Cette thèse a pour objet l'étude de la théorie des transitions de phases dans des systèmes désordonnés en dimension deux en utilisant les outils des théories conformes. Le premier chapitre est consacré a un exposé des différentes méthodes et techniques de calcul en théories conformes. Le deuxième chapitre est une présentation des différents types de systèmes désordonnés qui seront etudiés. On y trouvera aussi un bref resumé des résultats les plus connus dans l'étude de ces systèmes. L'application des méthodes de l'invariance conforme aux systèmes désordonnés se fera dans les chapitres 4, 5 et 6, ou l'on calcule les effets que produit un désordre faible dans les modèles d'Ising, de Potts et multicritiques respectivement. Finalement, dans le dernier chapitre, nous analysons les effets du désordre sur certains systèmes qui ont une transition de phases du premier ordre.

Nous présentons ici, l'analyse détaillée et l'étude de la stabilité de nouvelles solutions de type vortex dans le secteur bosonique de la théorie électrofaible. Les nouvelles solutions généralisent le plongement des solutions d'Abrikosov-Nielsen-Olesen dans la théorie électrofaible et reproduisent les résultats précédemment connus. Les vortex, génériquement porteurs d'un courant électrique, sont constitués d'un coeur massif de bosons chargés W entouré d'une superposition non-linéaire de champs Z et Higgs. Au loin la solution est purement électromagnétique avec un potentiel de Biot et Savart. Les solutions sont génériques de la théorie et existent en particulier pour les valeurs expérimentales des constantes de couplage. Il est en particulier démontré que le courant dont l'échelle typique est le milliard d'Ampères peut être arbitrairement grand. Dans un second temps la stabilité linéaire des vortex supraconducteurs vis-à-vis des perturbations génériques est considérée. Le spectre de l'opérateur de fluctuations est étudié qualitativement. Lorsque des modes instables sont détectés, ils sont explicitement construits ainsi que leurs relations de disperion. La plupart des modes instables sont supprimés par une périodisation du vortex. Il subsiste cependant un unique mode instable homogène. On peut espérer qu'un tel mode puisse être supprimé par des effets de courbure si une portion de vortex est refermée afin de former une boucle stabilisée par le courant électrique.

Le sujet de cette thèse est d'étudier la dynamique quantique d'une particule évoluant dans le plan sous l'influence de champs magnétique et électrique croisés. Dans le cas où ce système est actionné par un flux Aharonov-Bohm dépendant du temps, nous présenterons un théorème adiabatique basé sur une analyse spectrale fine en l'absence d'un potentiel électrique. Pour le cas sans champ extérieur et avec un petit potentiel électrique, nous présentons deux résultats. Premièrement, nous prouvons pour des potentiels arbitraires que la dynamique effective donne une approximation au premier ordre pour des temps longs. Ensuite, nous montrons que pour une classe de potentiels lisses et petits, nous pouvons construire une constante du mouvement non triviale. Pour cela, nous prouvons que l'hamiltonien est unitairement équivalent à un hamiltonien effectif commutant avec l'observable de l'énergie cinétique. Pour démontrer cela, nous utilisons un algorithme de diagonalisation partielle.

W.P. Allis. / "June 13, 1956." "Most of this report is identical with material prepared for Handbuch der Physik, volume XXI, 1956." / Bibliography: p. 98-100. / Army Signal Corps Contract DA36-039-sc-64637 Project 102B Dept. of the Army Projects 3-99-10-022 DA3-99-10-000

William H. Kautz. / "April 23, 1952." / Bibliography: p. 176-177. / Army Signal Corps Contract No. DA36-039 sc-100 Project No. 8-102B-0. Dept. of the Army Project No. 3-99-10-022.

