Séminaire: Problèmes Spectraux en Physique Mathématique

(ex-"séminaire tournant")

Prochain séminaire

Séminaires de l'année 2018-2019

Lundi 8 octobre 2018

 11h15 - 12h15 Mostafa Sabri (Orsay)
Lp norms and support of eigenfunctions on graphs

Recent years have seen much interest in the study of delocalization on graphs. Eigenvector delocalization is measured by many criteria : one can try to prove the eigenfunctions have a large support, or prove quantum ergodicity (a certain property of equi-distribution of eigenvectors on the graph), or give upper bounds on the supremum norms of the eigenfunctions, and more generally the
Lp-norms for p > 2.
In this talk, I will present some estimates on the
Lp-norms of eigenfunctions of Schrödinger operators on large finite graphs, and show that they have large supports. The results hold for any fixed, possibly irregular graph, in prescribed energy regions, and also for certain sequences of graphs such as N-lifts.

This is a joint work with Etienne Le Masson.

14h - 15h Joachim Kerner (Hagen)
On some recent developments in the study of many-particle models on quantum graphs

In this talk we investigate interacting many-particle systems on quantum graphs as well as quantum wires. We focus on a two-particle system on the half-line with three different types of interactions, one of which is characteristic for interacting quantum graphs.  A second type of interaction leads to a formation of a pair of particles, enabling us to establish interesting and novel connections to solid-state physics. We will also discuss connections of the model to the active area of many-particle quantum chaos, especially when considered on a generic compact graph. No preliminary knowledge of quantum graphs is necessary.
 15h15 - 16h15 Djalil Chafaï (Paris-Dauphine)
Dynamiques de gaz de particules en répulsion singulière

Cet exposé présentera des dynamiques aléatoires associés à des modèles de gaz de particules en interaction, issus ou inspirés des matrices aléatoires et de la physique statistique. Ces dynamiques sont à la fois des objets d'intérêt naturels et des outils pour l'échantillonnage. Les phénomènes de grande dimension y jouent un rôle important.

Lundi 12 novembre 2018

 11h15 - 12h15 François Nicoleau (Nantes)
Scattering matrices for dissipative quantum systems

We consider a quantum system S interacting with another system S' and susceptible of being absorbed by S'. The effective, dissipative dynamics of S is supposed to be generated by an abstract pseudo-Hamiltonian of the form H = H0 + V - i C* C. The generator of the free dynamics, H0 , is self-adjoint, V is symmetric and C is bounded. We study the scattering theory for the pair of operators (H,H0). We establish a representation formula for the scattering matrices and identify a necessary and sufficient condition to their invertibility. This condition rests on a suitable notion of spectral singularity. Our main application is the nuclear optical model, where H is a dissipative Schrödinger operator and spectral singularities correspond to real resonances.
This is a joint work with Jérémy Faupin.

14h - 15h Fabio Pizzichillo (Paris-Dauphine)
The Hardy inequality and the Dirac-Coulomb operator

The free Dirac operator on R3 is defined as
H0=-iα.∇ + m β (for a mass parameter m>0), where α=(α012) and β denote the Dirac matrices.
This talk aims to show the connection between Hardy-type inequalities and the Dirac-Coulomb operator. Firstly, I will prove some sharp Hardy-type inequalities related to the Dirac operator. Then I will show how to define the distinguished self-adjoint realisation of H0+V, with V a Hermitian matrix potential with Coulomb decay. Finally, I will focus on the spectral properties of Dirac-Coulomb operator. I will characterise its eigenvalues using the Birman-Schwinger principle. Moreover, I will show a bound from below of its discrete spectrum, and I will prove that this bound is reached if and only if V verifies some rigidity conditions.
This is a joint work with B. Cassano, and L. Vega.
 15h15 - 16h15 Christian Hainzl (Tübingen)
The interacting Fermi gas : A step beyond Hartree-Fock

We consider a homogenous Fermi gas interacting via a regular mean-field potential.We recover
the correlation energy to leading order in the coupling parameter and we will make general
remarks about the the correlation energy for Coulomb gases.

