that, for a Riemannian 4-manifold with periodic ends (X, g), verifying certain topological conditions, the Laplacian ∆+ : L2(Λ2+) → L2(Λ2+) is a Fredholm operator. This allows us to prove an important Hodge type decomposition for positively weighted Sobolev 1-forms on X.We prove, assuming non-negative scalar curvature on each end and certain technical topological conditions, that the associated Dirac operator associated with an end-periodic connection (which is ASD at infinity) is Fredholm.In the second part of the thesis we establish an isomorphism between be- tween the de Rham cohomology group, Hd1R(X,iR) (which is a topological in- variant of X) and the harmonic group intervening in the above Hodge type decomposition of the space of positively weighted 1-forms on X. We also prove two short exact sequences relating the gauge group of our Seiberg-Witten moduli problem and the cohomology group H1(X, 2πiZ).In the third part, we prove our main results: the coercivity of the Seiberg-Witten map and compactness of the moduli space for a 4-manifold with periodic ends (X,g,τ) verifying the above conditions.Finally, using our coercitivity property, we show that a Seiberg-Witten type cohomotopy invariant associated to (X, g, τ ) can be defined
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Zalczer, Sylvain. "Propriétés spectrales de modèles de graphène périodique et désordonné." Thesis, Toulon, 2020. http://www.theses.fr/2020TOUL0003.
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Cette thèse traite de différents aspects de la théorie spectrale d’opérateurs utilisés pour modéliser le graphène. Elle est constituée de deux parties. La première traite du cas périodique. Je commence par présenter la théorie générale des systèmes périodiques. J’introduis ensuite les différents modèles de graphène en les comparant. Enfin, je m’intéresse à différentes façons de rendre le graphène semi-conducteur. Je fais en particulier une étude de nanorubans de divers types et présente un résultat d’ouverture d’une lacune spectrale pour un opérateur pseudo-différentiel. La deuxième partie traite du cas désordonné. Je commence par présenter la théorie générale des opérateurs aléatoires. J’explique ensuite succinctement l’analyse multi-échelles qui est la méthode permettant de montrer le résultat essentiel de cette théorie, appelé localisation d’Anderson. Enfin, je donne la preuve de cette localisation pour un modèle de graphène ainsi qu’un résultat sur la densité d’états intégréeThis thesis deals with various aspects of spectral theory of operators used to model graphene. It is made of two parts.The first parts deals with the periodic case. I begin by presenting a general theory of periodic systems. I introduce then different models of graphene and compare them. Finally, I look at various ways to make graphene a semiconductor. In particular, I study different types of nanoribbons and I give a result of gap opening for a pseudodifferential operator. The second part deals with the disordered case. I begin by presenting a general theory of random operators. Then, I briefly explain multiscale analysis, which is the method used to prove the main result of this theory, which is called Anderson localization. Finally, I give a proof of this localization for a model of graphene and a result on the integrated density of states
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Dubuisson, Clement. "Etude du spectre discret de perturbations d'opérateurs de la physique mathématique." Thesis, Bordeaux, 2014. http://www.theses.fr/2014BORD0127/document.
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Le but de cette thèse est d’obtenir des informations sur le spectre discret d’opérateurs non auto-adjoints définis par des perturbations relativement compactes d’opérateurs auto-adjoints. Ces opérateurs auto-adjoints sont choisis parmi les opérateurs classiques de mécanique quantique. Il s’agit des opérateurs de Dirac, de Klein-Gordon et le laplacien fractionnaire qui généralise l’opérateur de Schrödinger habituellement considéré pour de tels problèmes. La principale méthode utilisée ici relève d’un théorème d’analyse complexe donnant une condition de type Blaschke sur les zéros d’une fonction holomorphe du disque unité. Cette condition traduit lecomportement des valeurs propres de l’opérateur perturbé sous forme d’inégalités de type Lieb-Thirring. Une autre méthode venant d’analyse fonctionnelle a été employée pour obtenir de telles inégalités et les deux méthodes sont comparées entre ellesThe topic of this thesis concerns the discrete spectrum of non-selfadjoint operators defined by relatively compact perturbation of selfadjoint operators. These selfadjoint operators are choosen among classical operators of quantum mechanics. These areDirac operator, Klein-Gordon operator, and the fractional Laplacian who generalize the Schrödinger operator. The main method is based on a theorem of complex analysis which gives Blaschke-type condition on the zeros of a holomorphic function on the unit disc. This Blaschke condition gives the information on the behaviour of eigenvalues of the perturbed operator by mean of Lieb-Thirring-type inequalities. Another method using functional analysis is also used to obtain these kind of inequalities and both methods are compared to each other
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Sok, Jérémy. "Etude d'un modèle de champ moyen en électrodynamique quantique." Phd thesis, Université Paris Dauphine - Paris IX, 2014. http://tel.archives-ouvertes.fr/tel-01070652.
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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).
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Özugurel, Umut Deniz [Verfasser], and Gernot [Akademischer Betreuer] Münster. "Polynomial preconditioning of the Dirac-Wilson operator of the N =1 SU(2) supersymmetric Yang-Mills theory / Umut Deniz Özugurel ; Betreuer: Gernot Münster." Münster : Universitäts- und Landesbibliothek Münster, 2014. http://d-nb.info/1138283053/34.
