Séminaire: Problèmes Spectraux en Physique Mathématique

(ex-"séminaire tournant")



Prochains séminaires


Séminaires de l'année 2013-2014

Lundi 7 octobre 2013


 11h15 - 12h15   Christophe Texier (Orsay)
Products of random matrices of SL(2,R) and 1D disordered systems

Résumé:
Random matrix products are involved in many models of statistical physics, like random spin chains or quantum localisation problem. A central property of a random matrix product is the Lyapunov exponent characterising the growth of the product; the physical interpretation is the free energy per spin of a spin chain, or the inverse localisation length in quantum localisation problems. I will consider products of i.i.d. random real 2x2 matrices and will identify a physical model of one-dimensional disordered quantum mechanics related to the most general matrix products. Using the Iwasawa decomposition of SL(2,R), we can identify a continuum regime where the mean values and the covariances of the three Iwasawa parameters are simultaneously small (matrices are close to the identity matrix). In this regime, the Lyapunov exponent of the product may be systematically obtained by considering the Hilbert transform of the invariant measure of the matrix product. The Lyapunov exponent is shown to present a scaling form. This general analysis allows us, among other things, to recover in a unified framework few results known previously from exactly solvable models of one-dimensional disordered systems and find several new ones, providing a classification of possible solutions for such 1D models.

Déjeuner
14h - 15h Tiphaine Jézéquel (ENS Cachan Bretagne)
Klein-Gordon non linéaire : dynamique près d'une orbite homocline

Résumé:
On s'intéresse à l'équation de Klein-Gordon non-linéaire
dt2 u - Δ u - m2 u + u2p+1=0, (x,t) dans RxM,
où M est une variété Riemanienne compacte de dimension 1, 2 ou 3. Si l'on s'intéresse d'abord au comportement de l'équation stationnaire en x, le portrait de phase est entièrement traçable, et notamment cette équation admet les équilibres 0 et m1/p, ainsi que 2 solutions ondes solitaires à 0 (appelées aussi orbites homoclines à 0). Notre principal résultat est l'existence d'un grand nombre de solutions non stationnaires en x proches de ces ondes solitaires. On souligne qu'on décrit ainsi des comportements à la fois non-linéaires et non locaux pour cette équation. Bien que non locale, la démonstration est inspirée d'une stratégie utilisée par Groves et Schneider dans l'étude d'une bifurcation locale de Klein-Gordon. Il s'agit d'un travail en collaboration avec Benoît Grébert et Laurent Thomann.



15h15 - 16h15 Thierry Daudé (Cergy-Pontoise)
Inverse scattering at fixed energy in black hole spacetimes

Résumé:
In this talk, we first describe a class of axisymmetric, electrically charged, spacetimes with positive cosmological constant, called Kerr-Newmann-de-Sitter black holes, which are exact solutions of the Einstein equations. The main question we address is the following: can we determine the metrics of such black holes by observing waves at the "infinities" of the spacetime? Precisely, the considered waves will be massless Dirac fields evolving in the outer region of Kerr-Newman-de-Sitter black holes. We shall define the corresponding scattering matrix, the object that encodes the far field behavior of these Dirac fields from the point of view of static observers. We finally shall show that the metrics of such black holes is uniquely determined by the knowledge of this scattering matrix at a fixed energy.
This result was obtained in collaboration with Fran\c cois Nicoleau (Nantes).

Lundi 4 novembre 2013


 11h15 - 12h15   Dmitry Jakobson (McGill Univ., Montréal) Averaging over manifold of metrics with the fixed volume form

Résumé:
We study the manifold of all metrics with the fixed volume form on a compact Riemannian manifold of dimension ≥3. We compute the characteristic function for the L2(Ebin) distance to the reference metric. Next, we study Lipschitz-type distance between Riemannian metrics, and give applications to the diameter and eigenvalue functionals.

This is joint work with Y. Canzani, B. Clarke, N. Kamran, L. Silberman and J. Taylor.


