| 11h15 - 12h15 | Mathieu Lewin (Cergy) |
Dérivation du
modèle de Bogoliubov Résumé: Nous considérons un système quantique de N bosons interagissant avec un potentiel d'intensité proportionnelle à 1/N. En supposant qu'il y a condensation de Bose-Einstein, nous donnons un développement à deux termes de l'énergie fondamentale et de l'énergie libre à température positive, dans la limite où N tend vers l'infini. Le premier terme est donné par le modèle de Hartree alors que le suivant est donné par la théorie de Bogoliubov. Nous donnons ensuite deux exemples: les atomes "bosoniques" et le gaz de Coulomb. Travail en collaboration avec Phan Thanh Nam (Cergy), Sylvia Serfaty (Paris 6) et Jan Philip Solovej (Copenhague). |
| Déjeuner | ||
| 14h - 15h | Georgi Popov (Nantes) | Invariants of
Isospectal Deformations and Spectral Rigidity Résumé: We consider a family of Laplace-Beltrami operators corresponding to a smooth deformation of Riemannian metrics on a compact manifold with or without boundary. We suppose that the initial metric is either completely integrable or close to a non-degenerate completely integrable metric (KAM system). If the deformation is isospectral we prove that the values of the corresponding Mather's α-function given by the average action on the KAM tori is constant along the deformation. As an application we obtain infinitesimal rigidity of Liouville billiard tables. The proof is based on a construction of quasi-modes associated with KAM tori. |
| 15h15 - 16h15 | Frédéric Hérau
(Nantes) |
Subelliptic
estimates for the inhomogeneous Boltzmann equation without cutoff Résumé: In this article we provide global subelliptic estimates for the Bolzmann equation without angular cutoff, and show that some global gain in the spatial coordinate is available although the corresponding operator is not elliptic in this direction. Due to the bad symbolic properties of the operator (in the microlocal sense), the proof uses the so-called Wick quantization and of some ideas coming from semi-classical analysis. This is a joint work with W.-X. Li (Wuhan) and R. Alexandre (Shanghai and Brest). |
| Semiclassical
analysis of discontinuous systems: Ray-Splitting Billiards Résumé: I will be looking at quantum systems that have metric jump-like discontinuities. For such systems it is expected that a ray-splitting billiard flow replaces the classical billiard flow. I will discuss some examples and counterexamples that show to which extent this is true. I will then present a theorem that allows to deduce quantum ergodicity from a property of a classical dynamics. (joint work with D. Jakobson and Y. Safarov) |
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| Déjeuner | ||
| 14h - 15h | Luc Hillairet (Orléans) | Spectre des
surfaces de translation Résumé: Les surfaces de translation sont des surfaces plates à singularités coniques qui possèdent de nombreuses propriétés. Elles apparaissent de façon naturelle dans plusieurs problèmes d'origine physique (lignes de flux magnétiques, théorie des cordes ...). Je présenterai quelques aspects de la théorie spectrale du laplacien plat défini sur ces surfaces, notamment les différentes extensions autoadjointes et les variations spectrales quand on se déplace dans l'espace des modules associé. |
| 15h15 - 16h15 | Frédéric Klopp
(Jussieu) |
Fermions
uni-dimensionnels en interaction dans un potentiel aléatoire Résumé: L'exposé sera consacré à un modèle simple de fermions uni-dimensionnels en interaction dans un milieu aléatoire. Nous considérerons N fermions dans un cube Λ de volume |Λ|. Nous supposerons que la densité ρ:=N/|Λ| est petite et que les interactions sont répulsives à courte portée. Nous donnerons une description précise de l'énergie fondamentale de ce système, ainsi que de l'état fondamental dans la limite |Λ|→∞. Collaboration avec N. Veniaminov (Paris 9) |
| 11h15 - 12h15 | Fabricio
Macià (Madrid) |
Comportement en
temps long des flots de Schrödinger complètement intégrables Résumé: Nous nous intéressons au comportement en temps long des solutions d'une équation de Schrödinger semi-classique sur le tore dont l'analogue classique est un système Hamiltonien complètement intégrable écrit en coordonnées action-angle. À cet effet, nous analysons les moyennes en temps des mesures semi-classiques associées aux solutions sur des intervalles dont la taille tend vers l'infini quand le paramètre semi-classique tend vers zéro. Nous présenterons des résultats précis sur la structure des limites de ces mesures, en particulier nous décrirons leur régularite et propagation ; finalement, nous montrerons comment ces propriétés dépendent des échelles de temps par rapport auxquelles nous moyennons. Il s’agit d’un travail en collaboration avec N.Anantharaman et C.Fermanian-Kammerer. |
