Search

The aim of this paper is to present new sparsity results about the so-called Lieb functional, which is a key quantity in Density Functional Theory for electronic structure calculations for molecules. The Lieb functional was actually shown by Lieb to be a convexification of the so-called L\'evy-Lieb functional. Given an electronic density for a system of $N$ electrons, which may be seen as a probability density on $\mathbb{R}^3$, the value of the Lieb functional for this density is defined as the solution of a quantum multi-marginal optimal transport problem, which reads as a minimization problem defined on the set of trace-class operators acting on the space of electronic wave-functions that are anti-symmetric $L^2$ functions of $\mathbb{R}^{3N}$, with partial trace equal to the prescribed electronic density. We introduce a relaxation of this quantum optimal transport problem where the full partial trace constraint is replaced by a finite number of moment constraints on the partial trace of the set of operators. We show that, under mild assumptions on the electronic density, there exist sparse minimizers to the resulting moment constrained approximation of the Lieb (MCAL) functional that read as operators with rank at most equal to the number of moment constraints. We also prove under appropriate assumptions on the set of moment functions that the value of the MCAL functional converges to the value of the exact Lieb functional as the number of moments go to infinity. We also prove some rates of convergence on the associated approximation of the ground state energy. We finally study the mathematical properties of the associated dual problem.

Let $A$ be an elliptic pseudo-differential operator of order $m$ on a closed manifold $\mathcal{X}$ of dimension $n>0$, formally positive self-adjoint with respect to some positive smooth density $d\mu_\mathcal{X}$. Then, the spectrum of $A$ is made up of a sequence of eigenvalues $(\lambda_k)_{k\geq 1}$ whose corresponding eigenfunctions $(e_k)_{k\geq 1}$ are $C^\infty$ smooth. Fix $s\in\mathbb{R}$ and define \[ K_L^s(x,y)=\sum_{0, Comment: major changes and corrections; 46 pages. arXiv admin note: text overlap with arXiv:1611.02018

We study geometric and spectral properties of typical hyperbolic surfaces of high genus, excluding a set of small measure for the Weil-Petersson probability. We first prove a quantitative estimate of the speed of Benjamini-Schramm convergence to the hyperbolic plane as the genus g goes to infinity. We then use the Selberg pretrace formula to find an estimate for the number of eigenvalues in an interval [a, b], as b and/or g go to infinity. This result proves that, as g → +∞, the spectrum approaches the continuous spectrum of the Laplacian on the hyperbolic plane, with the density appearing in the Selberg trace formula. It also leads to a probabilistic uniform Weyl law, when b → +∞. We also prove a probabilistic upper bound on the number of eigenvalues in the interval [a, b], and noticeably that the number of eigenvalues below 1/4 is O(g (log g)^(−3/4)).

We reprove the fact, due to Backus, that the Poincar{\'e} operator in ellipsoids admits a pure point spectrum with polynomial eigenfunctions.We then show that the eigenvalues of the Poincar{\'e} operator restricted to polynomial vector fields of fixed degree admitsa limit repartition given by a probability measure that we construct explicitely. For that, we use Fourier integral operators and ideas comingfrom Alan Weinstein and the first author in the seventies. The starting observation is that the orthogonal polynomials in ellipsoids satisfy a PDE.

We consider the magnetic Dirac operator with infinite mass boundary condition on an annulus and prove an explicit asymptotic expansion at the first order for the low-lying positive spectrum. An heuristic of proof is given in the last section for the case of the first negative eigenvalue.

[FeFe] hydrogenases are highly efficient metalloenyzmes for hydrogen conversion. Their active site cofactor (the H-cluster) is composed of a canonical [4Fe-4S] cluster ([4Fe-4S](H)) linked to a unique organometallic di-iron subcluster ([2Fe](H)). In [2Fe](H) the two Fe ions are coordinated by a bridging 2-azapropane-1,3-dithiolate (ADT) ligand, three CO and two CN- ligands, leaving an open coordination site on one Fe where substrates (H-2 and H+) as well as inhibitors (e.g. O-2, CO, H2S) may bind. Here, we investigate two new active site states that accumulate in [FeFe] hydrogenase variants where the cysteine (Cys) in the proton transfer pathway is mutated to alanine (Ala). Our experimental data, including atomic resolution crystal structures and supported by calculations, suggest that in these two states a third CN- ligand is bound to the apical position of [2Fe](H). These states can be generated both by "cannibalization" of CN- from damaged [2Fe](H) subclusters as well as by addition of exogenous CN-. This is the first detailed spectroscopic and computational characterisation of the interaction of exogenous CN- with [FeFe] hydrogenases. Similar CN--bound states can also be generated in wild-type hydrogenases, but do not form as readily as with the Cys to Ala variants. These results highlight how the interaction between the first amino acid in the proton transfer pathway and the active site tunes ligand binding to the open coordination site and affects the electronic structure of the H-cluster.