Déjeuner
14h - 15h Eric Dumas (Grenoble)
On the weak solutions to the Landau-Lifshitz equations

Résumé:
This joint work with Franck Sueur (Paris 6) deals with weak solutions to the Maxwell-Landau-Lifshitz and Hall-magnetohydrodynamics equations. I shall mainly present the results concerning the Landau-Lifshitz equations from ferromagnetism. We obtain a weak-strong uniqueness result.
We also give sufficient conditions on the regularity of weak solutions ensuring conservation of energy, so that no anomalous dissipation shows up. In addition, when anomalous dissipation is present, we address the question of its sign. With the point of view of dimensional analysis, a parallel can be established between our regularity conditions and Onsager's conjecture from hydrodynamics.
15h15 - 16h15 Thierry Ramond (Orsay)
Asymptotique des résonances engendrées par des orbites homoclines

Résumé:
On étudie les résonances semiclassiques d'opérateurs de Schrödinger sur
L2(Rn), dans le cas où l'ensemble capté correspondant consiste en un certain nombre de points fixes hyperboliques et d'orbites homoclines ou hétéroclines.
En utilisant une  approche quelque peu inhabituelle, nous obtenons des règles de quantification pour les résonances associées, et décrivons précisément leur position. Au passage,  nous démontrons des estimations polynomiales pour  la résolvante.
Il s'agit de résultats obtenus en collaboration avec J.-F. Bony, S. Fujiié et M. Zerzeri.


Lundi 2 décembre 2013 


 11h15 - 12h15   Gregory Eskin (UCLA) Spectral asymptotics and the Aharonov-Bohm effect

Abstract:

We study the magnetic Schrödinger operator in the domain Ω⊂R2 between a convex obstacle and the circle of radius R, where R is large. We assume that the magnetic field is zero in  but the magnetic flux α is not zero. The Aharonov-Bohm effect is the assertion that the magnetic flux has a physical impact. We compute explicitly the singularity of the wave trace corresponding to the inscribed equilateral triangles (or more generally n-gons) and we recover cos(α) from the singularities. This gives a proof of the Aharonov-Bohm effect since the magnetic flux influences the spectrum of the Schrödinger operator.
The first result on the Aharonov-Bohm effect and the spectrum of Schrödinger operator was proven by Bernard Helffer in the 1980ies. In the talk we also briefly describe other approaches to the proof of the Aharonov-Bohm effect.
This is a joint work with Jim Ralston.

Déjeuner
14h - 15h Nicolas Popoff (Marseille)
On the ground state of the magnetic Laplacian in polyhedral bodies

Abstract:
I will present recent results about the first eigenvalue of the magnetic Laplacian in polyhedral domains with Neumann boundary condition in the semi-classical limit. The use of singular chains show that the asymptotics of the first eigenvalue is governed by a hierarchy of model problems on the tangent cones of the domain. We provide estimations of the remainder depending on the geometry and the variations of the magnetic field.
This is a joint work with V.Bonnaillie-Noël and M.Dauge.

15h15 - 16h15 Victor Ivrii (Toronto)
Asymptotics of the ground state energy and related topics for heavy atoms and molecules: results: old, new, in progress and in perspective

Abstract:
We consider asymptotics of the ground state energy when the total charge of nuclei Z and the number of electrons N~Z tend to infinity (number of nuclei remains constant). We include cases of an external constant magnetic field with intensity B<<Z3 and of self-generated magnetic field. The other problems are number of extra electrons which system can bind, the excessive positive charge when two or more nuclei do not fly away, and the estimates from above and below for ionization energy.


Lundi 20 janvier 2014


 11h15 - 12h15   Roman Schubert (Bristol) Entropy of Eigenfunctions on Quantum Graphs

Abstract:
We are interested in the distribution of eigenfunctions on quantum graphs, in particular how they depend on the topology of the graph. As a measure for the distribution we consider the entropy; if an eigenfunction has a large entropy it implies that it cannot be concentrated on a small set of edges. We will focus on two classes of graphs, star graphs and regular graphs. For star graphs we show that the average of the entropies of eigenfunctions is small, indicating eigenfunctions which localise on few bonds. In contrast for regular graphs with large girth we show that the entropy of eigenfunctions is large. The strongest estimates we obtain for expanders where we choose the length of the bonds randomly, then we can show that with large probability the entropy is at least half as large as the maximal possible value. This is analogous to the results by Anantharaman and Nonnenmacher on the entropy of quantum limits on Anosov manifolds, and we in particular borrow one of their tools, the entropic uncertainty principle by Maassen and Uffink.
In the talk I will not assume any background in graph theory or on quantum graphs, all the necessary notions will be introduced.


