Dissertations / Theses on the topic 'Analyse microlocale et semi-classique'
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Prouff, Antoine. "Correspondance classique-quantique et application au contrôle d'équations d'ondes et de Schrödinger dans l'espace euclidien." Electronic Thesis or Diss., université Paris-Saclay, 2024. https://theses.hal.science/tel-04634673.
Wave and Schrödinger equations model a variety of phenomena, such as propagation of light, vibrating structures or the time evolution of a quantum particle. In these models, the high-energy asymptotics can be approximated by classical mechanics, as geometric optics. In this thesis, we study several applications of this principle to control problems for wave and Schrödinger equations in the Euclidean space, using microlocal analysis.In the first two chapters, we study the damped wave equation and the Schrödinger equation with a confining potential in the euclidean space. We provide necessary and sufficient conditions for uniform stability in the first case, or observability in the second one. These conditions involve the underlying classical dynamics which consists in a distorted version of geometric optics, due to the presence of the potential.Then in the third part, we analyze the quantum-classical correspondence principle in a general setting that encompasses the two aforementioned problems. We prove a version of Egorov's theorem in the Weyl--Hörmander framework of metrics on the phase space. We provide with various examples of application of this theorem for Schrödinger, half-wave and transport equations
Lablée, Olivier. "Autour de la dynamique semi-classique de certains systèmes complètement intégrables." Phd thesis, Grenoble 1, 2009. http://www.theses.fr/2009GRE10305.
The semi-classical dynamics of a pseudo-differential operator on a manifold is the quantum analogous of the classical flow of his main symbol on the manifold. This semi-classical dynamics is described by the Schrödinger equation of the operator whereas the classical Hamiltonian flow is given by the Hamilton's equations associated with the function. Thus the spectrum of the pseudo-differential operator enable to describe the general solutions of the associated Schrödinger equation. The long time behavior of these solutions remains in many ways mysterious. The semi-classical dynamics depends directly on the spectrum of the operator and consequently also on the underlying geometry into induced by the classical symbol. In this thesis, we first describe the long time semi-classical dynamics of an Hamiltonian in the one-dimensional case with a symbol function with no singularity or with non-degenerate elliptic singularity type : the associated fibers are closed elliptic orbits. The regular Bohr-Sommerfeld rules supply the spectrum of the operator. We are also interested in the elliptic case of the dimension 2 which leads to some discussion of numbers theory. Finally we consider the case of a one-dimensionnal pseudo-differential operator with a non-degenerate hyperbolic singularity : the singular fiber of in is a “ hyperbolic eight ” (this model is diffeomorphic to the Schrödinger operator with a double wells)
Lablée, Olivier. "Autour de la dynamique semi-classique de certains systèmes complètement intégrables." Phd thesis, Université Joseph Fourier (Grenoble), 2009. http://tel.archives-ouvertes.fr/tel-00439641.
Nectoux, Boris. "Analyse spectrale et analyse semi-classique pour l'étude de la métastabilité en dynamique moléculaire." Thesis, Paris Est, 2017. http://www.theses.fr/2017PESC1228/document.
This thesis is dedicated to the study of the sharp asymptotic behaviour in the low temperature regime of the exit event from a metastable domain $Omegasubset mathbb R^d$ (exit point and exit time) for the overdamped Langevin process. In practice, the overdamped Langevin dynamics can be used to describe for example the motion of the atoms of a molecule or the diffusion of interstitial impurities in a crystal. The obtention of sharp asymptotic approximations of the first exit point density in the small temperature regime is the main result of this thesis. These results justify the use of the Eyring-Kramers law to model the exit event. The Eyring-Kramers law is used for example to compute the transition rates between the states of a system in a kinetic Monte-Carlo algorithm in order to sample efficiently the state-to-state dynamics. The cornerstone of our analysis is the quasi stationary distribution associated with the overdamped Langevin dynamics in $Omega$. The proofs are based on tools from semi-classical analysis. This thesis is divided into three independent chapters. The first chapter (in French) is dedicated to an introduction to the mathematical results. The other two chapters (in English) are devoted to the precise statements and proofs
Raffaelli, Bernard. "Analyse semi-classique des phénomènes de résonance et d'absorption par des trous noirs." Phd thesis, Université Pascal Paoli, 2011. http://tel.archives-ouvertes.fr/tel-00653074.
