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We establish a central limit theorem for the fluctuations of the empirical measure in the β-ensemble of dimension N at a temperature proportional to N and with confining smooth potential. The space of test functions for which the CLT holds includes C1, vanishing functions at infinity. It is obtained by the inversion of an operator which is a pertubation of a Sturm-Liouville operator. The method that we use is based on a change of variables introduced by Bekerman, Guionnet and Figalli in arXiv:1311.2315 and by Shcherbina in arXiv:1310.7835.

International audience; This article introduces a random matrix framework for the analysis of clustering on high-dimensional data streams, a particularly relevant setting for a more sober processing of large amounts of data with limited memory and energy resources. Assuming data x_1, x_2, ... arrives as a continuous flow and a small number L of them can be kept in the learning pipeline, one has only access to the diagonal elements of the Gram kernel matrix: [K_L]_{i, j} = 1 / p x_i^⊤ x_j 1_{|i−j| < L}. Under a large-dimensional data regime, we derive the limiting spectral distribution of the banded kernel matrix K_L and study its isolated eigenvalues and eigenvectors, which behave in an unfamiliar way. We detail how these results can be used to perform efficient online kernel spectral clustering and provide theoretical performance guarantees. Our findings are empirically confirmed on image clustering tasks. Leveraging on optimality results of spectral methods for clustering, this work offers insights on efficient online clustering techniques for high-dimensional data.

We make a spectral analysis of the massive Dirac operator in a tubular neighborhood of an unbounded planar curve,subject to infinite mass boundary conditions. Under general assumptions on the curvature, we locate the essential spectrum and derive an effective Hamiltonian on the base curve which approximates the original operator in the thin-strip limit. We also investigate the existence of bound states in the non-relativistic limit and give a geometric quantitative condition for the bound states to exist., Some technical results have been moved to the Appendix. To appear on Ann. Henri Poincar\'e

We give a dynamical characterization of measures on the real line with finite logarithmic integral. The general case is considered in the setting of evolution groups generated by de Branges canonical systems. Obtained results are applied to the Dirac operators and Krein strings., typos fixed, presentation improved

International audience; In the present paper, we study spectral properties of Toeplitz operators with (quasi-) radial symbols on Bergman space. More precisely, the problem we are interested in is to understand when a given Toeplitz operator belongs to a Schatten-von Neumann class. The methods of the approximation theory (i.e., Legendre polynomials) are used to advance in this direction.

International audience; We study a parabolic Lotka-Volterra equation, with an integral term representing competition, and time periodic growth rate. This model represents a trait structured population in a time periodic environment. After showing the convergence of the solution to the periodic periodic solution of the problem, we study the influence of different factors on the mean limit population. As this quantity is the opposite of a certain eigenvalue, we are able to investigate the influence of the diffusion rate, the period length and the time variance of the environment fluctuations. We also give biological interpretation of the results in the framework of cancer, if the model represents a cancerous cells population under the influence of a periodic treatment. In this framework, we show that the population might benefit from a intermediate rate of mutation.

The conformal bootstrap hypothesis is a powerful idea in theoretical physics which has led to spectacular predictions in the context of critical phenomena. It postulates an explicit expression for the correlation functions of a conformal field theory in terms of its 3-point correlation functions. In this paper we give the first mathematical proof of the conformal bootstrap hypothesis in the context of Liouville theory, a 2-dimensional conformal field theory studied since the eighties in theoretical physics and constructed recently by F. David and the three last authors using probability theory. The proof is based on a probabilistic construction of the Virasoro algebra highest weight modules through spectral analysis of an associated self adjoint operator akin to harmonic analysis on non compact Lie groups but in an infinite dimensional setup., 91 pages. New version contains a proof of bootstrap for the whole possible range of parameters $\gamma\in (0,2)$

National audience; Cet article présente une analyse par matrices aléatoires du clustering sur des flux de données. En supposant que, du fait de limitations mémoire, l'on a accès qu'à un petit nombre de données, la matrice de Gram ne peut être calculée que partiellement. Dans un contexte où les données sont de grande dimension, on étudie sa distribution spectrale limite et ses valeurs et vecteurs propres isolés. Puis, on précise comment ces résultats permettent de réaliser un clustering spectral en ligne avec des garanties de performance théoriques. Des applications au clustering d'images confirment nos résultats.

In this PhD thesis we accomplish two objectives:- We show there is countable dense set at which the integrated density of sates of the Anderson-Bernoulli model on $\Z$ can be explicitly computed, provided the disorder parameter is large enough.- We give a partial proof a a conjecture, first stated in a 2012 article by Filoche and Mayboroda, concerning the product of principal eigenvalue and sup-norm of the landscape function of the Anderson model operator restricted to a large box of $\Z^d$. For the one dimensional case, we give a full proof of such conjecture., Dans cette thèse de doctorat, nous atteignons deux objectifs :- Nous montrons qu'il existe un ensemble dense dénombrable auquel la densité d'états intégrée du modèle d'Anderson-Bernoulli sur $\Z$ peut être explicitement calculée, à condition que le paramètre de désordre soit suffisamment grand.- Nous donnons une preuve partielle d'une conjecture, énoncée pour la première fois dans un article de 2012 par Filoche et Mayboroda, concernant le produit de la valeur propre principale et la sup-norme de la fonction landscape de l'opérateur du modèle d'Anderson restreint à une grande boîte de $\Z^d$. Pour le cas unidimensionnel, nous donnons une preuve complète de cette conjecture.