| Déjeuner | ||
| 14h - 15h | François David
(Saclay) |
Random matrix
ensembles for models of spins decoherence Résumé: I present a class of random matrix ensembles relevant for the study of the dynamics of quantum decoherence for quantum spins. These ensembles generalize the standard GUE and GOE ensemble. For a single large spin, they lead to exact solutions for the dynamics of decoherence and for quantum diffusion, including non-markovian regimes. Generalizations to closed systems of quantum spins lead to interesting problems, involving free probabilities. |
| 15h15 - 16h15 | Riccardo Adami
(Turin) |
On the stability
of the ground states of the NLS on simple graphs Résumé: The introduction of the nonlinear Schrödinger equation (NLS) on graphs is motivated by several physical reasons, e.g., the study of propagation of waves in ramified structures in nonlinear media, or the analysis of the dynamics of the Bose-Einstein condensates in the presence of impurities. The related mathematical problem can be formulated as a system of as many NLS as edges in the graph, coupled by suitable self-adjoint matching conditions at the vertices. The analysis is only at its beginning and mainly concerns the structure of the family of ground states of the NLS on star graphs; however, some results show the occurence of phenomena that are unexpected and far from the corresponding results for the standard NLS. Among them, the appearance of symmetry-breaking bifurcations, and the absence of ground states in the case of free (i.e. Kirchhoff’s) infinite graphs, that show the need for an adaptation of the rearrangement method and of techniques related with concentration-compactness theory. We give examples of such phenomena and related adaptations in the case of star graphs with delta and delta-prime vertex conditions. This is part of a research project with Claudio Cacciapuoti (Bonn), Domenico Finco (Rome), and Diego Noja (Milan). |
| 11h15 - 12h15 | Luc Miller (Nanterre) | Sur les inégalités spectrales pour le contrôle des EDP linéaires: groupe de Schrödinger contre semigroupe de la chaleur Résumé: Des inégalités spectrales furent introduites en théorie du contrôle par David Russell et George Weiss en 1994 pour généraliser le test de contrôlabilité de Hautus à la dimension infinie. Elles constituent un outil efficace pour le contrôle de l'équation de Schrödinger linéaire en temps arbitraire au moyen d''un terme source localisé, comme démontré par Nicolas Burq et Maciej Zworski en 2004 grâce à l'unitarité de la transformée de Fourier dans les espaces de Hilbert. Elle permettent aussi d'analyser le filtrage suffisant pour discrétiser en espace cette équation, comme initié par Sylvain Ervedoza en 2008. Parallèlement s'est développée une approche de la contrôlabilité de l'équation de la chaleur linéaire en temps arbitraire au moyen d'un terme source localisé partant d'un autre type d'inégalités spectrales, introduit par Gilles Lebeau, suivant la stratégie itérative qu'il avait conçue avec Luc Robbiano en 1995. Cet exposé relira ces deux approches spectrales, comparera le contrôle du groupe de Schrödinger et le semigroupe de la chaleur au niveau de l'analyse fonctionnelle abstraite, et l'illustrera avec des exemples de problèmes d'EDP. Il s'agit d'une collaboration avec Thomas Duyckaerts. |
| Déjeuner | ||
| 14h - 15h | Benoît Grébert (Nantes) | Autour du théorème KAM pour les EDP. Résumé: Je présenterai un rapide panorama des résultats de type KAM existants dans le contexte des EDP. Cela m'amènera au Théorème que j'ai obtenu récemment avec H.Eliasson et S.Kuksin pour certaines EDP multidimensionnelles. |
| 15h15 - 16h15 | Henrik Ueberschär (Saclay) | Quantum Chaos for point scatterers on flat tori Résumé: The Laplacian with a delta potential on a flat torus is a popular model to study the transition between integrability and chaos in quantum systems. Whereas the dynamics of the associated classical system is pseudo-integrable, the quantum system is believed to exhibit features of quantised chaotic systems. I will discuss joint work with Zeév Rudnick and Pär Kurlberg on the statistical properties of the wave functions of this system. |
| 11h15 - 12h15 | Laurent Thomann (Nantes) Annulé. Remplacé par S. Nonnenmacher |
Résonances: de la diffusion quantique aux flots d'Anosov |
| Déjeuner | ||
| 14h - 15h | Joe Viola (Nantes) |