Raffaelli, Bernard. "Analyse semi-classique des phénomènes de résonance et d’absorption par des trous noirs." Corte, 2011. https://tel.archives-ouvertes.fr/tel-00653074.
Beyond the mathematical definition of a black hole as a solution of Einstein equations in vacuum, there are some observational clues, as pointed out by Kip Thorne, from the first observation of the binary system Cygnus X1 to recent assumptions related to the presence of hypothetical supermassive black holes in the center of various galaxies, concerning their existence in our Universe and consequently encouraging their study. In physics, it is wellknown that in order to obtain information on interactions between fundamental particles, atoms, molecules, etc…, and on the structure of composite objects, we have to make collision experiments or, more precisely, scattering experiments. This is precisely the aim of this work. Indeed, studying how a black hole can interact with its environment, we should obtain fundamental information about those “invisible objects”. This work is also useful to understand the kind of signals one could detect by the future gravitational waves astronomy devices. This thesis is mainly focused on resonance and absorption phenomena by black holes. The originality of this study is about the use of a semiclassical method known as the “complex angular momentum theory”, which brings concepts like S matrix, Regge poles techniques, into high energy black hole physics as suggested implicitly by Chandrasekhar in the middle of the seventies. This approach allows us to have simple and quite intuitive physical interpretations of resonance and absorption phenomena related to the scattering of a scalar, massive or not, field by black holes
CHARLES, Laurent. "Aspects semi-classiques de la quantification géométrique." Phd thesis, Université Paris Dauphine - Paris IX, 2000. http://tel.archives-ouvertes.fr/tel-00001289.
Bouclet, Jean-Marc. "Distributions spectrales pour des operateurs perturbes." Phd thesis, Université de Nantes, 2000. http://tel.archives-ouvertes.fr/tel-00004025.
Stingo, Annalaura. "Problèmes d’existence globale pour les équations d’évolution non-linéaires critiques à données petites et analyse semi-classique." Thesis, Sorbonne Paris Cité, 2018. http://www.theses.fr/2018USPCD093.
In this thesis we study the problem of global existence of solutions to critical quasi-linear Klein-Gordon equations – or to critical quasi-linear coupled wave-Klein-Gordon systems – when initial data are small, smooth, decaying at infinity, in space dimension one or two. We first study this problem for Klein-Gordon equations with cubic non-linearities in space dimension one. It is known that, under a suitable structure condition on the non-linearity, the global well-posedness of the solution is ensured when initial data are small and compactly supported. We prove that this result holds true even when initial data are not localized in space but only mildly decaying at infinity, by combining the Klainerman vector fields’ method with a semi-classical micro-local analysis of the solution. The second and main contribution to the thesis concerns the study of the global existence of solutions to a quadratic quasilinear wave-Klein-Gordon system in space dimension two, again when initial data are small smooth and mildly decaying at infinity. We consider the case of a model non-linearity, expressed in terms of "nullforms". Our aim is to obtain some energy estimates on the solution when some Klainerman vector fieldsare acting on it, and sharp uniform estimates. The former ones are recovered making systematically use of normal forms’ arguments for quasi-linear equations, in their para-differential version. We derive the latter ones by deducing a system of ordinary differential equations from the starting partial differential system, this strategy maying leading us in the future to treat the case of the most general non-linearities
Quang, Sang Phan. "Monodromie d'opérateurs non auto-adjoints." Phd thesis, Université Rennes 1, 2012. http://tel.archives-ouvertes.fr/tel-00730517.
Le, Floch Yohann. "Théorie spectrale inverse pour les opérateurs de Toeplitz 1D." Phd thesis, Université Rennes 1, 2014. http://tel.archives-ouvertes.fr/tel-01065441.
Ingremeau, Maxime. "Ondes planes tordues et diffusion chaotique." Thesis, Université Paris-Saclay (ComUE), 2016. http://www.theses.fr/2016SACLS477/document.