The aim of this paper is to provide uniform estimates for the eigenvalue spacings of one-dimensional semiclassical Schr\"odinger operators with singular potentials on the half-line. We introduce a new development of semiclassical measures related to families of Schr\"odinger operators that provides a means of establishing uniform non-concentration estimates within that class of operators. This dramatically simplifies analysis that would typically require detailed WKB expansions near the turning point, near the singular point and several gluing type results to connect various regions in the domain., Comment: 22 pages, comments welcome!

Let $\Omega \subset \mathbb{R}^d$ be a bounded open connected set with Lipschitz boundary. Let $A^N$ and $A^D$ be the Neumann Stokes operator and Dirichlet Stokes operator on $\Omega$, respectively. Further let $\lambda_1^N \leq \lambda_2^N \leq \ldots$ and $\lambda_1^D \leq \lambda_2^D \leq \ldots$ be the eigenvalues of $A^N$ and $A^D$ repeated with multiplicity, respectively. Then \[ \lambda_{n+1}^N < \lambda_n^D \] for all $n \in \mathbb{N}$.

International audience; This paper is devoted to the semiclassical analysis of the spectrum of the Dirichlet-Pauli operator on an annulus. We assume that the magnetic field is strictly positive and radial. We give an explicit asymptotic expansion at first order of the lowest eigenvalues of this operator when the semiclassical parameter tends to zero. In particular, we describe the Ahranov-Bohm effect that has been revealed, for constant magnetic field, in a recent paper by Helffer and Sundqvist.

We consider a semiclassical linear Boltzmann model with a non local collision operator. We provide sharp spectral asymptotics for the small spectrum in the low temperature regime from which we deduce the rate of return to equilibrium as well as a metastability result. The main ingredients are resolvent estimates obtained via hypocoercive techniques and the construction of sharp Gaussian quasimodes through an adaptation of the WKB method., 41 pages, 1 figure

We implement numerically formulas of Isaev and Novikov (2022 J. Math. Pures Appl. 163 318–33) for finding a compactly supported function v on R d , d ⩾ 1, from its Fourier transform F [ v ] given within the ball B r . For the one-dimensional case, these formulas are based on the theory of prolate spheroidal wave functions, which arise, in particular, in the singular value decomposition of the aforementioned band-limited Fourier transform for d = 1. In multidimensions, these formulas also include inversion of the Radon transform. In particular, we give numerical examples of super-resolution, that is, recovering details beyond the diffraction limit.

In the present article, we study the discrete spectrum of certain bounded Toeplitz operators with harmonic symbol on a Bergman space. Using the methods of classical perturbaton theory and recent results by Borichev-Golinskii-Kupin and Favorov-Golinskii, we obtain a quantitative result on the distribution of the discrete spectrum of the operator in the unbounded (outer) component of its Fredholm set., Comment: 8 p

We prove two results concerning the nodal sets of eigenfunctions of sub-Laplacians. The first one asserts the validity in this setting of Courant's theorem on the number of nodal domains of eigenfunctions. The second one is the C/\sqrt{\lambda} density (with respect to the sub-Riemannian distance) of the nodal sets of eigenfunctions with eigenvalue \lambda.

In the first edition of this book, we discussed methods for determining the support of an inhomogeneous media from measured far field data as well as an extensive study of the central role played by the transmission eigenvalue problem in the mathematical development of these methods. In particular, we introduced the generalized linear sampling method (GLSM) and showed that this method provides a mathematical explanation of why it is permissible to use Tikhonov regularization to obtain an approximate solution of the far field equation associated with the linear sampling method.In the five years since the first edition of our book appeared, there has been considerable progress in both the development of GLSM as well as the theory of transmission eigenvalues. In this second edition, in addition to correcting typos in the first edition, we have added several highlights taken from these new developments. In particular, we have included new chapters on 1) the use of modified background media in the nondestructive testing of materials and in particular methods for determining the modified transmission eigenvalues that arise in such applications from measured far field data, 2) a study of a subset of transmission eigenvalues, called non-scattering wave numbers, through the use of techniques taken from the theory of free boundary value problems for elliptic partial differential equations and 3) the duality between scattering poles and transmission eigenvalues which, in addition to their intrinsic mathematical interest, leads to new methods for the numerical computation of scattering poles.