Lundi 10 décembre 2018

 11h15 - 12h15 San Vũ Ngọc (Rennes)
Analyticity of the Bergman projection

Recently, Rouby gave a precise description, in the semiclassical limit, of the spectrum of a class of non-selfadjoint pseudo-differential operators on the real line, with analytic symbols. This result opens interesting questions relating the complex spectrum to the symplectic geometry of
the phase space, and it was natural to investigate the generalization of the analytic microlocal tools to the setting of Berezin-Toeplitz quantization.
As a first step, I will present a recent result on the structure of the weighted Bergman projection on ℂn or on a compact Kähler manifold with high powers of a prequantum line bundle, when the weight (or curvature) is analytic. In this case, the projection is an elliptic Fourier integral operator acting on the class of analytic symbols. As a corollary, we prove a conjecture of Zelditch about the analyticity of the semiclassical asymptotics of the Bergman kernel.
This is a joint work with Ophélie Rouby and Johannes Sjöstrand.

14h - 15h Christopher Shirley (Sorbonne Univ.)
Propriétés de transport des opérateurs de Schrödinger stationnaires à petit désordre

Dans cet exposé, je reviendrai dans un premier temps sur les liens entre le spectre des opérateurs de Schrödinger et les propriétés de transport ainsi que sur la conjecture d’Anderson.
Dans un second temps, nous verrons comment nous pouvons obtenir des résultats de transport ballistique jusqu’à des temps qui dépendent du désordre et du type de stationnarité, en développant une théorie spectrale approchée.
 15h15 - 16h15 Alessandro Pizzo (Rome Tor Vergata)
Block-diagonalization and gapped quantum chains

I will present a new method to block-diagonalize some Hamiltonians describing quantum chains. The method is applied to study perturbations of the so called Kitaev Hamiltonian.
This is a joint work with J. Fröhlich.

Lundi 7 janvier 2019

 11h15 - 12h15 Catherine Donati-Martin (Versailles)
Fluctuations des outliers des matrices de Wigner déformées

On étudie le comportement des valeurs propres isolées du reste du spectre ("les outliers") de grandes matrices aléatoires, ainsi que leurs vecteurs propres associés. On s’intéresse en particulier aux propriétés de localisation des vecteurs propres et de non universalité des fluctuations.

14h - 15h Benjamin Küster (Orsay)
Quantum-classical correspondence for resonances on vector bundles

The geodesic flow on the unit (co-)sphere bundle of a closed Riemannian manifold can be regarded as a "classical dynamical system". When the manifold is negatively curved, the system is chaotic. In this case, there is a rich theory of "classical resonances" (often called Pollicott- Ruelle resonances), which form a spectral invariant associated to the chaotic geodesic flow. For a nice class of negatively curved manifolds, we consider a quantum-classical correspondence result which relates the classical resonances to the quantum energies of the system given by the eigenvalues of the Laplace-Beltrami operator. When lifting the geodesic flow to vector bundles using a connection, the questions related to quantum-classical correspondence become even more interesting, as we shall see in some examples.
 15h15 - 16h15 Thomas Duyckaerts (Paris-Nord)
Largeur des résonances pour un résonateur de Helmholtz

Un résonateur de Helmholtz est modélisé par une cavité dans un obstacle compact de l’espace euclidien, reliée à l’extérieur par un tube dont le diamètre de la section est petit. Le problème admet des résonances exponentiellement proches des valeurs propres du laplacien de Dirichlet dans la cavité. J’expliquerai dans cet exposé comment obtenir une borne supérieure du temps de vie d’une telle résonance.
C'est un travail en collaboration avec Alain Grigis et André Martinez.

Lundi 11 mars 2019

 11h15 - 12h15 Jean-Marie Barbaroux (Toulon)
Graphene antidot lattices : a mathematical approach

From the physics point of view, it is important to turn semimetallic graphene into a semiconductor. This can be achieved for example by considering graphene antidot lattices (GALs) that consists of a periodic array of perforations in a graphene sheet. This causes a band gap to open up at the Fermi level. In this talk, I will present some recent mathematical results on two-dimensional Dirac operators modeling Hamiltonians for GALs. I will mostly focus on operators with periodic mass potentials, as well as their random Anderson-like perturbations describing defects in the array of perforations.
This is joint work with H.Cornean, E.Stockmeyer and S.Zalczer.