This thesis deals with several problems of scattering theory in the semi-classical limit, that is to say, with properties of the generalised eigenfunctions of a Schrödinger operator at high frequencies. The generalised eigenfunctions of a Schrödinger operator on the Euclidean space, with a compactly supported smooth potential, may always be written as the sum of an incoming wave and an outgoing wave, plus a term which is negligible at infinity. The scattering matrix relates the incoming part with the outgoing part. The first part of this work deals with the spectrum of the scattering matrix. We show an equidistribution result for the eigenvalues of the scattering matrix, under the hypothesis that the sets of fixed points of some maps defined from the classical dynamics has measure zero. This result was previously known under the additional assumption that the classical dynamics has an empty trapped set.A second part of this work deals with the distorted plane waves, which are a particular family of generalized eigenfunctions of a Schrödinger operator, which can be written as the sum of a plane wave and a purely outgoing part. We make the hypothesis that the underlying classical dynamics has a hyperbolic trapped set, and that a certain topological pressure is negative. Under these assumptions, we obtain in the semiclassical limit a precise description of distorted plane waves as a convergent sum of Lagrangian states. In particular, we can deduce from this the semiclassical measure associated to distorted plane waves. If we furthermore assume that the manifold has non-positive curvature, and that the potential is zero, these Lagrangian states project on the base manifold without caustics. We deduce from this results on the C^l norms and on the nodal sets of distorted plane waves. We also obtain a lower bound on the number of nodal domains of the sum of two distorted plane waves with close enough incoming directions , for a small generic perturbation of a metric of negative curvature satisfying the topological pressure assumption
BONNAILLIE, Virginie. "Analyse mathématique de la supraconductivité dans un domaine à coins: méthodes semi-classiques et numériques." Phd thesis, Université Paris Sud - Paris XI, 2003. http://tel.archives-ouvertes.fr/tel-00005430.
Bonthonneau, Yannick. "Résonances du laplacien sur les variétés à pointes." Thesis, Paris 11, 2015. http://www.theses.fr/2015PA112141/document.
In this thesis, we study the resonances of the Laplace operator on cusp manifolds. They are manifolds whose ends are real hyperbolic cusps. The resonances were introduced by Selberg in the 50's for the constant curvature cusp surfaces. Their definition was later extended to the case of variable curvature by Lax and Phillips. The resonances are the poles of a meromorphic family of generalized eigenfunctions of the Laplace operator. They are associated to the continuous spectrum of the Laplace operator. To analyze this continuous spectrum, different directions of research are investigated.On the one hand, we obtain results on the localization of resonances. In particular, if the curvature is negative, for a generic set of metrics, they split into two sets. The first one is included in a band near the spectrum. The other is composed of resonances that are far from the spectrum. This leaves a log zone without resonances. On the other hand, we study the microlocal measures associated to certain sequences of spectral parameters. In particular we show that for some sequences of parameters that converge to the spectrum, but not too fast, the associated microlocal measure has to be the Liouville measure. This property holds when the curvature is negative
Richer, Dominic. "Transitions non-adiabatiques et dynamique de dissociation au-dessus du seuil critique en champ laser intense et ultra-court analyse semi-classique vs Calcul numérique exact." Mémoire, Université de Sherbrooke, 2003. http://savoirs.usherbrooke.ca/handle/11143/4606.
Richer, Dominic. "Transitions non-adiabatiques et dynamique de dissociation au-dessus du seuil critique en champ laser intense et ultra-court : analyse semi-classique vs Calcul numérique exact." Sherbrooke : Université de Sherbrooke, 2003.
Sun, Chenmin. "Contrôle et stabilisation pour des équations hyperboliques et dispersives." Thesis, Université Côte d'Azur (ComUE), 2018. http://www.theses.fr/2018AZUR4047/document.