In Press.-- Pre-publication available at: https://hal.archives-ouvertes.fr/hal-03469355v2, There are several phenomena in nature governed by simultaneous or intermittent diffusion and advection processes. Many of these systems are networked by their own nature. Here we propose a degree-biased advection processes to undirected networks. For that purpose we define and study the degree-biased advection operator. We then develop a degree-biased advection-diffusion equation on networks and study its general properties. We give computational evidence of the utility of this new model by studying random graphs as well as a real-life patched landscape network in southern Madagascar. In the last case we show that the foraging movement of the species L. catta in this environment occurs mainly in a diffusive way with important contributions of advective motions in agreement with previous empirical observations., M.M. thanks financial support PRE2020-092875 by MCIN/AEI /10.13039/501100011033 and by FSE invierte en tu futuro. E.E. thanks Grant PID2019-107603GB-I00 by MCIN/ AEI /10.13039/501100011033.

In this short note, we prove that for every bounded, planar and convex set Ω: λ1(Ω)T (Ω) |Ω| ≤ π 2 12 1 + 2 2(6 + π 2) − π 2 4 + π 2 2 ≈ 0.996613... where λ1, T and | • | are the first Dirichlet eigenvalue, the torsion and the measure. This bound improves the one provided in [8], which is also an improvement of Pólya's inequality λ 1 (Ω)T (Ω) |Ω| < 1 in the case of planar convex sets.

International audience; This article is the continuation of our first work on the determination of the cases where there is equality in Courant's Nodal Domain theorem in the case of a Robin boundary condition (with Robin parameter h). For the square, our first paper focused on the case where h is large and extended results that were obtained by Pleijel, Bérard-Helffer, for the problem with a Dirichlet boundary condition. There, we also obtained some general results about the behaviour of the nodal structure (for planar domains) under a small deformation of h, where h is positive and not close to 0. In this second paper, we extend results that were obtained by Helffer-Persson-Sundqvist for the Neumann problem to the case where h > 0 is small. MSC classification (2010): 35P99, 58J50, 58J37.

We equip the regular Fr\'echet Lie group of invertible, odd-class, classical pseudodifferential operators $Cl^{0,*}_{odd}(M,E)$ -- in which $M$ is a compact smooth manifold and $E$ a (complex) vector bundle over $M$ -- with pseudo-Riemannian metrics, and we use these metrics to introduce a large class of rigid body equations. We adapt to our infinite-dimensional setting Manakov's classical observation on the integrability of Euler's equations for the rigid body, and we show that our equations can be written in Lax form (with parameter) and that they admit an infinite number of integrals of motion. We also prove the existence of metric connections, we show that our rigid body equations determine geodesics on $Cl^{0,*}_{odd}(M,E)$, and we present rigorous formulas for the corresponding curvature and sectional curvature. Our main tool is the theory of renormalized traces of pseudodifferential operators on compact smooth manifolds without boundary., Comment: signs checked from the last version

We consider second order differential equations with real coefficients that are in the limit circle case at infinity. Using the semiclassical Ansatz, we construct solutions (the Jost solutions) of such equations with a prescribed asymptotic behavior for $x\to\infty$. It turns out that in the limit circle case, this Ansatz can be chosen common for all values of the spectral parameter $z$. This leads to asymptotic formulas for all solutions of considered differential equations, both homogeneous and non-homogeneous. We also efficiently describe all self-adjoint realizations of the corresponding differential operators in terms of boundary conditions at infinity and find a representation for their resolvents., Comment: arXiv admin note: text overlap with arXiv:2105.08641, arXiv:2104.13609

There exists many numerical methods to simulate wave propagation through complex media with a very good precision. However, taking into account the fluctuations of the propagation medium necessitates a statistical approach implying a prohibitive numerical cost.To have those studies affordable, we propose the construction of a metamodel based on a polynomial chaos decomposition of normal modes. This approach presents the great advantage to give statistics of signals propagating in a random medium at an affordable numerical cost.Those results are illustrated with acoustic propagation in the atmosphere. In fact, meteorological fluctuations have a critical impact on the propagation, it is therefore essential to take them into account. The numerical cost of a simulation over thousands of kilometers fully justifies the use of a metamodel. An application to source localization is proposed to illustrate the joint use of a metamodel and a bayesian inversion. The bayesian framework allows a resolution of the inverse problem in a probabilistic context able to take into account the fluctuations of the medium and uncertainties due to unknown source localization., De nombreuses méthodes permettent de simuler numériquement la propagation d'une onde dans un milieu complexe avec une excellente précision. Cependant, la prise en compte des fluctuations du milieu de propagation requiert un traitement statistique nécessitant un grand nombre d'appel a des codes de calcul souvent coûteux.Afin de rendre accessible ces études nous proposons la construction d'un métamodèle basé sur une décomposition en polynômes de chaos des modes normaux. Cette approche permet de restituer les statistiques des signaux se propageant dans un milieu aléatoire avec un coût calcul moindre.Les applications proposées dans cette thèse concernent la propagation d'ondes acoustiques dans l'atmosphère terrestre. En effet, les fluctuations météorologiques modifiant considérablement les conditions de propagation, leur prise en compte est indispensable. Le coût numérique de la simulation sur un domaine de plusieurs centaines de milliers de kilomètres carrés justifie pleinement l'utilisation d'un métamodèle.Une application à la localisation de source couplant ces techniques de métamodèlisation avec une approche bayésienne est aussi proposée. En effet, le cadre bayésien permet une résolution du problème inverse dans un cadre probabiliste capable de prendre en compte les fluctuations du milieu et l'incertitude sur la localisation de la source.

