Séminaire: Problèmes Spectraux en Physique Mathématique

(ex-"séminaire tournant")

Prochains séminaires

Séminaires de l'année 2016-2017

Lundi 17 octobre 2016

 11h15 - 12h15 Constanza Rojas-Molina  (Bonn)
Characterization of the metal-insulator transport transition for the two-particle Anderson model

In dimensions higher than two it is expected that a disordered system, modeled by an Anderson operator, undergoes a metal-insulator transition from a region of localization to delocalization. For the one-particle Anderson model, F. Germinet and A. Klein showed that the transport exponent in these regions can be related to the applicability of the multiscale analysis method used in the proof of localization. In this talk we will present a recent generalization of this characterization to the two-particle Anderson model with short-range interactions. We show that, for any fixed number of particles, the slow spreading of wave packets in time implies the initial estimate of a modified version of the Bootstrap Multiscale Analysis. In the case of two particles, this gives a characterization of the metal-insulator transport transition.
This is joint work with A. Klein and Son T. Nguyen.

14h - 15h Alberto Maspero (Nantes)
On time dependent  Schrödinger equations: global well-posedness and growth of Sobolev norms


We study well-posedness and growth of Sobolev norms for
time dependent Schrödinger equations of the form
i∂tψ =(H+V(t))ψ, where H is a self-adjoint positive operator, while V(t) is a perturbation smoothly depending on time.
Under the assumptions that the spectrum of H can be enclosed in spectral clusters whose distance is increasing and
V(t) is a relatively bounded perturbation of H, we prove that the Sobolev norms of the solution grow at most as
tε when t increases, for any ε>0. If V(t) is analytic in time we improve the bound to (log t)γ, for some γ>0.
The proof follows the strategy, due to Joye and Nenciu, of the adiabatic approximation of the flow. We apply our result to Schrödinger operators on R and Schrödinger operators on Zoll manifolds.

This is a joint work with Didier Robert.

15h15 - 16h15 Isabelle Tristani (X)
On the inhomogeneous Landau equation.

This work deals with the inhomogeneous Landau equation on the torus (only in the cases for which we have the existence of a spectral gap for the linearized problem). We first investigate the linearized equation and we prove exponential decay estimates for the associated semigroup. We then turn to the nonlinear equation and we use the linearized semigroup decay in order to construct perturbative solutions (close to the equilibrium). Finally, we prove the exponential stability of these solutions.

This is a joint work with K. Carrapatoso and K.-C. Wu.      

Lundi 7 novembre

 11h15 - 12h15 Gueorgui Popov (Nantes)
K.A.M. tori isospectral deformations and spectral rigidity

We are interested in the spectral rigidity of the Laplace-Beltrami operator in the case when the corresponding classical Hamiltonian system is either completely integrable, or close to a nondegenerate completely integrable system. By the Kolmogorov-Arnold-Moser (K.A.M.) theorem, there exists a large family of invariant tori of the classical system with Diophantine rotation vectors. We show that the average action on the K.A.M. tori is an isospectral invariant. As an application, we obtain spectral rigidity for two-dimensional billiard tables in the presence of elliptic "bouncing ball" geodesics. We obtain as well infinitesimal rigidity of Liouville billiard tables in dimension two and three. The proof is based on a construction of smooth families of quasi-modes associated with the K.A.M. tori.

14h - 15h Mostafa Sabri (Strasbourg)
Quantum ergodicity for the Anderson model on regular graphs

I will discuss a result of delocalization for the Anderson model on the regular tree (Bethe lattice). The Anderson model is a random Schrödinger operator, where we add a random i.i.d. perturbation to the adjacency matrix. Localization at high disorder is well understood today for a wide variety of models, both in the sense of a.s. pure point spectrum with exponentially decaying eigenfunctions, and in a dynamical sense. Delocalization remains a great challenge; only for tree models, it was proved by Klein, Aizenman, Warzel and generalized by others, that for weak disorder, large parts of the spectrum are purely absolutely continuous, and the dynamical transport is ballistic. A form of delocalization was also obtained by Geisinger. In this work, we try to complete the picture by proving that in such a regime, the eigenfunctions are also delocalized in space, in
the sense that if we consider a sequence of regular graphs converging to the regular tree, then the eigenfunctions become asymptotically uniformly distributed (as opposed to the exponential decay in the localization regime). The precise result is a quantum ergodicity theorem.
This is a joint work with Nalini Anantharaman.