Déjeuner
14h - 15h Lysianne Hari (Cergy)
Nonlinear propagation of coherent states through an avoided crossing

Abstract:
We study the propagation of a coherent state for a one-dimensional system of two coupled Schrödinger equations in the semiclassical limit. Couplings are induced by a cubic nonlinearity and a matrix valued potential, whose eigenvalues present an “avoided crossing" : at one given point, the gap between them reduces as the semiclassical parameter becomes smaller. We show that when an initial coherent state polarized along an eigenvector of the potential propagates through the avoided crossing point, there are transitions between the modes at leading order. In the regime we consider, we observe a nonlinear propagation far from the crossing region, while the transition probability can be computed with the linear Landau–Zener formula.
15h15 - 16h15 Suresh Eswarathasan (IHES)
Expected values of eigenfunction periods

Abstract:
Let (M,g) be a compact Riemannian surface. A period of an eigenfunction of the Laplace- Beltrami operator is the integral of this eigenfunction over a given curve on M. The study of these periods complements that of Lp norms and semiclassical limit measures, and helps to understand the structure of eigenfunctions. Using the Kuznecov trace formula and an idea of Burq–Lebeau, we obtain estimates on the expected values of eigenfunction periods. If time permits, we will also present probabilistic results on restricted Lp norms over curves.


Lundi 10 mars 2014

 11h15 - 12h15   Vesselin Petkov (Bordeaux) Grandes déviations exactes pour des systèmes dynamiques hyperboliques

Abstract:
Soit f : X → X un difféomorphisme hyperbolique, Ψ et Φ deux fonctions höldériennes sur X. On note Ψ(n) la somme de Birkhoff de Ψ au temps n, et mΦ la mesure de probabilité f-invariante ("état d’équilibre") associée à Φ. On étudie les grandes déviations de Ψ(n), par rapport à cette mesure mΦ, pour des largeurs d’intervalle δn = exp(−δn), avec δ > 0 assez petit. Précisément, on prouve l’asymptotique suivant en n→+∞:

mΦ ({x∈X : Ψ(n)(x)/n ∈(p−δn,p+δn)}) ∼ Cδn√n exp(−nJ(p)),

où C > 0 et J(p) ≥ 0 est la fonction de grande déviation habituelle. Le cas de largeurs δn = n−κ, κ > 0, avait été traité par M.Pollicott et R.Sharp. Notre preuve repose sur des estimations spectrales associées à un opérateur de transfert de Ruelle.
C’est un travail en collaboration avec L. Stoyanov.


Déjeuner
14h - 15h Jose Luis Jaramillo (Brest)
A perspective on black hole horizons from the quantum charged particle

Abstract:
We point out a formal similarity between the characterization of black hole apparent horizons in terms of light rays trapped in spacetime, on the one hand, and the quantum description of a non-relativistic charged particle moving in given magnetic and electric fields on a closed surface, on the other hand. Such quantum analogy may provide clues to the study of the spectrum of the relevant (non-selfadjoint) apparent-horizon stability operator, in comparison with the (selfadjoint) Hamiltonian of the charged particle. In particular, this might open an avenue to the use of semiclassical tools to explore qualitative aspects of the black hole spectral problem.
15h15 - 16h15 Laurent Bruneau (Cergy)
Mixing properties of the one-atom maser

Abstract:
We study the relaxation properties of the quantized electromagnetic field in a cavity under repeated interactions with single two-level atoms, so-called one-atom maser. We prove that, whenever the atoms are initially distributed according to the canonical ensemble at temperature T > 0, all the invariant states are mixing. Under some non-resonance condition the invariant state is unique and known to be thermal equilibirum at some renormalized temperature T*. We prove that the mixing is then arbitrarily slow, in other words that there is no lower bound on the relaxation speed. These results are based on the spectral analysis of a suitable representation of the one-step map L which describes the effects of a single interaction between an atom and the cavity field.


Lundi 7 avril 2014

 11h15 - 12h15   Benjamin Texier
(Jussieu)
Approximations of pseudo-differential flows

Abstract:
I will put forward a microlocal approach to stability of reference solutions in systems of partial differential equations. The idea is to look for unstable spectrum at the symbolic level, as opposed to unstable spectrum for the associated differential operators. I will give model results in this direction, based on an approximation lemma for pseudo-differential flows, and draw a comparison with Gårding's inequality.

Déjeuner
14h - 15h Michał Wrochna
(Orsay)

Microlocal analysis of quantum fields on curved spacetime

Abstract:
In Quantum Field Theory on curved spacetime, a key problem is the existence of states whose two-point functions are distributions with a specified wave front set. I will show that this is equivalent to the existence of a parametrix (for the given hyperbolic differential equation), that has good properties with respect to the conserved charge. I will then present a construction of such parametrix for the (vectorial) Klein-Gordon and wave equation, based on pseudodifferential calculus rather than Fourier integral operators.
The talk is based on a joint work with Christian Gérard.