In the field of enumeration of walks in cones, it is known how to compute asymptotically the number of excursions (finite paths in the cone with fixed length, starting and ending points, using jumps from a given step set). As it turns out, the associated critical exponent is related to the eigenvalues of a certain Dirichlet problem on a spherical domain. An important underlying question is to decide whether this asymptotic exponent is a (non-)rational number, as this has important consequences on the algebraic nature of the associated generating function. In this paper, we ask whether such an excursion sequence might admit an asymptotic expansion with a first rational exponent and a second non-rational exponent. While the current state of the art does not give any access to such many-term expansions, we look at the associated continuous problem, involving Brownian motion in cones. Our main result is to prove that in dimension three, there exists a cone such that the heat kernel (the continuous analogue of the excursion sequence) has the desired rational/non-rational asymptotic property. Our techniques come from spectral theory and perturbation theory. More specifically, our main tool is a new Hadamard formula, which has an independent interest and allows us to compute the derivative of eigenvalues of spherical triangles along infinitesimal variations of the angles., 23 pages, 5 figures

We study the band structure of self-adjoint elliptic operators $\mathbb{A}_g= -\nabla \cdot \sigma_{g} \nabla$, where $\sigma_g$ has the symmetries of a honeycomb tiling of $\mathbb{R}^2$. We focus on the case where $\sigma_{g}$ is a real-valued scalar: $\sigma_{g}=1$ within identical, disjoint "inclusions", centered at vertices of a honeycomb lattice, and $\sigma_{g}=g \gg1 $ (high contrast) in the complement of the inclusion set (bulk). Such operators govern, e.g. transverse electric (TE) modes in photonic crystal media consisting of high dielectric constant inclusions (semi-conductor pillars) within a homogeneous lower contrast bulk (air), a configuration used in many physical studies. Our approach, which is based on monotonicity properties of the associated energy form, extends to a class of high contrast elliptic operators that model heterogeneous and anisotropic honeycomb media. Our results concern the global behavior of dispersion surfaces, and the existence of conical crossings (Dirac points) occurring in the lowest two energy bands as well as in bands arbitrarily high in the spectrum. Dirac points are the source of important phenomena in fundamental and applied physics, e.g. graphene and its artificial analogues, and topological insulators. The key hypotheses are the non-vanishing of the Dirac (Fermi) velocity $v_D(g)$, verified numerically, and a spectral isolation condition, verified analytically in many configurations. Asymptotic expansions, to any order in $g^{-1}$, of Dirac point eigenpairs and $v_D(g)$ are derived with error bounds. Our study illuminates differences between the high contrast behavior of $\mathbb{A}_g$ and the corresponding strong binding regime for Schroedinger operators., Comment: 63 pages, 13 figures

We investigate an application of a mathematically robust minimization method -- the gradient method -- to the consistencization problem of a pairwise comparisons (PC) matrix. Our approach sheds new light on the notion of a priority vector and leads naturally to the definition of instant priority vectors. We describe a sample family of inconsistency indicators based on various ways of taking an average value, which extends the inconsistency indicator based on the "$\sup$"- norm. We apply this family of inconsistency indicators both for additive and multiplicative PC matrices to show that the choice of various inconsistency indicators lead to non-equivalent consistencization procedures., Comment: 1 figure, several corrections and precisions