15h15 - 16h15 Andrea Mantile (Reims)
On the simultaneous identification of scattering parameters for classical waves with potential and transmission conditions.

We consider the stationary waves scattering for 3D Schrödinger-type operators with singular perturbations supported on surfaces. This is a multiple scattering problem from obstacles and potentials, whose solutions depend on locations and shapes of the obstacles, the related transmission impedances and the background potentials. The corresponding inverse problem consists in determining these scattering parameters from a complete set of far-field data at a fixed energy. In this talk we provide, under suitable a priori bounds assumptions, a uniqueness result for this problem.
This is a joint work with A. Posilicano and M. Sini.

Lundi 12 décembre

 11h15 - 12h15 Viet Dang (Lyon)
Equidistribution du cycle conormal d’ensembles nodaux aléatoires

A toute fonction f sur une variété riemannienne compacte M, on associe un ensemble dans le cotangent T*M, appelé la conormale au lieu des zéros de f ; c’est la version duale de la normale à la sous-variété {f = 0}. Dans un premier temps, j’expliquerai comment interpréter cet objet géométrique comme un courant au sens de De Rham. Dans un second temps, je calculerai l’espérance de ce courant, dans le cas où f est une combinaison linéaire aléatoire de fonctions propres du Laplacien sur M, en introduisant des techniques d’intégration supersymétrique à la Berezin. Enfin je montrerai un phénomène d’équidistribution de ce courant dans T*M. Il s’agit d’un travail en commun avec Gabriel Rivière.

14h - 15h Virginie Ehrlacher (ENPC)
Greedy algorithms for electronic structure calculations for molecules

Greedy algorithms will be presented in order to compute the lowest eigenvalue and an associated eigenstate for high-dimensional problems and their numerical behaviour will be illustrated for the computation of the ground state electronic wavefunction of a molecule, solution of the many-body Schrödinger equation. The resolution of the many-body Schrödinger equation is a difficult task in practice when the number of electrons is significant, due to the curse of dimensionality. The principle of these methods is to compute an approximation of the wavefunction through an iterative procedure, at each iteration of which, only one Slater determinant is computed, which leads effectively
to small-dimensional problems at each stage. As the number of iterations increases, the obtained approximation can be proved to converge to an eigenstate of the many-body Schrödinger operator.
Usually, these algorithms are implemented in practice using the Alternating Least-Square algorithm ; this leads to some computational difficulties in this particular situation, due to the antisymmetry of the ground state wavefunction. A computational strategy to overcome this difficulty will be
presented and illustrated on several molecules.
This is a joint work with Eric Cancès and Tony Lelièvre.

15h15 - 16h15 Konstantin Pankrashkin (Orsay)
Opérateurs autoadjoints du type div sgn grad

On va discuter la définition et les propriétés spectrales des opérateurs donnés par l’expression différentielle div h grad dans un domaine borné U, où la fonction h est égale à 1 sur une partie de U et à b < 0 sur le reste de U. On verra que les propriétés de cet opérateur dépendent fortement du paramètre b, de la dimension et de la géométrie du domaine. En particulier, on peut avoir un spectre essentiel non vide. Le travail est motivé par l’étude mathématique des métamatériaux à indice négatif.
Les résultats sont obtenus en collaboration avec Claudio Cacciapuoti et Andrea Posilicano.

Lundi 9 janvier 2017

 11h15 - 12h15 Gueorgui Raykov (Santiago du Chili)
Discrete spectrum of Schrödinger operators with oscillating decaying potentials

I will consider the Schrödinger operator H, whose potential is a product of a nonconstant almost periodic factor and a function which decays slowly and regularly at infinity. I will discuss the asymptotic behaviour of the discrete spectrum of H near the origin. Due to the irregular decay of the potential, there exist some non-semiclassical phenomena; in particular, H has less eigenvalues than suggested by the semiclassical intuition.

14h - 15h Maher Zerzeri (Paris-Nord)
Ensemble de trajectoires captées et asymptotique des résonances

Nous nous intéressons à l’étude des résonances semiclassiques d’opérateurs de Schrödinger sur L2(RN). Après avoir rappelé "rapidement" quelques résultats antérieurs (zone sans résonances, cas d’un puits dans une île correspondant aux résonances de forme, cas d’un sommet, d’une trajectoire périodique hyperbolique), nous donnerons l’asymptotique des résonances dans le cas où l’ensemble
des trajectoires captées est constitué d’un nombre fini de points fixes hyperboliques et d’orbites homoclines ou hétéroclines. Pour cela, nous établissons des règles
de quantification pour les (pseudo-)résonances associées et nous décrivons précisément leur position. Nous donnerons l’exemple de la trajectoire homocline et celui du potentiel de trois bosses. L’approche utilisée est quelque peu inhabituelle,
faisant intervenir un problème de Cauchy microlocal qui permet de décrire la fonction résonnante associée microlocalement près du point fixe hyperbolique.