14h - 15h Serena Cenatiempo (Gran Sasso Science Institute, L'Aquila)
Bogoliubov theory in the Gross-Pitaevskii regime

In 1947 Bogoliubov suggested a heuristic theory to compute the excitation spectrum of a weakly interacting Bose gas. Remarkably, such a theory predicts a linear excitation spectrum (in sharp contrast with the quadratic dispersion of free bosons) and provides expressions for the thermodynamic functions which are believed to be correct in the dilute limit. However, so far there are only a few cases where the predictions of Bogoliubov theory can be obtained
through a rigorous mathematical analysis. In particular, a major challenge is to recover the physical intuition that the correct parameter to appear in the expressions of the physical quantities is the scattering length of the interaction.
In this talk I will discuss how the validity of the predictions of Bogolibov theory can be established for a system of N interacting bosons trapped in a box in the Gross-Pitaevskii
limit, where the scattering length of the potential is of the order 1/N and N tends to infinity.
Joint work with C. Boccato, C. Brennecke and B. Schlein.
 15h15 - 16h15 Gabriel Stoltz (Ecole des Ponts + Inria Paris)
Longtime convergence of evolution semigroups in molecular dynamics

I will present exponential decay estimates for the evolution operators associated with paradigmatic stochastic evolutions in molecular dynamics. The first case I will consider is a nonequilibrium Langevin dynamics. The associated generator is hypoelliptic and not coercive. It can however be shown to be hypocoercive for equilibrium dynamics, through the use of
a modified
L2 scalar product ; this property persists for nonequilibrium dynamics provided the external forcing is not too large. The second case corresponds to a nonlinear Feynman-Kac dynamics, whose convergence can be studied by Lyapunov techniques, once the existence and uniqueness of the eigenvector associated with the dominant eigenvalue of the evolution operator has been proved through a Krein-Rutman theorem.

Lundi 8 avril 2019

 11h15 - 12h15 Victor Kleptsyn (Rennes)
Furstenberg theorem : now with a parameter !

For a random product of i.i.d. matrices Ai, randomly chosen from SL(2;R), Tn = An...A2A1, the classical Furstenberg theorem states that the norm of such a product almost surely grows exponentially in n.  What happens if each of these matrices Ai depends on an additional parameter s, and hence
so does their product Tn?  For each individual s, the Furstenberg theorem is still applicable. However, what can be said almost surely for the random products Tn(s)? In particular, what can be said about the limit (Lyapunov exponent) limn(1/n) log ||Tn(s)|| ? Does it exist for all (and not only almost all) parameter values s ?
Under a few (physically reasonable) assumptions, we show that :
– “For the limsup, everything is OK”. Almost surely, for all parameter values, the upper Lyapunov exponent equals the Furstenberg one. This can be considered as a dynamical analog of the result by Craig and Simon from spectral theory.
– “Sometimes, the limit does not exists”. However, in the no-uniform-hyperbolicity parameter region, there exists a dense subset of parameters, where the lower Lyapunov exponent vanishes.
– “The disaster is limited”. Almost surely there is a zero Hausdorff dimension (random) set in the
space of parameters, outside which the Lyapunov exponent exists and equals to the Furstenberg one.
This theorem is proven via a geometric description of the (“highly probable”) behavior of finite-length products; these results are applicable to the setting of the one-dimensional Anderson localization, providing a purely dynamical viewpoint on its proof.
This is a joint work with Anton Gorodetski.

14h - 15h Marco Falconi (Tübingen)
Quasi-Classical Dynamics

Quasi-classical systems are physical systems combining a quantum and a classical part, often used to model the interaction between matter and radiation (or other forces). The quantum part is seen as an open system, that is subjected to the macroscopic force field, seen as an environment. Typical examples are Magnetic Laplacians, atoms in a (harmonic) trap, and optical lattices. In this talk, I
will introduce quasi-classical systems from a mathematical standpoint, and discuss their derivation from more fundamental, but complicated, quantum systems.
The talk is based on a joint work with M. Correggi and M. Olivieri.
 15h15 - 16h15 Emmanuel Schenck (Paris-Nord)
Séparations exponentielles dans le spectre des longueurs

Sur une variété riemannienne compacte de courbure négative, il en général difficile de contrôler précisément la distribution locale des longueurs des géodésiques fermées, ce qui est un obstacle dans les problèmes spectraux qui utilisent la trace du groupe des ondes. On présentera dans cet exposé un résultat de densité pour des métriques avec de bonnes propriétés de séparations dans leur spectre de longueurs, et une application possible pour la loi de Weyl sur les surfaces.