The current Ph.D. Thesis is devoted to the study of the spectral properties of a specifis class of compact Toeplitz operators on the bergman space of holomorphic functions of the disk of the complex plane. Recall that a Toeplitz operator in question is defined as the composition of the Riesz projection on space and the multiplication by a given bounded function, Which is colled the operator symnbol.More precisely, we consider Toeplitz operators with continuous symbols decaying logarithmically toward the boundary of the disk and we obtain the spectral asymptotics (i.e., the singular value asymptotics) for its objects. These results are then applied to the stady of the spectral properties of certain compact “ band “ matrices.The organization of the thesis is a follows. The first chapter is an introduction to subject which gives, among other things, the formulation of the problem of the thesis and presents the known results on the subject. The second chapter is devoted to the presentation of Bergman spaces of holomorphic functions. We study their general properties, as well as the integral operators “ of the Riesz projector type “ which will play a certain role thereafter.The third chapter of the thesis contains the necessary reminders on operator theory, and in particular on compact operators. In particular, we introduce the important notion of “ asymptotic orthogonality “ of a family of compact operators. The basic tools on Toeplitz operators in Bergman spaces are given in the fourth chapter. Finally, the fifth (and the last) chapter of the manuscript synthesizes the techniques of previous chapters and is dedicated to the detailed demonstration of the main theorem of the thesis., La présente thèse est consacrée à l’étude des propriétés spectrales d’une classe spécifique d’opérateurs de Toeplitz compacts sur l’espace de Bergman de fonctions holomorphes du disque unité du plan complexe. Rappelons qu’un opérateur de Toeplitz en question est défini comme la composition de la projection de Riesz sur l’espace et la multiplication par une fonction bornée donnée, que l’on appelle le symbole de l’opérateur.Plus précisément, nous considérons des opérateurs de Toeplitz avec des symboles continus à la décroissance logarithmique vers le bord du disque et nous obtenons les asymptotiques spectrales (i.e., les asymptotiques des valeurs singulières) pour ses objets. Ces résultats sont ensuite appliqués à l’étude des propriétés spectrales de certaines matrices “à bandes” compactes.L’organisation de la thèse est la suivante. Le premier chapitre est une introduction au sujet qui donne, entre outre, la formulation du problème de la thèse et présente les résultats connus sur le sujet. Le second chapitre est consacré à la présentation d’espaces de Bergman des fonctions holomorphes. On y étudie leurs propriétés générales, ainsi que les opérateurs intégraux "du type de projecteur de Riesz" qui joueront un certain rôle par la suite. Le troisième chapitre de la thèse contient les rappels nécessaires sur la théorie des opérateurs, et notamment sur les opérateurs compacts. En particulier, on y introduit la notion importante d'"orthogonalité asymptotique" d'une famille d'opérateurs compacts. Les outils de base sur les opérateurs de Toeplitz dans les espaces de Bergman sont donnés au quatrième chapitre. Enfin, le cinquième (et le dernier) chapitre du manuscrit fait la synthèse des techniques de chapitres précédentes et il est dédié à la démonstration détaillée du théorème principal de la thèse.

We consider the abstract scattering structure of the non homogeneous linearized Vlasov-Poisson equations from the viewpoint of trace class properties which are emblematic of the abstract scattering theory [13, 14, 15, 19]. In dimension 1+1, we derive an original reformulation which is trace class. It yields the existence of the Moller wave operators. The non homogeneous background electric field is periodic with 4 + ε bounded derivatives. Mathematics Subject Classification (2010). Primary: 47A40; Secondary: 35P25.

In this paper, we establish a general result for the existence and the uniqueness of the quasi-stationary distribution of a strongly Feller Markov process killed when it exits a domain D, under some Lyapunov function condition. Our result covers the case of hypoelliptic damped Hamiltonian systems. Our method is based on the characterization of the essential spectral radius by means of Lyapunov functions and measures of non-compactness.

In this article, we consider algebras $\mathcal{A}$ of non-formal pseudodifferential operators over $S^1$ which contain $C^\infty(S^1),$ understood as multiplication operators. We apply a construction of Chern-Weil type forms in order to get $2k-$closed cocycles. For $k=1,$ we obtain a cocycle on the algebra of (maybe non classical) pseudodifferential operators with the same cohomology class as the Schwinger cocycle on the algebra of Classical pseudodifferential operators, previously extended and studied by the author on algebras of the same type., A result deleted in this new shorter version. arXiv admin note: text overlap with arXiv:2007.00387

This thesis achieves a theoretical and numerical studies of cavities partially filled with a negative material, that is a material for which the magnetic permeability and/or the electric permittivity (or at least their real part) become negative in some frequency ranges. This study is part of the main thrust of the work started in our team focusing on the electromagnetic wave propagation in presence of such negative materials, at a given fraquency. The purpose of this thesis is to take into account the frequency dispersion, that is the frequency dependence of the permeability and/or the permittivity, considering the frequency as the spectral parameter. We highlight the essential spectrum arising from the presence of negative material, as well as the resulting resonance phenomena, for different models describing this material. The theoretical study focuses on the case of polygonal bi-dimensional cavities for the Drude and the Lorentz models (with dissipation or not). The theoretical study of the simplest model (the non dissipative Drude model) is extended to the case of a curved (but regular) interface. This model is also the subject of a numerical study, aimed at exploring the effect of a finite element discretization of the theoretical problem, and thus highlight the difficulties to numerically notice some of the resonance phenomena., Cette thèse réalise une étude théorique et numérique du spectre de cavités partiellement composées de matériau négatif, c'est-à-dire de matériau pour lequel la perméabilité magnétique et/ou la permittivité électrique (ou au moins leur partie réelle) deviennent négatives dans certaines plages de fréquences. Cette étude s'inscrit dans la continuité des travaux engagés dans notre laboratoire qui se concentrent sur la propagation des ondes électromagnétiques en présence de matériau négatif, à fréquence fixée. L'objectif de cette thèse est de prendre en compte la dispersion fréquentielle, autrement dit la dépendance en fréquence de la perméabilité et de la permittivité, en considérant la fréquence comme paramètre spectral. Nous mettons en évidence le spectre essentiel résultant de la présence de matériau négatif ainsi que les phénomènes de résonance qui en découlent, pour différents modèles décrivant ce matériau.L'étude théorique se concentre sur le cas de cavités bidimensionnelles polygonales pour les modèles de Drude et de Lorentz (avec et sans dissipation). L'étude théorique du modèle le plus simple (Drude non dissipatif) est étendue au cas d'une interface courbe (mais régulière).Ce modèle fait également l'objet d'une étude numérique, visant à explorer l'effet d'une discrétisation éléments finis du problème théorique, et ainsi mettre en avant les difficultés à observer numériquement certains des phénomènes de résonance.