Il s’agit d’un travail en collaboration avec J-F. Bony, S. Fujiié et T. Ramond.

15h15 - 16h15 Evelyne Miot (Grenoble)
Collisions de filaments de tourbillon

On étudie la problématique de collisions de filaments tourbillonnaires dans les fluides en dimension trois, à partir d’un système d’équations introduit par Klein, Majda et Damodaran. Pour des configurations symétriques de filaments, le système se ramène à une seule équation dispersive, pour laquelle on démontre l’existence d’une collision auto-similaire en temps fini.
Il s’agit d’un travail en collaboration avec Valeria Banica et Erwan Faou.

Lundi 13 mars 2017

 11h15 - 12h15 Vincent Duchêne (Rennes)
Opérateur de Schrödinger avec un potentiel fortement oscillant

On s’intéressera au comportement asymptotique de l’opérateur de Schrödinger sur la droite réelle avec un potentiel localisé et oscillant, lorsque la longueur caractéristique d’oscillation tend vers zéro. On mettra en avant un potentiel effectif permettant de décrire avec précision le spectre, et en particulier la bifurcation d’une valeur propre, à basse énergie.
Travail en collaboration avec Michael Weinstein, Iva Vukicevic, Nicolas Raymond.

14h - 15h Oana Ivanovici (Nice)
Estimations de dispersion pour les ondes à l’extérieur d’un obstacle strictement convexe et contre-exemples

L’objet de cet exposé est de démontrer des estimations de dispersion pour l’équation des ondes et de Schrödinger à l’extérieur d’un obstacle strictement convexe de Rd. Si d = 3,
on démontre que, pour chacune des deux équations, le flot linéaire vérifie les estimations de dispersion comme dans
de R3. En dimension d > 3, on démontre que des pertes dans la
dispersion apparaissent à l’extérieur d’une boule
de Rd, et cela arrive au point de Poisson.
Il s’agit d’un travail en collaboration avec Gilles Lebeau.

15h15 - 16h15 Dominique Spehner (Grenoble)
Bosons en interaction dans un double puits de potentiel : régime localisé

Nous nous intéressons dans cet exposé à l’état fondamental d’un système de N bosons piégés dans un double puits de potentiel symétrique, dans la limite de champ moyen (N est grand) et pour des grandes distances L entre les puits (L diverge avec N). On s’attend pour de tels gaz de bosons à une transition induite par les interactions entre un état fondamental délocalisé (dans lequel les particules sont indépendantes et toutes dans le même état délocalisé dans les deux puits) et un état localisé (pour lequel on a la moitié des particules localisées dans chaque puits). En partant du Hamiltonien à N corps, nous montrons que si l’énergie tunnel reste petite devant l'énergie d’interaction, les fluctuations du nombre de particules dans chaque puits sont considérablement réduites par rapport au cas de particules indépendantes. Ceci confirme l’existence du régime localisé et de corrélations entre les particules.
Travail en collaboration avec Nicolas Rougerie.

Lundi 24 avril 2017

 11h15 - 12h15 Giambattista Giacomin (Paris 7)
Singular behavior of the leading Lyapunov exponent of a product of 2x2 random matrices

We consider a certain infinite product of random 2×2 matrices appearing in the solution of some 1 and 1+1 dimensional disordered models in statistical mechanics.
B.Derrida and H.J.Hilhorst (J. Phys. A 16:2641, 1983) pointed out that this Lyapunov exponent has a singular behavior in a suitable parameter and they provide a sharp prediction about this singularity. Their analysis is based on a two-scale analysis of the integral equation for the invariant distribution of the Markov chain associated to the matrix product, and  tney obtained a probability measure that is expected to be close to the invariant one near the singularity. We introduce suitable norms and exploit contractivity properties to show that such a probability measure is indeed close to the invariant one, in a sense which implies the sharp control of the Lyapunov exponent.