Resolvent bounds and spectral projections for partially elliptic quadratic operators. Résumé: We discuss upper bounds on the exponentially rapid resolvent growth of quadratic differential operators under a hypothesis of partial ellipticity. We also obtain much more precise information on the norms of spectral projections for these operators, and discuss how both these quantities are connected to the geometric properties of anisotropic weights for Bargmann-Fock-type spaces. Examples include the complex harmonic oscillator studied by Davies and Kramers-Fokker-Planck operators with quadratic potentials. |
| 15h15 - 16h15 | Sergey Morozov (Munich) |
High energy asymptotics of integrated density of states for multidimensional (almost-)periodic operators. Résumé: I will review some recent results and techniques concerning complete asymptotic expansion of the integrated density of states for large values of energy parameter. The results are valid for a large class of periodic and almost-periodic multidimensional selfadjoint operators. In particular, the case of magnetic Schrödinger operators will be considered. The talk is based on a joint work with Leonid Parnovski and Roman Shterenberg. |
| 11h15 - 12h15 | Zeév Rudnick (Tel-Aviv) |
Nodal Intersections Abstract: We study the fine structure of nodal lines for eigenfunctions of the Laplacian on a a surface by examining the number of intersection of the nodal lines with a fixed reference curve. It is expected that in many cases the number of these intersections is bounded above by the wave number k (the square root of the eigenvalue). Very little is known concerning lower bounds. For the flat torus, we prove the expected upper bound of k and give a lower bound of almost the same quality. To do so, we connect this problem to bounds on the Lp norms of the restriction of the eigenfunctions to the curve, and to a problem in Number Theory. (joint work with Jean Bourgain). |
| Déjeuner | ||
| 14h - 15h | Eric Séré (Paris-Dauphine) |
Kink solutions in a simplified model of Polyacetylene. Abstract: We consider a simplified model of Polyacetylene introduced by Su, Schrieffer and Cheeger in 1979, which belongs to the class of Peierls models at half-filling. In 1987 Kennedy and Lieb studied finite chains and proved that if the number N of nuclei is even, the energy has exactly two minimisers which are periodic of period 2, and are translates of one another by a translation of one unit in the lattice. We study rigorously the case of an odd number of atoms. We prove that if N is odd and converges to infinity, the global minimizer of the energy converges to a "kink" soliton in the infinite chain. This soliton is asymptotic to one of the periodic minimizers found by Kennedy-Lieb in one direction of the chain, and to the other solution in the other direction. This is joint work with Mauricio Garcia Arroyo. |
| 15h15 - 16h15 | Francis Nier (ENPC & Rennes) |
Asymptotique basse température pour des distributions quasi-stationnaires en domaine borné. Résumé: Après avoir introduit les notions de mesures quasi-stationnaires sur des ouverts et précisé les questions qui se posent naturellement sur ces objets à partir de la mise en oeuvre d'algorithmes de dynamique moléculaire, j'expliquerai comment l'analyse semi-classique des Laplaciens de Witten à bord permet de résoudre ces problèmes. Travail en commun avec T. Lelièvre |
| 11h15 - 12h15 | Jens Bolte (Royal Holloway) |
Many-particle quantum systems on graphs and Bose-Einstein condensation. Abstract: Quantum graphs have become popular systems in many areas of Physics and Mathematics, e.g., in quantum chaos and spectral geometry. We first discuss many-particle quantum systems on graphs and introduce two types of singular two-particle interactions. The first type is associated with the vertices, and the second type is a realisation of delta-type contact interactions on the edges. In the second part of the talk we discuss under what circumstances Bose-Einstein condensation (BEC) occurs for non-interacting Bose gases on graphs. We also address the problem of BEC for interacting systems and prove the absence of BEC for repulsive hardcore interactions |
| Déjeuner | ||
| 14h - 15h | Virginie Bonnaillie-Noël (Rennes) |
Opérateurs de Schrödinger avec champ magnétique dans les domaines à coins. Résumé: Dans cet exposé, nous présenterons quelques résultats récents sur le spectre d'opérateurs de Schrödinger avec champ magnétique sur des géométries modèles en dimension 2 et 3. Cela permet d'en déduire les asymptotiques des premiers modes propres des opérateurs avec paramètre semi-classique pour des domaines à coin généraux. Nous mentionnerons en particulier des travaux réalisés en collaboration avec M.Dauge, S.Fournais, B.Helffer, D.Martin, N.Popoff, N.Raymond, G.Vial. |