In this thesis, we deal with the control and stabilization for certain hyperbolic and dispersive partial differential equations. The first part of this work is devoted to the stabilization of hyperbolic Stokes equation. The propagation of singularity for semi-classical Stokes system is established in Chapter 1. This will be done by adpating the strategy of Ivrii and Melrose-Sjöstrand. However,compared to the Laplace operator, the difficulty is caused by the pressure term which has non-trivial impact to solutions concentrated near the boundary. We apply parametrix construction to resolve the issue in elliptic and hyperbolic regions. We next adapte a fine micro-local decomposition for solutions concentrated near the glancing set. The impact of pressure to the solution is then well controled by geometric considerations. As a consequence of the main theorem in Chapter 1, we prove the stabilization of hyperbolic Stokes equation under geometric control condition in Chapter 2. The second part is devoted to the controllability of Kadomtsev–Petviashvili(KP in short) equations. In Chapter 3, the controllability in L 2 (T) from vertical strip is proved using semi-classical analysis. Additionally, a negative result for the controllability in L^2 (T) from horizontal strip is also showed. In Chapter 4, we prove the exact controllability of linear KP-I equation if the control input is added on a vertical domain. It is an interesting model in which the group velocity may degenerate. More generally, we have obtained the least dispersion needed to insure observability for fractional linear KP I equation. Finally in Chapter 5, we prove exact controllability and stabilization of KP-II equation and fifth order KP-II equation for any size of initial data in Sobolev spaces with additional partial compactness conditions. This extends the exact controllability for small data obtained in Chapter 3.compactness condition. This extends the exact controllability for small data obtained in Chapter 3
Spielfiedel, Annie. "Analyse spectroscopique d'un complexe collisionnel et redistribution du rayonnement." Grenoble 2 : ANRT, 1988. http://catalogue.bnf.fr/ark:/12148/cb37618683p.
Cassanas, Roch. "Hamiltoniens quantiques et symétries." Phd thesis, Université de Nantes, 2005. http://tel.archives-ouvertes.fr/tel-00009289.
Le, Masson Etienne. "Ergodicité et fonctions propres du laplacien sur les grands graphes réguliers." Phd thesis, Université Paris Sud - Paris XI, 2013. http://tel.archives-ouvertes.fr/tel-00866843.
Faraj, Ali. "Méthodes asymptotiques et numériques pour le transport quantique résonant." Toulouse 3, 2008. http://thesesups.ups-tlse.fr/363/.
Numerical methods to simulate resonant tunneling diodes are proposed. A decomposition of the wave functions, solution of the Schrödinger equation, in a resonant part and a non -resonant part gives, with a large frequency mesh, results in agreement with a nig number of frequency points computation. The real improvement was to adapt the algorithm to the unsteady case. An asymptotic analysis is performed on a steady Schrödinger-Poisson system. The semi-classical limit leads to different behaviours understood with the help of a spectral renormalisation and depending on the dimension of the space variable
Raymond, Nicolas. "Méthodes spectrales et théorie des cristaux liquides." Phd thesis, Université Paris Sud - Paris XI, 2009. http://tel.archives-ouvertes.fr/tel-00424859.
Le premier et principal aspect de ce travail concerne l'analyse semi-classique de la plus petite valeur propre $\la_1(B,\A)$ de la réalisation de Neumann de l'opérateur de Schrödinger magnétique $(i\nabla+B\A)^2$ dans le cas où le champ magnétique $\bbeta=\nabla\times\A$ n'est pas uniforme. Plus précisément, en dimension 2, nous établissons un développement asymptotique à deux termes de $\la_1(B,\A)$ lorsque $B$ tend vers l'infini et démontrons simultanément des résultats de localisation pour les premières fonctions propres correspondantes ; pour ce qui est du problème en dimension 3, nous étudions d'une part des estimations uniformes pour une famille de champs magnétiques d'intensité constante (en vue de l'application à une famille spéciale apparaissant à l'occasion de la théorie des cristaux liquides) et d'autre part nous nous plaçons dans des hypothèses génériques sur le champ magnétique et prouvons une majoration qui laisse conjecturer l'expression des deuxième et troisième termes du développement asymptotique.
Le deuxième aspect de cette thèse est l'étude de la transition de phase en théorie des cristaux liquides. Nous mettons en évidence une température critique pour la fonctionnelle de Landau-de Gennes qui permet de déterminer, lorsque certains coefficients de la fonctionnelle appelés constantes d'élasticité explosent, la phase dans laquelle se trouve le cristal liquide (nématique ou smectique). Par ailleurs, nous sommes amenés à introduire une nouvelle fonctionnelle (en imposant une condition de Dirichlet non homogène) en vue d'obtenir des informations plus quantitatives.