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Witzel, Oliver. "Non-Hermitian polynomial hybrid Monte Carlo." Doctoral thesis, Humboldt-Universität zu Berlin, Mathematisch-Naturwissenschaftliche Fakultät I, 2008. http://dx.doi.org/10.18452/15805.
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In dieser Dissertation werden algorithmische Verbesserungen und Varianten für Simulationen der zwei-Flavor Gitter QCD mit dynamischen Fermionen studiert. Der O(a)-verbesserte Dirac-Wilson-Operator wird im Schrödinger Funktional mit einem Update des Hybrid Monte Carlo (HMC)-Typs verwendet. Sowohl der Hermitische als auch der nicht-Hermitische Operator werden betrachtet. Für den Hermitischen Dirac-Wilson-Operator untersuchen wir die Vorteile des symmetrischen gegenüber dem asymmetrischen Gerade-Ungerade-Präkonditionierens, wie man von einem mehr Zeitskalen-Integrator profitieren kann, sowie die Auswirkungen der kleinsten Eigenwerte auf die Stabilität des HMC Algorithmus. Im Fall des nicht-Hermitischen Operators leiten wir eine (semi)-analytische Schranke für das Spektrum her und zeigen eine Methode, um Informationen über den spektralen Rand zu gewinnen, indem wir komplexe Eigenwerte mit dem Lanczos-Algorithmus abschätzen. Diese spektralen Ränder erlauben es, Vorzüge des symmetrischen Gerade-Ungerade-Präkonditionierens oder den Effekt des Sheikholeslami-Wohlert-Terms für das Spektrum des nicht-Hermitischen Operators zu zeigen. Unter Verwendung der Informationen des spektralen Randes konstruieren wir angepasste, komplexe, skalierte und verschobene Tschebyschow Polynome zur Approximation des inversen Dirac-Wilson-Operators. Basierend auf diesen Polynomen entwickeln wir eine neue HMC-Variante, genannt nicht-Hermitischer polynomialer Hybrid Monte Carlo (NPHMC). Sie erlaubt, vom Importance Sampling unter Kompensation mit einem Gewichtungsfaktor abzuweichen. Zudem wird eine Erweiterung durch Anwendung des Hasenbusch-Tricks abgeleitet. Erste Größen der Leistungsfähigkeit, die die Abhängingkeit von den Eingabeparametern als auch einen Vergleich mit unserem Standard-HMC zeigen, werden präsentiert. Im Vergleich der beiden ein-Pseudofermion-Varianten ist der neue NPHMC etwas besser; eine eindeutige Aussage im Fall der zwei-Pseudofermion-Variante ist noch nicht möglich.In this thesis algorithmic improvements and variants for two-flavor lattice QCD simulations with dynamical fermions are studied using the O(a)-improved Dirac-Wilson operator in the Schrödinger functional setup and employing a hybrid Monte Carlo-type (HMC) update. Both, the Hermitian and the Non-Hermitian operator are considered. For the Hermitian Dirac-Wilson operator we investigate the advantages of symmetric over asymmetric even-odd preconditioning, how to gain from multiple time scale integration as well as how the smallest eigenvalues affect the stability of the HMC algorithm. In case of the non-Hermitian operator we first derive (semi-)analytical bounds on the spectrum before demonstrating a method to obtain information on the spectral boundary by estimating complex eigenvalues with the Lanzcos algorithm. These spectral boundaries allow to visualize the advantage of symmetric even-odd preconditioning or the effect of the Sheikholeslami-Wohlert term on the spectrum of the non-Hermitian Dirac-Wilson operator. Taking advantage of the information of the spectral boundary we design best-suited, complex, scaled and translated Chebyshev polynomials to approximate the inverse Dirac-Wilson operator. Based on these polynomials we derive a new HMC variant, named non-Hermitian polynomial Hybrid Monte Carlo (NPHMC), which allows to deviate from importance sampling by compensation with a reweighting factor. Furthermore an extension employing the Hasenbusch-trick is derived. First performance figures showing the dependence on the input parameters as well as a comparison to our standard HMC are given. Comparing both algorithms with one pseudo-fermion, we find the new NPHMC to be slightly superior, whereas a clear statement for the two pseudo-fermion variants is yet not possible.
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Santos, Júnior Valdecir Alves dos. "Representação Tipo Weierstrass para Superfícies Imersas em Espaços de Heisenberg." Universidade Federal da Paraíba, 2011. http://tede.biblioteca.ufpb.br:8080/handle/tede/7364.
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In this work we obtain Weierstrass-type representations for immersed surfaces in Heisenberg space, endowed with a left-invariant metric. We will consider the Riemannian and Lorentzian case. We will define two complex functions (spinors) satisfying a linear Dirac-type equation, obtaining thus a representation for immersed surfaces with prescribed mean curvature. The same will enable us write a representation of minimal immersion in terms of a harmonic Gauss map.
Neste trabalho obtemos uma representações tipo Weierstrass para superfícies imersas no espaço de Heisenberg, dotado com uma métrica invariante à esquerda. Consideraremos os casos Riemanniano e Lorentziano. Definimos duas funções complexas (spinors), satisfazendo uma equação linear tipo Dirac que usamos para obter uma representação para superfícies imersas com curvatura média prescrita. A mesma possibilita escrever uma representação de imersões mínimas em termos de uma aplicação de Gauss harmônica.
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Flamencourt, Brice. "On some problems in spectral analysis, spin geometry and conformal geometry." Thesis, université Paris-Saclay, 2022. http://www.theses.fr/2022UPASM014.