| 15h15 - 16h15 | Gabriel Stoltz (ENPC) |
A mathematical formulation of the random phase approximation for crystals Résumé: We study the response of crystals to an external, time-dependent forcing by considering the time evolution of localized defects. The aim of this work is to extend previous results on the static (time-independent) response of crystals. The linear response allows to define the frequency-dependent polarization operator. On the other hand, the nonlinear response allows to derive, in some spatial homogenization limit, the expression of the frequency-dependent permittivity of a crystal in terms of its band structure. This is joint work with Eric Cances. |
| 10h00 - 11h00 | Gianluca Panati
(Univ. Roma "La Sapienza") |
Topological invariants of eigenvalue
intersections and decay of Wannier functions in graphene. Abstract: The lack of electron localization in graphene is conveniently reformulated in terms of the asymptotic decay of the Wannier functions corresponding to the valence and the conduction band. To quantify the decay of the Wannier functions,we introduce a topological invariant for the family of Bloch eigenspaces, baptized eigenspace vorticity. This invariant characterizes the behavior of such eigenspaces around an eigenvalue intersection. If time permits, a comparison with the pseudospin winding number of the physics literature will be also outlined. For each value n∈ Z of the eigenspace vorticity, a canonical model for the local topology of the eigenspaces is exhibited, and a suitable universality theorem for these models is stated. This allows us to extract the asymptotic decrease of the Wannier functions for the valence and conduction band of graphene, both in the monolayer and the bilayer case. We show that that the single band Wannier function satisfies, in a suitable weak sense, w(x)≍|x|-3/2 as |x|→∞. In particular, the expectation value of the modulus of the position operator is infinite, yielding the expected delocalization of the electrons. The talk is based on a joint work with D. Monaco. |
| 11h15 - 12h15 | Riccardo
Weder (Univ. Nacional Autonoma, Mexico) |
High- and
low- energy analysis and Levinson’s theorem for the selfadjoint matrix
Schrödinger operator on the half line Abstract: The matrix Schrödinger equation with a selfadjoint matrix potential is considered on the half line with the general selfadjoint boundary condition at the origin. When the matrix potential is integrable, the high-energy asymptotics are established for the related Jost matrix, the inverse of the Jost matrix, and the scattering matrix. Under the additional assumption that the matrix potential has a first moment, it is shown that the scattering matrix is continuous at zero energy. An explicit formula is provided for the scattering matrix at zero energy. The small-energy asymptotics are established also for the related Jost matrix, its inverse, and various other quantities relevant to the corresponding direct and inverse scattering problems. Furthermore, Levinson’s theorem is derived, relating the number of bound states to the change in the argument of the determinant of the scattering matrix. |
| Déjeuner | ||
| 14h - 15h | Laurent Thomann
(Nantes) |
Random
weighted Sobolev inequalities. Abstract: We extend a randomisation method, introduced by Burq-Lebeau on compact manifolds, to the case of the harmonic oscillator. We construct measures, under Log-Sobolev type assumptions, on the support of which we prove optimal weighted Sobolev estimates on Rd . As an application we can prove almost sure global well posedness results for the nonlinear Schrödinger equation with harmonic potential. This is a joint work with Aurélien Poiret and Didier Robert. |
| 15h30 - 16h30 | Lyonell Boulton
(Heriot-Watt Univ.) |
Mistuned Toeplitz operators and
applications. Abstract: The study of a new family of highly non-self-adjoint operators which is reminiscent of the classical Toeplitz operators arises naturally in the context of evolution problems connected to sandpiles/slow-fast diffusion. The aim of this talk is to describe the difficulties involved in determining general spectral properties of operators in this family and show how to overcome these difficulties. In turns we will establish basisness/non-basisness theorems for general families of 1D periodic functions. We will then apply the latter, in order to examine non-orthogonal projection methods for the p-poisson parabolic time-evolution initial value problem with stochastic forcing. |