14h - 15h Quentin Liard (Paris 13)
Dérivation de problèmes à N corps par les mesures de Wigner

Dans cet exposé, j'aborderai des problèmes d'évolution à N corps pour des systèmes de bosons confinés ou localement confinés. J'évoquerai en parallèle la stratégie basée sur la résolution de hiérarchies de type BBGKY ou Gross-Pitaevskii et celle, plus récente, basée sur la résolution de l'équation de Liouville.

15h15 - 16h15 Oliver Butterley (ICTP Trieste)
Open sets of exponentially mixing Anosov flows

If a flow is sufficiently close to a volume-preserving Anosov
flow and the stable and unstable subspaces satisfy dim
Es = 1 and dim Eu ≥ 2, then the flow mixes exponentially fast whenever the stable and unstable foliations are not jointly integrable (and similarly if the requirements on stable and unstable bundle are reversed). This implies the existence of non-empty open sets of exponentially mixing Anosov flows.
This is a joint work with Khadim War.

Lundi 22 mai 2017

 11h15 - 12h15 Norbert Peyerimhoff (Durham)
Spectral properties of Laplacians on the Kagome Lattice.


The Kagome lattice is one of the 11 Archimedean Tilings of the plane. In this talk, we use this example to illustrate spectral properties, and in particular properties of the Integrated Density of States (IDS), of Laplacians on combinatorial and metric graphs (both with equilateral and with random edge lengths).
The talk is based on joint results with Daniel Lenz, Olaf Post, Matthias Taeufer, and Ivan Veselic.

14h - 15h Normann Mertig (Tokyo Metropolitan University)
Dynamical tunneling in systems with a mixed phase space

In certain 1-dimensional Schrödinger Hamiltonian systems, the lifetimes of long-lived mestable states are associated with tunneling through potential barriers. In particular, the lifetime is semiclassically related to the action of a complex path joining trajectories of energy E in the bounded and unbounded region, along their common analytic continuation through the complex domain. While this picture generalizes to integrable systems, it breaks down for Hamiltonian systems whose classical limit exhibits a mixed phase space (namely, the phase space splits into disjoint regions of regular and chaotic motion).
We will consider the situation where long-lived metastable states are associated with bounded trajectories of regular motion, while their lifetimes are determined by so-called dynamical tunneling towards trajectories of chaotic motion. Since for mixed systems the analytic continuation of the tori terminates at natural boundaries, it is not possible to determine the lifetimes of long-lived states from complex paths along the analytic continuation of tori.
I will present how we handle such situations in a physicist’s approach, by reducing the system to an integrable approximation and solving for the decay rates in the reduced system.

15h15 - 16h15 Faizan Nazar (Paris-Dauphine)
Locality of the TFW equations


In this talk, I will provide an overview of Density Functional Theory (DFT), which provides a collection of approximations to the ground state of the Schrödinger equation. I will focus
on one such example, the Thomas-Fermi-von Weizsäcker model, and introduce pointwise exponential estimates for this model, that demonstrate the stability of ground states with
respect to the nuclear configuration. We consider several applications of this results, as well as comparing this result to existing exponential decay results shown for quantum systems.

Lundi 19 juin 2017

 11h15 - 12h15 André Martinez (Bologne)
Resonances and molecular dynamics near an energy level crossing


We study the location and the width of the resonances of a
semiclassical 2x2 system of interacting 1d Schrödinger operators, near an energy level where the two potentials cross. One of the potentials is supposed to be bonding, and the other one anti-bonding. Then, we describe the asymptotic behaviour (both semiclassical and at large times) of the survival amplitude of the corresponding bound state.

This talk is based on joint works with Ph. Briet, S. Fujiié, and T. Watanabe.

14h - 15h Günter Stolz (U.of Alabama, Birmingham)
Many-body localization of the droplet spectrum in the random XXZ spin chain

We report on joint work with A. Elgart and A. Klein in which we provide the first rigorous results on many-body localization for the XXZ quantum spin chain in random field. In the Ising phase of the model, we prove a strong form of exponential clustering (i.e. exponential decay of correlations of local observables) for eigenstates in the droplet spectrum, both for the large disorder regime as well as for a strong Ising phase at fixed disorder. We will also discuss other MBL properties which hold in these regimes, such as a zero-velocity Lieb-Robinson bound on many-body transport, as well as a uniform area law for the entanglement of eigenstates.
15h15 - 16h15 Nicolas Raymond (Rennes)
Semiclassical Robin Laplacians: Miscellaneous of linear and nonlinear results

In this talk, I will describe recent results related to the semiclassical Robin Laplacians: Weyl asymptotics, tunneling effect and electro-magnetic Sobolev embeddings.

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