We prove dispersive estimates for two models : the adjacency matrix on a discrete regular tree, and the Schrödinger equation on a metric regular tree with the same potential on each edge/vertex. The latter model can be thought of as an extension of the case of periodic Schrödinger operators on the real line. We establish a t^{−3/2}-decay for both models which is sharp, as we give the first-order asymptotics.

In this article, we propose to build a tensor that generalizes the concept of "Stess-energy" tensor in general relativity. An extension of this tensor is built by coupling a Koszul connection and a section of Dirac Lorentzienne. The connection and section are obtained by minimizing a Yang-Mills type Lagrangian., Dans cet article, on propose de construire un tenseur qui généralise la notion de tenseur "énergie-impulsion" de la relativité générale. On construit une extension de ce tenseur par couplage d'une connexion de Koszul et d'une section de Dirac Lorentzienne. La connexion et la section sont obtenues en minimisant un Lagrangien du type Yang-Mills.

International audience; We address the problem of filtering out localized time-frequency components in signals. The problem is formulated as a minimization of a suitable quadratic form, that involves a data fidelity term on the short-time Fourier transform outside the support of the undesired component, and an energy pe-nalization term inside the support. The minimization yields a linear system whose solution can be expressed in closed form using Gabor multipliers. We provide an analysis of the solution and its approximation by truncated eigenvalue expansion, and illustrate its performance on synthetic mixtures of audio signals. The proposed approach outperforms approaches that are routinely used in applications.

International audience; The aim of this work is to study the controllability of the bilinear Schr\"odinger equation on compact graphs. In particular, we consider the equation (BSE) $i\partial_t\psi=-\Delta\psi+u(t)B\psi$ in the Hilbert space $L^2(\mathscr{G},\mathbb{C})$, with $\mathscr{G}$ being a compact graph. The Laplacian $-\Delta$ is equipped with self-adjoint boundary conditions, $B$ is a bounded symmetric operator and $u\in L^2((0,T),\mathbb{R})$ with $T>0$. We provide a new technique leading to the global exact controllability of the (BSE) in $D(|\Delta|^{s/2})$ with $s\geq 3$. Afterwards, we introduce the "energetic controllability", a weaker notion of controllability useful when the global exact controllability fails. In conclusion, we develop some applications of the main results involving for instance star graphs.

Dénoncer la fausse science épidémiologique : réquisitoire contre l'article "Estimating the burden of SARS-CoV-2 in France" : 17 chercheurs de 10 instituts ne comprennent ni les probabilités ni les mathématiques et inventent l'équation générale de la vérité qu'ils résolvent en double aveugle avant d'en maquiller piteusement la présentation et de se suicider sur la théorie du R 0

Let $${ \rm I\!K}$$ be an ultrametric complete valued field and $${ \rm I\!E}$$ be an ultrametric space. We examine some Banach algebras $$S$$ of bounded continuous functions from $${ \rm I\!E}$$ to $${ \rm I\!K}$$ with the use of ultrafilters, particularly the relation of stickness. We recall and deepen results obtained in a previous paper by N. Mainetti and the third author concerning the whole algebra $$\cal A$$ of all bounded continuous functions from $${ \rm I\!E}$$ to $${ \rm I\!K}$$ . Every maximal ideal of finite codimension of $$\cal A$$ is of codimension $$1$$ and we show that this property also holds for every algebra $$S$$ , provided $${ \rm I\!K}$$ is perfect. If $$S$$ admits the uniform norm on $${ \rm I\!E}$$ as its spectral norm, then every maximal ideal is the kernel of only one multiplicative semi-norm, the Shilov boundary is equal to the whole multiplicative spectrum and the Banaschewski compactification of $${ \rm I\!E}$$ is homeomorphic to the multiplicative spectrum of $$S$$ .