Ourmières-Bonafos, Thomas. "Quelques asymptotiques spectrales pour le Laplacien de Dirichlet : triangles, cônes et couches coniques." Thesis, Rennes 1, 2014. http://www.theses.fr/2014REN1S143/document.
This thesis deals with the spectrum of the Dirichlet Laplacian in various two or three dimensional domains. First, we consider asymptotically flat triangles and cones with small aperture. These problems admit a semi-classical formulation and we provide asymptotic expansions at any order for the first eigenvalues and the associated eigenfunctions. These type of results is already known for thin domains with smooth profiles. For triangles and cones, we show that the problem admits now two different scales. Second, we study a family of conical layers parametrized by their aperture. Again, we consider the semi-classical limit when the aperture tends to zero: We provide a two-term asymptotics of the first eigenvalues and we prove a localization result about the associated eigenfunctions. We also estimate, for each chosen aperture, the number of eigenvalues below the threshold of the essential spectrum
El, Hajj Raymond. "Etude mathématique et numérique de modèles de transport : application à la spintronique." Phd thesis, Université Paul Sabatier - Toulouse III, 2008. http://tel.archives-ouvertes.fr/tel-00342139.
Henkel, Carsten. "Réflexion et diffraction d'atomes lents par un miroir à onde évanescente." Phd thesis, Université Paris Sud - Paris XI, 1996. http://tel.archives-ouvertes.fr/tel-00006757.
El, Hajjj Raymond. "Etude mathématique et numérique de modèles de transport." Toulouse 3, 2008. http://thesesups.ups-tlse.fr/353/.
This thesis is decomposed into three parts. The main part is devoted to the study of spin polarized currents in semiconductor materials. An hierarchy of microscopic and macroscopic models are derived and analyzed. These models takes into account the spin relaxation and precession mechanisms acting on the spin dynamics in semiconductors. We have essentially two mechanisms : the spin-orbit coupling and the spin-flip interactions. We begin by presenting a semiclassical analysis (via the Wigner transformation) of the Schrödinger equation with spin-orbit hamiltonian. At kinetic level, the spinor Vlasov (or Boltzmann) equation is an equation of distribution function with 2x2 hermitian positive matrix value. Starting then from the spinor form of the Boltzmann equation with different spin-flip and non spin-flip collision operators and using diffusion asymptotic technics, different continuum models are derived. We derive drift-diffusion, SHE and Energy-Transport models of two-components or spin-vector types with spin rotation and relaxation effects. Two numerical applications are then presented : the simulation of transistor with spin rotational effect and the study of spin accumulation effect in inhomogenous semiconductor interfaces. In the second part, the diffusion limit of the linear Boltzmann equation with a strong magnetic field is performed. The Larmor radius is supposed to be much smaller than the mean free path. The limiting equation is shown to be a diffusion equation in the parallel direction while in the orthogonal direction, the guiding center motion is obtained. The diffusion constant in the parallel direction is obtained through the study of a new collision operator obtained by averages of the original one. Moreover, a correction to the guiding center motion is derived. .
Lassoued, Dhaou. "Fonctions presque-périodiques et Équations Différentielles." Phd thesis, Université Panthéon-Sorbonne - Paris I, 2013. http://tel.archives-ouvertes.fr/tel-00942969.
VU, NGOC San. "Systèmes intégrables semi-classiques: du local au global." Habilitation à diriger des recherches, 2003. http://tel.archives-ouvertes.fr/tel-00007113.
ROY, Nicolas. "Sur les déformations des systèmes complètement intégrables classiques et semi-classiques." Phd thesis, 2003. http://tel.archives-ouvertes.fr/tel-00003400.
Faure, F. "Approche géométrique de la limite semi-classique par les états cohérents et mécanique quantique sur le tore." Phd thesis, 1993. http://tel.archives-ouvertes.fr/tel-00383065.