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Cette thèse se divise en deux grandes parties. Dans la première, on s'intéresse à deux problèmes d'analyse spectrale portant sur la convergence des valeurs propres d'opérateurs à paramètres. D'une part, on considère l'opérateur de Schrödinger dans le plan, avec un potentiel singulier supporté par une courbe fermée Γ admettant un point de rebroussement. Ce potentiel s'écrit formellement −αδ(x−Γ), et l'on décrit le comportement du spectre de l'opérateur dans la limite α→∞. D'autre part, on étudie l'opérateur de Dirac qui apparaît dans le modèle MIT Bag, en le généralisant aux variétés spin. Lorsque le paramètre de masse de cet opérateur tend vers l'infini, on observe une convergence des valeurs propres. Dans la seconde partie, on discute différents problèmes de géométrie. On démontre tout d'abord des résultats de structure et de classification en dimension 3 pour une classe particulière de spineurs, appelés spineurs de Cauchy, qui apparaissent naturellement comme restrictions de spineurs parallèles à des hypersurfaces orientées de variétés spin. Enfin, on s'intéresse aux connexions de Weyl sur les variétés conformes. On définit les structures localement conformément produits (LCP) par la donnée d'une structure de Weyl fermée, non-exacte, non-plate et à holonomie réductible sur une variété conforme compacte. On analyse les variétés LCP afin d'initier une classificationThis thesis is divided into two main parts. In the first one, we focus on two problems of spectral analysis concerning the convergence of eigenvalues of operators with parameters. On the one hand, we consider the Schrödinger operator in the plane, with a singular potential supported by a closed curve Γ admitting a cusp. This potential is formally written −αδ(x−Γ), and we describe the behaviour of the spectrum of the operator as α→∞. On the other hand, we study the Dirac operator which appears in the MIT Bag model, by generalizing it from Euclidean spaces to spin manifolds. We observe a convergence of the eigenvalues of this operator when the mass parameter tends to infinity. In the second part, we discuss two different geometric problems. First, we prove structure and classification results in dimension 3 for a particular class of spinors, called Cauchy spinors, arising as restrictions of parallel spinors to oriented hypersurfaces of spin manifolds. Finally, we focus on Weyl connections on conformal manifolds. We define a locally conformally product (LCP) structure as a closed, non-exact, non-flat Weyl structure with reducible holonomy on a compact conformal manifold. We analyse the LCP manifolds in order to initiate a classification
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Desmonts, Christophe. "Surfaces à courbure moyenne constante via les champs de spineurs." Thesis, Université de Lorraine, 2015. http://www.theses.fr/2015LORR0073/document.
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Les travaux présentés dans cette thèse portent sur le rôle que peuvent jouer les différentes courbures extrinsèques d’une hypersurface dans l’étude de sa géométrie, en particulier dans le cas des variétés spinorielles. Dans un premier temps, nous nous intéressons au cas de la courbure moyenne et construisons une nouvelle famille de surfaces minimales non simplement connexes dans le groupe de Lie Sol3, en adaptant une méthode déjà utilisée par Daniel et Hauswirth dans Nil3 et utilisant les propriétés de l’application de Gauss d’une surface. Ensuite, nous démontrons le Théorème d’Alexandrov généralisé aux Hr-courbures dans l’espace euclidien Rn+1 et dans l’espace hyperbolique Hn+1 en testant un spineur adéquat dans des inégalités de type holographiques établies récemment par Hijazi, Montiel et Raulot. Grâce à ces inégalités, nous démontrons également l'Inégalité de Heintze-Karcher dans l'espace euclidien. Enfin, nous donnons des majorations extrinsèques de la première valeur propre de l’opérateur de Dirac des surfaces de S2 x S1(r) et des sphères de Berger Sb3 (τ) grâce aux restrictions de spineurs ambiants construits par Roth, et nous en caractérisons les cas d’égalitéIn this thesis we are interested in the role played by the extrinsic curvatures of a hypersurface in the study of its geometry, especially in the case of spin manifolds. First, we focus our attention on the mean curvature and construct a new family of non simply connected minimal surfaces in the Lie group Sol3, by adapting a method used by Daniel and Hauswirth in Nil3 based on the properties of the Gauss map of a surface. Then we give a new spinorial proof of the Alexandrov Theorem extended to all Hr-curvatures in the euclidean space Rn+1 and in the hyperbolic space Hn+1, using a well-chosen test-spinor in the holographic inequalities recently obtained by Hijazi, Montiel and Raulot. These inequalities lead to a new proof of the Heintze-Karcher Inequality as well. Finally we use restrictions of particular ambient spinor fields constructed by Roth to give some extrinsic upper bounds for the first nonnegative eigenvalue of the Dirac operator of surfaces immersed into S2 x S1(r) and into the Berger spheres Sb3 (τ), and we describe the equality cases
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Phillips, Michael James. "A random matrix model for two-colour QCD at non-zero quark density." Thesis, Brunel University, 2011. http://bura.brunel.ac.uk/handle/2438/5084.
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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.
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Desmonts, Christophe. "Surfaces à courbure moyenne constante via les champs de spineurs." Electronic Thesis or Diss., Université de Lorraine, 2015. http://www.theses.fr/2015LORR0073.