15h15 - 16h15 Sandrine Péché
(Paris 7)

Deformed ensembles of random matrices

Abstract:
We review more or less recent results on deformations of random matrices. We focus on additive deformations of random matrices, by considering the impact of adding a given matrix to a standard Wigner random matrix. The asymptotic behavior of the spectrum is then described in terms of the deformation. Applications to statistics, physical mathematics are also given.

Lundi 12 mai 2014

 11h15 - 12h15   Jérémy Faupin (Univ. de Lorraine) On quantum electrodynamics of atomic resonances

Abstract:
We consider a simple model of an atom interacting with the quantized electromagnetic field. The atom has a finite mass, finitely many excited states, and an electric dipole moment proportional to the elementary electric charge. We establish the existence of resonances associated to the excited states of the atom, and we prove that these resonances are analytic functions of the total momentum p and of the coupling constant, provided |p| < mc (where m is the mass of the atom and c is the speed of light) and assuming the coupling constant is small enough. The proof relies on a somewhat novel inductive construction involving a sequence of ‘smooth
Feshbach-Schur maps’ applied to a complex dilatation of the original Hamiltonian, which yields an algorithm for the calculation of resonance energies that converges superexponentially fast.
Joint work with M. Ballesteros, J. Fröhlich and B. Schubnel.

Déjeuner
14h - 15h Nicolas Rougerie (Grenoble)
The Ginzburg-Landau model in the surface superconductivity regime

Abstract:
The Ginzburg-Landau functional is a phenomenological model describing the response of a superconductor to an applied magnetic field. In this talk I will present new results about the ground state of the functional for type II superconductors in magnetic fields varying between the second and third critical fields. In this regime, superconductivity is a surface phenomenon, restricted to a thin layer along the boundary of the sample. Our results show that the Ginzburg-Landau energy is, to subleading order, entirely determined by the minimization of simplified 1D functionals. The leading order of the energy is given by a universal, sample-independent, problem, whereas corrections depend on the curvature of the sample. Refined estimates on the Ginzburg-Landau minimizer follow from these energy estimates.
Joint work with Michele Correggi.
15h15 - 16h15 Hakim Boumaza (Paris-Nord)
Random scattering zippers

Abstract:
A scattering zipper is a system obtained by concatenation of scattering events with equal even number of incoming and out going channels. The associated scattering zipper operator is the unitary equivalent of Jacobi matrices with matrix entries and it also generalizes CMV matrices. Using the formalism of transfer matrices, one can get an explicit expression for the resolvent of the operator which is used to prove a bijection between the set of semi-infinite scattering zipper operators and matrix valued probability measures on the unit circle. We then consider a random version of the scattering zipper where the randomness appears through random phases. For this random model, Lyapunov exponents positivity is proved and yields to the absence of absolutely
continuous spectrum.
Joint work with Laurent Marin.

Lundi 16 juin 2014

 11h15 - 12h15 Pär Kurlberg (KTH Stockholm)
Nodal length statistics for arithmetic random waves

Abstract:
Using spectral multiplicities of the Laplacian acting on the standard two-torus, we endow each eigenspace with a Gaussian probability measure. This induces a notion of a random eigenfunction on the torus, and we study the statistics of nodal lengths of the eigenfunctions in the high energy limit. In particular, we determine the variance for a generic sequence of energy levels, and also find that the variance can be different for certain “degenerate" subsequences. (These degenerate subsequences are closely related to circles on which integer lattice points are very badly distributed.)

Déjeuner
14h - 15h Mario Sigalotti (INRIA)
Exploiting conical eigenvalue intersections for controlling quantum mechanical systems

Abstract:
In this talk we will present the controllability problem for a closed quantum system driven by external fields. We will recall some known necessary and sufficient conditions for controllability.
We will present a control algorithm based on adiabatic approximation, exploiting the presence of conical eigenvalue intersections. We will conclude by comparing the hypothesis
ensuring controllability via adiabatic evolution with the classical controllability results.
15h15 - 16h15 Xavier Blanc (Paris 7)
Existence of the thermodynamic limit for Coulomb disordered quantum systems

Abstract:
We study a system of atoms in which the nuclei are classical point particles, while electrons are quantum particles described by the N-body Schrödinger equation. The positions of the nuclei are assumed to be randomly distributed and stationary. The interaction between particles are of electrostatic type. We prove, that, as the number of particles
tend to infinity, the energy per particle has a finite limit.
This is a joint work with M. Lewin.




Dernière mise à jour: 10 septembre 2014
Page maintenue par Stéphane Nonnenmacher