This work is focused on the different supersymmetric extensions of the Landau model. We aim to fully solve each model and describe its energy levels, wavefunctions, Hilbert space and define a norm on it, as well as find symmetry generators and transformations with respect to them. Several possible generalizations were considered before. It was found for Landau model on the so called Superflag manifold as well as planar Superflag and Superplane Landau models that standard norm on the Hilbert space is not positive definite. Later for planar cases it was found that it is possible to fix this by introducing a new norm which will be invariant and positive definite. Surprisingly this procedure brings up "hidden" symmetries for the known super Landau models. In the dissertation we apply the same procedure for Landau model on superpshere and Superflag manifolds. It turns out that superpsherical Landau model is equivalent to the Superflag model with one of the parameters fixed. Because the model on superpshere can be recovered from the Superflag we will do calculations of corrected norm only for the Superflag. After this we develop a different generalization of the Superplane Landau model. Starting with Lagrangian in a superfield form we introduce two arbitrary functions of superfields K(Φ) and V(Φ) into the Lagrangian. We follow with the component form of Lagrangian. The quantization of the model is possible, and we will show that there is a reparametrization which turn equation of motion of the first scheme into the second set. Standard metric is again non-positive definite and we apply already known procedure to correct it. It will not be possible to solve Schrodinger equations in general with undefined K and V, so we consider one specific case which give us Landau model on a sphere with N = 2 supersymmetry, which put it apart from the superspherical Landau model, which have a superpshere for a target space but do not possess supersymmetry.

In this thesis, which consists of four papers, we study the discrete spectrum of Schrödinger operators in waveguides. In these domains the quadratic form of the Dirichlet Laplacian operator does not satisfy any Hardy inequality. If we include an attractive electric potential in the model or curve the domain, then bound states will always occur with energy below the bottom of the essential spectrum. We prove that a magnetic field stabilises the threshold of the essential spectrum against small perturbations. We deduce this fact from a magnetic Hardy inequality, which has many interesting applications in itself. In Paper I we prove the magnetic Hardy inequality in a two-dimensional waveguide. As an application, we establish that when a magnetic field is present, a small local deformation or a small local bending of the waveguide will not create bound states below the essential spectrum. In Paper II we study the Dirichlet Laplacian operator in a three-dimensional waveguide, whose cross-section is not rotationally invariant. We prove that if the waveguide is locally twisted, then the lower edge of the spectrum becomes stable. We deduce this from a Hardy inequality. In Paper III we consider the magnetic Schrödinger operator in a three-dimensional waveguide with circular cross-section. If we include an attractive potential, eigenvalues may occur below the bottom of the essential spectrum. We prove a magnetic Lieb-Thirring inequality for these eigenvalues. In the same paper we give a lower bound on the ground state of the magnetic Schrödinger operator in a disc. This lower bound is used to prove a Hardy inequality for the magnetic Schrödinger operator in the original waveguide setting. In Paper IV we again study the two-dimensional waveguide. It is known that if the boundary condition is changed locally from Dirichlet to magnetic Neumann, then without a magnetic field bound states will occur with energies below the essential spectrum. We however prove that in the presence of a magnetic field, there is a critical minimal length of the magnetic Neumann boundary condition above which the system exhibits bound states below the threshold of the essential spectrum. We also give explicit bounds on the critical length from above and below. / QC 20101007

In type II noncommutative geometry, the geometry on a C∗-algebra A is given by an unbounded Breuer–Fredholm module (ρ,N,D) over A. Here ρ:A→N is a ∗-homomorphism from A to the semi-finite von Neumann algebra N⊂B(H), and D is an unbounded Breuer–Fredholm operator affiliated with N that satisfies certain axioms. Each Breuer–Fredholm module assigns an index to a given element in the K-theory of A. The Breuer–Fredholm index provides a real-valued pairing between the K-homology and the K-theory of A. When N=B(H), a construction of Jaffe-Lesniewski-Osterwalder associates to the module (ρ,N,D) a cocycle in the entire cyclic cohomology group of A for D is theta-summable. The JLO character and the K-theory character intertwine the K-theoretical pairing with the pairing of entire cyclic theory. If (ρ,N,F) is a finitely summable bounded Breuer–Fredholm module, Benameur-Fack defined a cocycle generalizing the Connes's cocycle for bounded Fredholm modules. On the other hand, given a finitely-summable unbounded Breuer–Fredholm module, there is a canonically associated bounded Breuer–Fredholm module. The first main result of this thesis extends the JLO theory to Breuer–Fredholm modules (possibly N does not equal B(H)) in the graded case, and proves that the JLO cocycle and Connes cocycle define the same class in the entire cyclic cohomology of A. This extends a result of Connes-Moscovici for Fredholm modules. An example of an unbounded Breuer–Fredholm module is given by the noncommutative space of G-connections due to Aastrup-Grimstrup-Nest. In their original work, the authors limit their construction to the case that the group G=U(1) or G=SU(2). Another main result of the thesis extends AGN’s construction to any connected compact Lie group G; and generalizes by considering connections defined on sequences of graphs, using limits of spectral triples. Our construction makes it possible to equip the module (ρ,N,D) with a Z_2-grading. The last part of this thesis studies the JLO character of the Breuer–Fredholm module of AGN. The definition of this Breuer–Fredholm module depends on a divergent sequence. A concrete condition on possible perturbations of the sequence ensuring that the resulting JLO class remains invariant is established. The condition implies a certain functoriality of AGN’s construction.