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Les travaux présentés dans cette thèse portent sur le rôle que peuvent jouer les différentes courbures extrinsèques d’une hypersurface dans l’étude de sa géométrie, en particulier dans le cas des variétés spinorielles. Dans un premier temps, nous nous intéressons au cas de la courbure moyenne et construisons une nouvelle famille de surfaces minimales non simplement connexes dans le groupe de Lie Sol3, en adaptant une méthode déjà utilisée par Daniel et Hauswirth dans Nil3 et utilisant les propriétés de l’application de Gauss d’une surface. Ensuite, nous démontrons le Théorème d’Alexandrov généralisé aux Hr-courbures dans l’espace euclidien Rn+1 et dans l’espace hyperbolique Hn+1 en testant un spineur adéquat dans des inégalités de type holographiques établies récemment par Hijazi, Montiel et Raulot. Grâce à ces inégalités, nous démontrons également l'Inégalité de Heintze-Karcher dans l'espace euclidien. Enfin, nous donnons des majorations extrinsèques de la première valeur propre de l’opérateur de Dirac des surfaces de S2 x S1(r) et des sphères de Berger Sb3 (τ) grâce aux restrictions de spineurs ambiants construits par Roth, et nous en caractérisons les cas d’égalitéIn this thesis we are interested in the role played by the extrinsic curvatures of a hypersurface in the study of its geometry, especially in the case of spin manifolds. First, we focus our attention on the mean curvature and construct a new family of non simply connected minimal surfaces in the Lie group Sol3, by adapting a method used by Daniel and Hauswirth in Nil3 based on the properties of the Gauss map of a surface. Then we give a new spinorial proof of the Alexandrov Theorem extended to all Hr-curvatures in the euclidean space Rn+1 and in the hyperbolic space Hn+1, using a well-chosen test-spinor in the holographic inequalities recently obtained by Hijazi, Montiel and Raulot. These inequalities lead to a new proof of the Heintze-Karcher Inequality as well. Finally we use restrictions of particular ambient spinor fields constructed by Roth to give some extrinsic upper bounds for the first nonnegative eigenvalue of the Dirac operator of surfaces immersed into S2 x S1(r) and into the Berger spheres Sb3 (τ), and we describe the equality cases
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Ginoux, Nicolas. "Dirac operators on Lagrangian submanifolds." Universität Potsdam, 2004. http://opus.kobv.de/ubp/volltexte/2005/562/.
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We study a natural Dirac operator on a Lagrangian submanifold of a Kähler manifold. We first show that its square coincides with the Hodge - de Rham Laplacian provided the complex structure identifies the Spin structures of the tangent and normal bundles of the submanifold. We then give extrinsic estimates
for the eigenvalues of that operator and discuss some examples.
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Axelsson, Andreas, and kax74@yahoo se. "Transmission problems for Dirac's and Maxwell's equations with Lipschitz interfaces." The Australian National University. School of Mathematical Sciences, 2002. http://thesis.anu.edu.au./public/adt-ANU20050106.093019.
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The aim of this thesis is to give a mathematical framework for scattering of electromagnetic waves by rough surfaces. We prove that the Maxwell transmission problem with a weakly Lipschitz interface,in finite energy norms, is well posed in Fredholm sense for real frequencies. Furthermore, we give precise conditions on the material constants ε, μ and σ and the frequency ω when this transmission problem is well posed. To solve the Maxwell transmission problem, we embed Maxwells equations in an elliptic Dirac equation. We develop a new boundary integral method to solve the Dirac transmission problem. This method uses a boundary integral operator, the rotation operator, which factorises the double layer potential operator. We prove spectral estimates for this rotation operator in finite energy norms using Hodge decompositions on weakly Lipschitz domains. To ensure that solutions to the Dirac transmission problem indeed solve Maxwells equations, we introduce an exterior/interior derivative operator acting in the trace space. By showing that this operator commutes with the two basic reflection operators, we are able to prove that the Maxwell transmission problem is well posed. We also prove well-posedness for a class of oblique Dirac transmission problems with a strongly Lipschitz interface, in the L_2 space on the interface. This is shown by employing the Rellich technique, which gives angular spectral estimates on the rotation operator.
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Levitt, Antoine. "Etude théorique et numérique de modèles non linéaires en mécanique quantique." Phd thesis, Université Paris Dauphine - Paris IX, 2013. http://tel.archives-ouvertes.fr/tel-00881031.
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Dans cette thèse, on étudie plusieurs modèles et problèmes issus de la mécanique quantique. Ces modèles interviennent naturellement en chimie quantique pour le calcul de la structure électronique de la matière. Ils présentent des difficultés théoriques liées aux problèmes d'existence de solutions et à leur calcul numérique. Cette thèse est une contribution à l'étude de ces problèmes.
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Calixto, Alexandre Pitangui [UNESP]. "Operador quaterniônico de Klein-Gordon-Dirac." Universidade Estadual Paulista (UNESP), 2002. http://hdl.handle.net/11449/92025.
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calixto_ap_me_sjrp.pdf: 2312676 bytes, checksum: 5de410aa23e48f51d5aa6d67e78c03b6 (MD5)Nesta dissertação é apresentada uma aproximação da Teoria de Variáveis Complexas de duas para quatro dimensões. Procura-se definir diferenciabilidade de funções quaterniônico, a partir da qual se estabelece uma relação com a teoria de regularidade de funções hipercomplexos [9]. Observa-se que após definir o operados quaterniônico T, é possível reescrever equações clássicas da Física de forma concisa, utilizando a definição de regularidade, que resulta na decomposição de uma equação diferencial de segunda ordem em duas equações diferenciais lineares de primeira ordem.