In this work, we study the controllability of the bilinear Schr\"odinger equation on infinite graphs for periodic quantum states. We consider the bilinear Schr\"odinger equation $i\partial_t\psi=-\Delta\psi+u(t)B\psi$ in the Hilbert space $L^2_p$ composed by functions defined on an infinite graph $\mathscr{G}$ verifying periodic boundary conditions on the infinite edges. The Laplacian $-\Delta$ is equipped with specific boundary conditions, $B$ is a bounded symmetric operator and $u\in L^2((0,T),\mathbb{R})$ with $T>0$. We present the well-posedness of the system in suitable subspaces of $D(|\Delta|^{3/2})$ . In such spaces, we study the global exact controllability and we provide examples involving for instance tadpole graphs and star graphs with infinite spokes., Comment: arXiv admin note: text overlap with arXiv:1811.04273

We study the resonances of the Laplacian acting on the compactly supported sec- tions of a homogeneous vector bundle over a Riemannian symmetric space of the non- compact type. The symmetric space is assumed to have rank-one but the irreducible representation τ of the maximal compact K defining the vector bundle is arbitrary. We determine the resonances. Under the additional assumption that τ occurs in the spherical principal series, we determine the resonance representations. They are all irreducible. We find their Langlands parameters, their wave front sets and determine which of them are unitarizable., On étudie les résonances de l’opérateur de Laplace agissant sur les sections d’un fibré vectoriel homogène sur un espace symétrique Riemannien de type non-compact. On suppose que l’espace symétrique est de rang un, mais la représentation irréductible τ du compact maximal K, qui définit le fibré vectoriel, est quelconque. On détermine alors les résonances. Si on suppose de plus que τ apparaît dans les représentations de la série principale sphérique, on détermine les représentations issues des résonances. Elles sont toutes irréductibles. On trouve leurs paramètres de Langlands, leurs fronts d’onde et lesquelles sont unitarisables.

Based on the example of stochastic cosurfaces, we develop the notion of groups of series indexed by a graded small category, with coefficients in a Fr\"olicher algebra. The presence of the notion of small category is motivated by the cobordism-like relations of stochastic cosurfaces, while the use of Fr\"olicher algebra is a framework adapted to convolution of probabilities. Basic description of these groups, which generalize classical groups of series, is given, and the example raised by non-abelian stochatic cosurfaces of any dimension is studied.MSC(2010): 22E65, 22E66, 58B25, 70G65.

International audience; We characterise those complete Kähler manifolds supporting a nonconstant real-valued function with critical points whose Hessian is nonnegative, complex linear, has pointwise two eigenvalues and whose gradient is a Hessian-eigenvector.

In this thesis, we study the spectrum of particular Schrödinger operators that are associated to magnetic fields with geometrical invariances. More precisely we are interested on the case of a constant magnetic fields on the half-plane with Neumann boundary condition and on the case of a magnetic field associated with an axysymmetric potential in high dimension.In each configuration, the Schrödinger operator is fibered and its spectrum is described by a family of real function, called band functions. Inside the spectrum, some energy level enjoy remarkable properties and are refered to as thresholds. They can match with the critical values of the band functions or with their limits at infinity.In the case of a constant magnetic field on the half plane, we study the existence of resonances at the neighborhood of the atteined thresholds that are the critical values of the band functions. The operator is perturbed by a magnetic potential of by a deformation of the border of the half plane. To deal with each of these cases, we extend the free resolvant followed by the resolvant of the perturbed operator through the spectrum. Therefore we identify the resonancs with the poles of this extension. Using this method we also provide the asymptotic behavior of these resonances as the perturbation vanishes.The second model we study is the case of a unitary magnetic fields associated with an asysymmetric potential in the whole space. The magnetic fields is translationally invariant. We study the associated band funtions, in particular the influence of a the limit of the band functions that are a new kind of thershold. Moreover each of these limits concenr an infinity of band function and every energy level is intersected by an infinity of band functions. Our main goal is to look for some consequences of this accumilation. We prove that the band functions that intersect a given energy level may have an arbitrarily small derivative. We apply this result to prove that quantum particle affected by this fields can be localised in energy away from the thersholds but also carry a arbitrarily small current. This situation is diffrent as the one that existed before on this subject., Cette thèse analyse le spectre d’opérateurs de Schrödinger particuliers, associés à des champs magnétiques possédant des invariances géométriques. Nous nous intéressons aux cas d’un champ magnétique constant sur le demi-plan, avec une condition au bord de Neumann, puis d’un champ magnétique en dimensions supérieures, associé à un potentiel axisymétrique.Dans ces deux configurations, l’opérateur de Schrödinger a une structure fibrée et son spectre est décrit par une famille de fonctions réelles appelées fonctions de bande. Au sein du spectre, certains niveaux d’énergie, appelés seuils, sont remarquables. Ils peuvent correspondre aux valeurs critiques de ces fonctions de bande ou à leurs limites en l'infini.Dans le cas d'un champ magnétique constant sur le demi-plan, nous étudions l'existence de résonances au voisinage des seuils correspondant aux valeurs critiques. Nous perturbons l’opérateur soit par l’ajout d’un potentiel électrique soit par une déformation de la frontière du demi-plan. Nous prolongeons la résolvante de l’opérateur libre à travers le spectre puis nous identifions les résonances avec les pôles de cette extension. Cette méthode nous permet également d’analyser le comportement asymptotique de ces résonances lorsque la perturbation est petite. Le second modèle que nous étudions est celui d’un champ magnétique engendré par un potentiel axisymétrique dans l’espace entier. Le champ magnétique généré par ce potentiel est invariant par translation. Nous étudions les fonctions de bande associées, en particulier l’influence d’un autre type de seuil : les limites finies des fonctions de bande. Chacune de ces limites concerne une infinité de fonctions de bande et chaque niveau d’énergie au dessus d'un seuil est intersecté par une infinité de fonctions de bande. Nous nous intéressons aux conséquences de cette accumulation. Nous démontrons que les fonctions de bande qui intersectent un niveau d’énergie donné peuvent avoir une dérivée arbitrairement petite et nous appliquons ce résultat pour démontrer qu'il peut exister des particules quantiques soumises à ce champ magnétique, localisées en énergie loin des seuils, mais portant tout de même un courant arbitrairement faible, ce qui contraste avec les études préexistantes sur le sujet.