Prochain séminaire le lundi 13 mai 2019

 11h15 - 12h15 Vedran Sohinger (Warwick)
Gibbs measures of nonlinear Schrödinger equations as limits of many-body quantum states

Gibbs measures of nonlinear Schrödinger equations are a fundamental object used to study low-regularity solutions with random initial data. In the dispersive PDE community, this point of view was pioneered by Bourgain in the 1990s. We study the problem of the derivation of Gibbs measures as high-temperature limits of thermal states in many-body quantum mechanics.
The first result on this problem was obtained by Lewin, Nam and Rougerie by using variational methods. In our work, we apply a perturbative expansion in the interaction. This expansion is then analysed by means of Borel resummation techniques. In two and three dimensions, we need to apply a Wick-ordering renormalisation procedure. In the first part of the talk, we study the regime when the interaction potential is bounded and defocussing. In the second part, we extend this result to the optimal range of
Lp defocussing interaction potentials.
This is based partly on joint work with Jürg Fröhlich, Antti Knowles and Benjamin Schlein.

14h - 15h Loïc Le Treust (Marseille) On the semiclassical spectrum of the Dirichlet-Pauli operator

This talk is devoted to semiclassical estimates of the eigenvalues of the Pauli operator on a bounded open set whose boundary carries Dirichlet conditions. Assuming that the magnetic field is positive and a few generic conditions, we establish the simplicity of the eigenvalues and provide accurate asymptotic estimates involving Bergman-Hardy spaces associated with the magnetic field.
 15h15 - 16h15 Dan Mangoubi (Jérusalem)
Multiplicity of Eigenvalues for the circular clamped plate problem

A celebrated theorem of C.L. Siegel from 1929 shows that the multiplicity of eigenvalues for the Laplace eigenfunctions on the unit disk is at most two. More precisely, Siegel shows that positive zeros of Bessel functions are transcendental. We study the fourth order clamped plate problem, showing that the multiplicity of eigenvalues is uniformly bounded (by not more than six). Our method is based on new recursion formulas and Siegel-Shidlovskii theory.
The talk is based on a joint work with Yuri Lvovski.

Lundi 17 juin 2019

 11h15 - 12h15 Takuya Watanabe (Ritsumeikan)
Widths of resonances above an energy-level crossing


We study the existence and location of the resonances of a 2x2 semiclassical system of coupled Schrödinger operators, in the case where the two electronic levels cross at some point, and one of them is bonding, while the other one is anti-bonding. Considering energy levels just above that of the crossing, we find the asymptotics of both the real parts and the imaginary parts of the resonances close to such energies. In this talk, we would like give the result above and compare it with our previous works where we considered energy levels around that of the crossing.
This talk is based on joint works with S. Fujiié (Ritsumeikan) and A. Martinez (Bologna).

14h - 15h Mona Ben Said (Paris-Nord)
Compactness of the resolvent for Kramers-Fokker-Planck operators

This talk is devoted to the study of some spectral properties and compactness criteria for the resolvent of Kramers-Fokker-Planck operators, which will be denoted by KV . I will speak in particular about the results obtained from a joint work with Francis Nier and Joe Viola on the case of a polynomial potential V of degree less than 3. Then based on this work, I will present my recent results which concerns compactness criteria for the resolvent of the operator
KV with more general potentials.
 15h15 - 16h15 Stephan De Bièvre (Lille)
Measuring the Non-Classicality of the Quantum States of a Bosonic Field with Ordering Sensitivity

In quantum optics, a classical state of the quantized electromagnetic field is a mixture of coherent states. When a state is not classical, it is of importance to establish how strongly nonclassical it is. We will present a recently introduced distance-based measure for the nonclassicality of the states of such fields, and show its advantages over existing such measures.
We define for that purpose the operator ordering sensitivity of the state which evaluates the sensitivity to operator ordering of the Renyi entropy of its quasi-probabilities and which measures the oscillations in its Wigner function. Through a sharp control on the operator ordering sensitivity of classical states we obtain a precise geometric image of their location in the density matrix space allowing us to introduce the aforementioned measure of nonclassicality. Its
properties will be illustrated with a variety of examples.
Joint work with D.Horoshko, G.Patera and M.Kolobov

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