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Yang, Fangyun Ph D. Massachusetts Institute of Technology. "Dirac operators and monopoles with singularities." Thesis, Massachusetts Institute of Technology, 2007. http://hdl.handle.net/1721.1/41723.
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Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2007.Includes bibliographical references (p. 75-77).
This thesis consists of two parts. In the first part of the thesis, we prove an index theorem for Dirac operators of conic singularities with codimension 2. One immediate corollary is the generalized Rohklin congruence formula. The eta function for a twisted spin Dirac operator on a circle bundle over a even dimensional spin manifold is also derived along the way. In the second part, we study the moduli space of monopoles with singularities along an embedded surface. We prove that when the base manifold is Kahler, there is a holomorphic description of the singular monopoles. The compactness for this case is also proved.
by Fangyun Yang.
Ph.D.
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Savale, Nikhil Jr (Nikhil A. ). "Spectral asymptotics for coupled Dirac operators." Thesis, Massachusetts Institute of Technology, 2012. http://hdl.handle.net/1721.1/77804.
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Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2012.Cataloged from PDF version of thesis.
Includes bibliographical references (p. 137-139).
In this thesis, we study the problem of asymptotic spectral flow for a family of coupled Dirac operators. We prove that the leading order term in the spectral flow on an n dimensional manifold is of order r n+1/2 followed by a remainder of O(r n/2). We perform computations of spectral flow on the sphere which show that O(r n-1/2) is the best possible estimate on the remainder. To obtain the sharp remainder we study a semiclassical Dirac operator and show that its odd functional trace exhibits cancellations in its first n+3/2 terms. A normal form result for this Dirac operator and a bound on its counting function are also obtained.
by Nikhil Savale.
Ph.D.
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Calixto, Alexandre Pitangui. "Operador quaterniônico de Klein-Gordon-Dirac /." São José do Rio Preto : [s.n.], 2002. http://hdl.handle.net/11449/92025.
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Orientador: Manoel Ferreira Borges NetoResumo: Nesta dissertação é apresentada uma aproximação da Teoria de Variáveis Complexas de duas para quatro dimensões. Procura-se definir diferenciabilidade de funções quaterniônico, a partir da qual se estabelece uma relação com a teoria de regularidade de funções hipercomplexos [9]. Observa-se que após definir o operados quaterniônico T, é possível reescrever equações clássicas da Física de forma concisa, utilizando a definição de regularidade, que resulta na decomposição de uma equação diferencial de segunda ordem em duas equações diferenciais lineares de primeira ordem.
Mestre
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Anghel, Nicolae. "L²-index theorems for perturbed Dirac operators /." The Ohio State University, 1989. http://rave.ohiolink.edu/etdc/view?acc_num=osu1487598303839391.
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Zucca, Alessandro. "Dirac Operators on Quantum Principal G-Bundles." Doctoral thesis, SISSA, 2013. http://hdl.handle.net/20.500.11767/4108.
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In this thesis I discuss some results on the noncommutative (spin) geometry of quantum principal G-bundles. The first part of the thesis is devoted to the study of spectral triples over toral bundles; extending some recent results by L. Dabrowski and A. Sitarz, we introduce the notion of projectable spectral triple for T^n-bundles. Moreover, we work out twisted Dirac operators. We discuss, in particular, the application of these results to noncommutative tori. In the second part of the thesis, instead, we work out a method for constructing real spectral triples over cleft quantum principal G-bundles and we study the properties of these triples and their behaviour under gauge transformations.
Some of the results discussed in this thesis can also be found in the following papers:
arXiv:1305.6185
arXiv:1308.4738
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Afentoulidis-Almpanis, Spyridon. "Noncubic Dirac Operators for finite-dimensional modules." Electronic Thesis or Diss., Université de Lorraine, 2021. http://www.theses.fr/2021LORR0035.