This thesis is concerned with various aspects of large deviations theory in relation with statistical physics. Both theoretical and numerical considerations are dealt with. The first part of the work studies long time large deviations properties of diffusion processes. First, we prove new ergodicity results for Feynman-Kac dynamics, both in continuous and discrete time. This leads to new fine results (in the sense of topology) for large deviations of empirical measures of diffusion processes. Various numerical problems are then covered. We first provide precise error estimates on discretizations of Feynman-Kac dynamics, for which the nonlinear features of the dynamics demand new tools. In order to reduce the variance of naive estimators, we provide an adaptive algorithm relying on the technique of stochastic approximation. We finally consider a problem concerning low temperature systems. We present a new method for constructing an approximation of the optimal control from the instanton (or reaction path) theory. The last part of the thesis is concerned with the different topic of Coulomb gases, which appear both in physics and random matrix theory. We first present an efficient method for simulating such gases, before turning to gases under constraint. For such gases, we prove new concentration results in the limit of a large number of particles, under some conditions on the constraint. We also present a simulation algorithm, which confirms the theoretical expectations., Cette thèse s’intéresse à différents problèmes de grandes déviations en rapport avec la physique statistique, qu’elle aborde sous l’angle théorique aussi bien que numérique. La première partie concerne l’étude de grandes déviations en temps long pour les processus de diffusion. Tout d’abord, de nouveaux résultats d’ergodicité sont montrés pour les dynamiques de Feynman-Kac, en temps discret et en temps continu. Ceci conduit à de nouveaux résultats fins (au sens de la topologie considérée) sur les grandes déviations de mesures empiriques de processus de diffusion. Divers aspects numériques sont ensuite abordés. Tout d’abord, des estimées d’erreur précises sont fournies pour les discrétisations de processus de Feynman-Kac, la non-linéarité de la dynamique demandant le développement de nouveaux outils. Afin de réduire la variance des estimateurs classiques de grandes déviations, un algorithme adaptatif est ensuite présenté, qui utilise les techniques dites d’approximation stochastique. Enfin, nous abordons une problème numérique concernant les systèmes à basse température, et présentons une méthode pour construire une approximation du contrôle optimal à partir de la théorie du chemin de réaction. La dernière partie de cette thèse porte sur un sujet légèrement différent, celui des gaz de Coulomb, qui apparaissent en physique mais aussi dans la théorie des matrices aléatoires. Nous présentons d’abord une méthode efficace pour la simulation de tels gaz, avant de nous tourner vers l’étude des gaz sous contrainte. Pour ceux-ci, nous prouvons de nouveaux résultats de concentration dans la limite d’un grand nombre de particules, sous certaines conditions sur la contrainte. Nous présentons également un algorithme de simulation qui confirme les attentes théoriques.

We study unitary representations of semidirect products of a compact quantum group with a finite group. We give a classification of all irreducible unitary representations, a description of the conjugate representation of irreducible unitary representations in terms of this classification, and the fusion rules for the semidirect product., Comment: version submitted for publication (compiled with pdflatex to fix overful lines with references)

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Let φ : D → D be a holomorphic map with a fixed point α ∈ D such that 0 ≤ | φ ′ ( α ) | 1 . We show that the spectrum of the composition operator C φ on the Frechet space Hol ( D ) is { 0 } ∪ { φ ′ ( α ) n : n = 0 , 1 , ⋯ } and its essential spectrum is reduced to {0}. This contrasts the situation where a restriction of C φ to Banach spaces such as H 2 ( D ) is considered. Our proofs are based on explicit formulae for the spectral projections associated with the point spectrum found by Koenigs. Finally, as a byproduct, we obtain information on the spectrum for bounded composition operators induced by a Schroder symbol on arbitrary Banach spaces of holomorphic functions.

The asymptotic variance is a natural indicator of the efficiency for a Markov Chain Monte Carlo algorithm. In this note, we prove that the asymptotic variance of a drift accelerated diffusion converges to zero uniformly if and only if there are no non-trivial first order Sobolev functions in the kernel of the drift generating operator. Its proof is based on spectral analysis in the first order Sobolev space of mean zero functions.