In type II noncommutative geometry, the geometry on a C∗-algebra A is given by an unbounded Breuer–Fredholm module (ρ,N,D) over A. Here ρ:A→N is a ∗-homomorphism from A to the semi-finite von Neumann algebra N⊂B(H), and D is an unbounded Breuer–Fredholm operator affiliated with N that satisfies certain axioms. Each Breuer–Fredholm module assigns an index to a given element in the K-theory of A. The Breuer–Fredholm index provides a real-valued pairing between the K-homology and the K-theory of A. When N=B(H), a construction of Jaffe-Lesniewski-Osterwalder associates to the module (ρ,N,D) a cocycle in the entire cyclic cohomology group of A for D is theta-summable. The JLO character and the K-theory character intertwine the K-theoretical pairing with the pairing of entire cyclic theory. If (ρ,N,F) is a finitely summable bounded Breuer–Fredholm module, Benameur-Fack defined a cocycle generalizing the Connes's cocycle for bounded Fredholm modules. On the other hand, given a finitely-summable unbounded Breuer–Fredholm module, there is a canonically associated bounded Breuer–Fredholm module. The first main result of this thesis extends the JLO theory to Breuer–Fredholm modules (possibly N does not equal B(H)) in the graded case, and proves that the JLO cocycle and Connes cocycle define the same class in the entire cyclic cohomology of A. This extends a result of Connes-Moscovici for Fredholm modules. An example of an unbounded Breuer–Fredholm module is given by the noncommutative space of G-connections due to Aastrup-Grimstrup-Nest. In their original work, the authors limit their construction to the case that the group G=U(1) or G=SU(2). Another main result of the thesis extends AGN’s construction to any connected compact Lie group G; and generalizes by considering connections defined on sequences of graphs, using limits of spectral triples. Our construction makes it possible to equip the module (ρ,N,D) with a Z_2-grading. The last part of this thesis studies the JLO character of the Breuer–Fredholm module of AGN. The definition of this Breuer–Fredholm module depends on a divergent sequence. A concrete condition on possible perturbations of the sequence ensuring that the resulting JLO class remains invariant is established. The condition implies a certain functoriality of AGN’s construction.

<p>In this thesis, which consists of four papers, we study the discrete spectrum of Schrödinger operators in waveguides. In these domains the quadratic form of the Dirichlet Laplacian operator does not satisfy any Hardy inequality. If we include an attractive electric potential in the model or curve the domain, then bound states will always occur with energy below the bottom of the essential spectrum. We prove that a magnetic field stabilises the threshold of the essential spectrum against small perturbations. We deduce this fact from a magnetic Hardy inequality, which has many interesting applications in itself.</p><p>In Paper I we prove the magnetic Hardy inequality in a two-dimensional waveguide. As an application, we establish that when a magnetic field is present, a small local deformation or a small local bending of the waveguide will not create bound states below the essential spectrum.</p><p>In Paper II we study the Dirichlet Laplacian operator in a three-dimensional waveguide, whose cross-section is not rotationally invariant. We prove that if the waveguide is locally twisted, then the lower edge of the spectrum becomes stable. We deduce this from a Hardy inequality.</p><p>In Paper III we consider the magnetic Schrödinger operator in a three-dimensional waveguide with circular cross-section. If we include an attractive potential, eigenvalues may occur below the bottom of the essential spectrum. We prove a magnetic Lieb-Thirring inequality for these eigenvalues. In the same paper we give a lower bound on the ground state of the magnetic Schrödinger operator in a disc. This lower bound is used to prove a Hardy inequality for the magnetic Schrödinger operator in the original waveguide setting.</p><p>In Paper IV we again study the two-dimensional waveguide. It is known that if the boundary condition is changed locally from Dirichlet to magnetic Neumann, then without a magnetic field bound states will occur with energies below the essential spectrum. We however prove that in the presence of a magnetic field, there is a critical minimal length of the magnetic Neumann boundary condition above which the system exhibits bound states below the threshold of the essential spectrum. We also give explicit bounds on the critical length from above and below.</p>