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Cette thèse porte sur l’étude des opérateurs de Dirac non-cubiques dans le cadre de la théorie des représentations des groupes de Lie. Après avoir présenté des notions de la théorie de Lie et des algèbres de Clifford, nous rappelons les propriétés principales des opérateurs de Dirac cubiques D introduits par Kostant en 1999. Ces résultats ont rapidement suscité un vif intérêt. En particulier, à la fin des années 2000, Vogan introduit une cohomologie définie par l’opérateur de Kostant et suggère une classification cohomologique des représentations. La cohomologie de Dirac a été calculée pour diverses familles de représentations, telles que les séries discrètes, les modules Aq(>) ou les modules de dimension finie. Pour les modules de dimension finie, la cohomologie de Dirac coïncide avec le noyau de D. Il apparait que l’opérateur de Dirac de Kostant est une version algébrique d’un opérateur différentiel issu d’une famille continue d’opérateurs de Dirac géométriques introduits par Slebarski dans les années 1980 dans le cadre de fibrés au- dessus d’espaces homogènes G/H de groupes compacts. Ce qui distingue l’opérateur de Dirac de Kostant est qu’il est le seul membre de cette famille dont le carré, généralisant une formule de Parthasarathy, diffère de l’opérateur de Casimir à un scalaire près. Cette propriété a des applications importantes en théorie des représentations des groupes de Lie. Le carré des opérateurs de Dirac non-cubiques, i.e des autres membres de la famille d’opérateurs de Slebarski, a été calculé par Agricola qui a également établit des liens précis entre ces opérateurs non-cubiques et la théorie des cordes en physique. Par ailleurs, les opérateurs de Dirac non-cubiques sont des opérateurs différentiels invariants, et donc leur noyau est le siège de représentations (de dimension finie) de groupes compacts. Dans cette thèse nous étudions le noyau des opérateurs de Dirac non-cubiques, et nous montrons, sous certaines conditions sur les espaces homogènes G/H, que ce noyau contient le noyau de l’opérateur de Dirac cubique. Nous obtenons en fait une formule explicite pour le noyau que nous appliquons aux cas des algèbres de Lie classiques et des algèbres de Lie exceptionnelles. Nous constatons que certaines propriétés des opérateurs non-cubiques sont analogues à celles de l’opérateur de Dirac de Kostant, tel que l’indice. Nous déduisons également quelques observations sur les opérateurs de Dirac géométrique non-cubiquesThis thesis focuses on the study of noncubic Dirac operators within the framework of representation theory of Lie groups. After recalling basic notions of Lie theory and Clifford algebras, we present the main properties of cubic Dirac operators D introduced by Kostant in 1999. These results quickly aroused great interest. In particular, in the late 1990’s, Vogan introduced a cohomology defined by Kostant operator D and suggested a cohomological classification of representations. Dirac cohomology was computed for various families of representations, such as the discrete series, Aq(>) modules or finite dimensional representations. It turns out that for finite dimensional modules, Dirac cohomology coincides with the kernel of D. It appears that Kostant’s Dirac operator is an algebraic version of a specific member of a continuous family of geometric Dirac operators introduced by Slebarski in the mid 1980’s in the context of bundles over homogeneous spaces G/H of compact groups. What distinguishes the cubic Dirac operator is that it is the only member of this family whose square, generalizing Parthasarathy’s formula, differs from the Casimir operator up to a scalar. This property has important applications in representation theory of Lie groups. The square of the noncubic Dirac operators, i.e. of the other members of Slebarski’s family, was calculated by Agricola who also established precise links between these noncubic operators and string theory in physics. Actually, noncubic Dirac operators are invariant differential operators, and therefore their kernels define (finite-dimensional) representations of compact groups. In this thesis we study the kernel of noncubic Dirac operators, and we show that, under certain conditions on the homogeneous spaces G/H, the kernel contains the kernel of the cubic Dirac operator. We obtain an explicit formula for the kernel which we apply to the case of classical Lie algebras and of exceptional Lie algebras. We remark that some properties of noncubic operators are analogous to those of Kostant cubic Dirac operator, such as the index. We also deduce some observations on noncubic geometric Dirac operators
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Pfäffle, Frank [Verfasser]. "Eigenwertkonvergenz für Dirac-Operatoren / Frank Pfäffle." Aachen : Shaker, 2003. http://d-nb.info/1179023595/34.
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Nita, A. "Essential Self-Adjointness of the Symplectic Dirac Operators." Thesis, University of Colorado at Boulder, 2016. http://pqdtopen.proquest.com/#viewpdf?dispub=10108819.
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The main problem we consider in this thesis is the essential self-adjointness of the symplectic Dirac operators D and D constructed by Katharina Habermann in the mid 1990s. Her constructions run parallel to those of the well-known Riemannian Dirac operators, and show that in the symplectic setting many of the same properties hold. For example, the symplectic Dirac operators are also unbounded and symmetric, as in the Riemannian case, with one important difference: the bundle of symplectic spinors is now infinite-dimensional, and in fact a Hilbert bundle. This infinite dimensionality makes the classical proofs of essential self-adjointness fail at a crucial step, namely in local coordinates the coefficients are now seen to be unbounded operators on L2(Rn). A new approach is needed, and that is the content of these notes. We use the decomposition of the spinor bundle into countably many finite-dimensional subbundles, the eigenbundles of the harmonic oscillator, along with the simple behavior of D and D with respect to this decomposition, to construct an inductive argument for their essential self-adjointness. This requires the use of ancillary operators, constructed out of the symplectic Dirac operators, whose behavior with respect to the decomposition is transparent. By an analysis of their kernels we manage to deduce the main result one eigensection at a time.
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Araujo, Oslenne Nogueira de. "Estimativas para os autovalores do operador de Dirac." Universidade Federal do CearÃ, 2012. http://www.teses.ufc.br/tde_busca/arquivo.php?codArquivo=8444.
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CoordenaÃÃo de AperfeiÃoamento de Pessoal de NÃvel SuperiorEste trabalho tem como objetivo apresentar algumas estimativas para os autovalores do operador de Dirac em variedades Riemannianas Spin compactas com curvatura escalar positiva. Para isto, utilizaremos algumas ferramentas clÃssicas de geometria Riemanniana e algumas de suas propriedades tais como Ãlgebra de Clifford, grupos spin, conexÃes,derivada covariante e operador de Dirac.
The aim of this work is to present some estimates for the eigenvalues of the Dirac operator on compact Riemannian Spin manifolds with positive scalar curvature. For this, we use some tools of classical Riemannian geometry and some of its properties as Clifford algebra, spin groups, connections, covariant derivative and Dirac operator.
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Araújo, Oslenne Nogueira de. "Estimativas para os autovalores do operador de Dirac." reponame:Repositório Institucional da UFC, 2012. http://www.repositorio.ufc.br/handle/riufc/4076.
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ARAÚJO, Oslenne Nogueira de. Estimativas para os autovalores do operador de Dirac. 2012. 50 f. Dissertação (Mestrado em Matemática)- Centro de Ciências, Universidade Federal do Ceará, Programa de Pós-Graduação em Matemática, Fortaleza, 2012.Submitted by Rocilda Sales (rocilda@ufc.br) on 2012-11-27T13:17:08Z No. of bitstreams: 1 2012_dis_onaraújo.pdf: 350911 bytes, checksum: 13484f2e29b5af876a1526fb8dc26aa6 (MD5)
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The aim of this work is to present some estimates for the eigenvalues of the Dirac operator on compact Riemannian Spin manifolds with positive scalar curvature. For this, we use some tools of classical Riemannian geometry and some of its properties as Clifford algebra, spin groups, connections, covariant derivative and Dirac operator.
Este trabalho tem como objetivo apresentar algumas estimativas para os autovalores do operador de Dirac em variedades Riemannianas Spin compactas com curvatura escalar positiva. Para isto, utilizaremos algumas ferramentas clássicas de geometria Riemanniana e algumas de suas propriedades tais como álgebra de Clifford, grupos spin, conexões,derivada covariante e operador de Dirac.
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Zreik, Mahdi. "Spectral properties of Dirac operators on certain domains." Electronic Thesis or Diss., Bordeaux, 2024. http://www.theses.fr/2024BORD0085.
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Cette thèse se focalise sur l'étude spectrale des modèles de perturbations de l'opérateur de Dirac libre en dimensions 2 et 3.Le premier chapitre de cette thèse étudie la perturbation de l'opérateur de Dirac par une grande masse M, supportée sur un domaine. Notre objectif principal est d'établir, sous la condition d'une masse M suffisamment grande, la convergence de l'opérateur perturbé vers l'opérateur de Dirac avec la condition au bord MIT bag, au sens de la norme de la résolvante. Pour se faire, nous introduisons ce que nous appelons les opérateurs Poincaré-Steklov (PS) (comme un analogue des opérateurs Dirichlet-to-Neumann pour l'opérateur de Laplace) et les analysons d'un point de vue microlocal, afin de comprendre précisément le taux de convergence de la résolvante. D'une part, nous montrons que les opérateurs PS s'intègrent dans le cadre des opérateurs pseudodifférentiels et nous déterminons leurs symboles principaux. D'autre part, comme nous nous intéressons principalement aux grandes masses, nous traitons notre problème du point de vue semiclassique, où le paramètre semiclassique est h = M^{-1}. Enfin, en établissant une formule de Krein reliant la résolvante de l'opérateur perturbé à celle de l'opérateur MIT bag, et en utilisant les propriétés pseudodifférentielles des opérateurs PS combinées aux structures matricielles des symboles principaux, nous établissons la convergence requise avec un taux de convergence de O(M^{-1}.Dans le chapitre 2, nous définissons un voisinage tubulaire de la frontière d'un domaine régulier donné. Nous considérons la perturbation de l'opérateur de Dirac libre par une grande masse M, supportée dans ce voisinage d'épaisseur varepsilon:=M^{-1}. Notre objectif principal est d'étudier la convergence de l'opérateur de Dirac perturbé lorsque M tend vers l'infini. En comparaison avec la première partie, nous obtenons ici deux opérateurs limites MIT bag, qui agissent en dehors de la frontière. Il est intéressant de noter que le découplage de ces deux opérateurs MIT bag peut être considéré comme la version confinée de delta-interaction scalaire de Lorentz de l'opérateur de Dirac, supportée sur une surface fermée. La méthodologie suivie, comme au problème précédent, porte sur l'étude des propriétés pseudodifférentielles des opérateurs PS. Cependant, la nouveauté de ce problème réside dans le contrôle de ces opérateurs en suivant la dépendance du paramètre varepsilon, et par conséquent, dans la convergence lorsque varepsilon tend vers 0 et M tend vers l'infini. Avec ces ingrédients, nous prouvons que l'opérateur perturbé converge au sens de la norme de la résolvante vers l'opérateur de Dirac couplé à une delta-interaction scalaire de Lorentz.Dans le chapitre 3, nous généralisation une approximation de l'opérateur de Dirac tridimensionnel couplé à une combinaison singulière de delta-interactions électrostatiques et scalaires de Lorentz supportée sur une surface fermée, par un opérateur de Dirac avec un potentiel régulier localisé dans une couche mince contenant la surface. Dans les cas non-critiques et non-confinants, nous montrons que l'opérateur de Dirac perturbé régulier converge au sens de la résolvante forte vers la delta-interaction singulière de l'opérateur de Dirac.Dans le dernier chapitre, notre étude porte sur l'opérateur de Dirac bidimensionnel couplé à une delta-interaction électrostatique et scalaire de Lorentz. Nous traitons dans des espaces de Sobolev d'ordre un-demi l'auto-adjonction de certaines réalisations de ces opérateurs dans divers contextes de courbes. Le cas le plus important se présente lorsque les courbes considérées sont des polygones curvilignes. Sous certaines conditions sur les constantes de couplage, en utilisant la propriété de Fredholm de certains opérateurs intégraux de frontière, et en exploitant la forme explicite de la transformée de Cauchy sur des courbes non lisses, nous établissons l'auto-adjonction de l'opérateur perturbéThis thesis mainly focused on the spectral analysis of perturbation models of the free Dirac operator, in 2-D and 3-D space.The first chapter of this thesis examines perturbation of the Dirac operator by a large mass M, supported on a domain. Our main objective is to establish, under the condition of sufficiently large mass M, the convergence of the perturbed operator, towards the Dirac operator with the MIT bag condition, in the norm resolvent sense. To this end, we introduce what we refer to the Poincaré-Steklov (PS) operators (as an analogue of the Dirichlet-to-Neumann operators for the Laplace operator) and analyze them from the microlocal point of view, in order to understand precisely the convergence rate of the resolvent. On one hand, we show that the PS operators fit into the framework of pseudodifferential operators and we determine their principal symbols. On the other hand, since we are mainly concerned with large masses, we treat our problem from the semiclassical point of view, where the semiclassical parameter is h = M^{-1}. Finally, by establishing a Krein formula relating the resolvent of the perturbed operator to that of the MIT bag operator, and using the pseudodifferential properties of the PS operators combined with the matrix structures of the principal symbols, we establish the required convergence with a convergence rate of mathcal{O}(M^{-1}).In the second chapter, we define a tubular neighborhood of the boundary of a given regular domain. We consider perturbation of the free Dirac operator by a large mass M, within this neighborhood of thickness varepsilon:=M^{-1}. Our primary objective is to study the convergence of the perturbed Dirac operator when M tends to +infty. Comparing with the first part, we get here two MIT bag limit operators, which act outside the boundary. It's worth noting that the decoupling of these two MIT bag operators can be considered as the confining version of the Lorentz scalar delta interaction of Dirac operator, supported on a closed surface. The methodology followed, as in the previous problem study the pseudodifferential properties of Poincaré-Steklov operators. However, the novelty in this problem lies in the control of these operators by tracking the dependence on the parameter varepsilon, and consequently, in the convergence as varepsilon goes to 0 and M goes to +infty. With these ingredients, we prove that the perturbed operator converges in the norm resolvent sense to the Dirac operator coupled with Lorentz scalar delta-shell interaction.In the third chapter, we investigate the generalization of an approximation of the three-dimensional Dirac operator coupled with a singular combination of electrostatic and Lorentz scalar delta-interactions supported on a closed surface, by a Dirac operator with a regular potential localized in a thin layer containing the surface. In the non-critical and non-confining cases, we show that the regular perturbed Dirac operator converges in the strong resolvent sense to the singular delta-interaction of the Dirac operator. Moreover, we deduce that the coupling constants of the limit operator depend nonlinearly on those of the potential under consideration.In the last chapter, our study focuses on the two-dimensional Dirac operator coupled with the electrostatic and Lorentz scalar delta-interactions. We treat in low regularity Sobolev spaces (H^{1/2}) the self-adjointness of certain realizations of these operators in various curve settings. The most important case in this chapter arises when the curves under consideration are curvilinear polygons, with smooth, differentiable edges and without cusps. Under certain conditions on the coupling constants, using the Fredholm property of certain boundary integral operators, and exploiting the explicit form of the Cauchy transform on non-smooth curves, we achieve the self-adjointness of the perturbed operator
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Wittmann, Anja [Verfasser], and Sebastian [Akademischer Betreuer] Goette. "Eta-forms and adiabatic limits for fibrewise Dirac operators with varying kernel dimension = Eta-Formen und adiabatische Limiten für faserweise Dirac Operatoren mit variierender Kern-Dimension." Freiburg : Universität, 2016. http://d-nb.info/1122647476/34.
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Li, Liangpan. "Local spectral asymptotics and heat kernel bounds for Dirac and Laplace operators." Thesis, Loughborough University, 2016. https://dspace.lboro.ac.uk/2134/23004.
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In this dissertation we study non-negative self-adjoint Laplace type operators acting on smooth sections of a vector bundle. First, we assume base manifolds are compact, boundaryless, and Riemannian. We start from the Fourier integral operator representation of half-wave operators, continue with spectral zeta functions, heat and resolvent trace asymptotic expansions, and end with the quantitative Wodzicki residue method. In particular, all of the asymptotic coefficients of the microlocalized spectral counting function can be explicitly given and clearly interpreted. With the auxiliary pseudo-differential operators ranging all smooth endomorphisms of the given bundle, we obtain certain asymptotic estimates about the integral kernel of heat operators. As applications, we study spectral asymptotics of Dirac type operators such as characterizing those for which the second coefficient vanishes. Next, we assume vector bundles are trivial and base manifolds are Euclidean domains, and study non-negative self-adjoint extensions of the Laplace operator which acts component-wise on compactly supported smooth functions. Using finite propagation speed estimates for wave equations and explicit Fourier Tauberian theorems obtained by Yuri Safarov, we establish the principle of not feeling the boundary estimates for the heat kernel of these operators. In particular, the implied constants are independent of self-adjoint extensions. As a by-product, we affirmatively answer a question about upper estimate for the Neumann heat kernel. Finally, we study some specific values of the spectral zeta function of two-dimensional Dirichlet Laplacians such as spectral determinant and Casimir energy. For numerical purposes we substantially improve the short-time Dirichlet heat trace asymptotics for polygons. This could be used to measure the spectral determinant and Casimir energy of polygons whenever the first several hundred or one thousand Dirichlet eigenvalues are known with high precision by other means.