In this work, we analyse the metastability of non-reversible diffusion processes $$dX_t=\boldsymbol{b}(X_t)dt+\sqrt h\,dB_t$$ on a bounded domain $\Omega$ when $\mathbf{b}$ admits the decomposition $\mathbf{b}=-(\nabla f+\mathbf{\ell})$ and $\nabla f \cdot \mathbf{\ell}=0$. In this setting, we first show that, when $h\to 0$, the principal eigenvalue of the generator of $(X_t)_{t\ge 0}$ with Dirichlet boundary conditions on the boundary $\partial\Omega$ of $\Omega$ is exponentially close to the inverse of the mean exit time from $\Omega$, uniformly in the initial conditions $X_0=x$ within the compacts of $\Omega$. The asymptotic behavior of the law of the exit time in this limit is also obtained. The main novelty of these first results follows from the consideration of non-reversible elliptic diffusions whose associated dynamical systems $\dot X=\mathbf{b}(X)$ admit equilibrium points on $\partial\Omega$. In a second time, when in addition $\div \mathbf{\ell} =0$, we derive a new sharp asymptotic equivalent in the limit $h\to 0$ of the principal eigenvalue of the generator of the process and of its mean exit time from $\Omega$. Our proofs combine tools from large deviations theory and from semiclassical analysis, and truly relies on the notion of quasi-stationary distribution.

In this paper we investigate the spectrum of the Dirac operator posed in a tubular neighborhood of a planar loop with infinite mass boundary conditions. We show that when thewidth of the tubular neighborhood goes to zero the asymptotic expansion of the eigenvalues isdriven by a one dimensional operator of geometric nature involving the curvature of the loop.

On an asymptotically conical manifold we prove time decay estimates for the flow of the Schrödinger, wave and Klein-Gordon equations via some continuity properties of the spectral measure. To keep the paper at a reasonable length we limit ourselves to the low energy part of the spectrum, which is the one that dictates the decay rates. With this paper we extend sharp estimates that are known in the asymptotically flat case (see Bouclet and Burq in [BB21]) to this more general geometric framework and therefore recover the same decay properties as for the euclidean case. The first step is to prove some resolvent estimates via a limiting absorption principle. It is at this stage that the proof of the previously mentioned authors fails, in particular when we try to recover a low frequency positive commutator estimate. Once the resolvent estimates are established we derive regularity for the spectral measure that in turn is applied to obtain the decay of the flows.

International audience; In this article, we introduce an infinite-dimensional deterministic SIS model which takes into account the heterogeneity of the infections and the social network among a large population. We study the long-time behavior of the dynamic. We identify the basic reproduction number $R_0$ which determines whether there exists a stable endemic steady state (super-critical case: $R_0>1$) or if the only equilibrium is disease-free (critical and sub-critical case: $R_0\leq1$). As an application of this general study, we prove that the so-called ``leaky'' and ``all-or-nothing'' vaccination mechanism have the same effect on $R_0$. This framework is also very natural and intuitive to model lockdown policies and study their impact.

We consider a many-body system of 2D anyons, free quantum particles with general statistics parameter \alpha \in ]0,2[. In the magnetic gauge picture they are described as bosons attached to Aharonov-Bohm fluxes of intensity 2 \pi \alpha generating long-range magnetic forces. A dimensional reduction to 1D is obtained by imposing a strongly anisoptropic trapping potential. This freezes the motion in the direction of strong trapping, leading to 1D physics along the weak direction. The latter is governed to leading order by the Tonks-Girardeau model of impenetrable bosons, independently of \alpha.

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.

Anyons with a statistical phase parameter $\alpha\in(0,2)$ are a kind of quasi-particles that, for topological reasons, only exist in a 1D or 2D world. We consider the dimensional reduction for a 2D system of anyons in a tight wave-guide. More specifically, we study the 2D magnetic-gauge picture model with an imposed anisotropic harmonic potential that traps particles much stronger in the $y$-direction than in the $x$-direction. We prove that both the eigenenergies and the eigenfunctions are asymptotically decoupled into the loose confining direction and the tight confining direction during this reduction. The limit 1D system for the $x$-direction is given by the impenetrable Tonks-Girardeau Bose gas, which has no dependency on $\alpha$, and no trace left of the long-range interactions of the 2D model.

International audience; We establish some properties of quantum limits on a product manifold, proving for instance that, under appropriate assumptions, the quantum limits on the product of manifolds are absolutely continuous if the quantum limits on each manifolds are absolutely continuous. On a product of Riemannian manifolds satisfying the minimal multiplicity property, we prove that a periodic geodesic can never be charged by a quantum limit.

In this work, we revisit the Krein-Rutman theory for semigroups of positive operators in a Banach lattice framework and we provide some very general, efficient and handy results with constructive estimates about- the existence of a solution to the first eigentriplet problem;- the geometry of the principal eigenvalue problem;- the asymptotic stability of the first eigenvector with possible constructive rate of convergence.This abstract theory is motivated and illustrated by several examples of differential, intro-differential and integral operators. In particular, we revisit the first eigenvalue problem and the asymptotic stability of the first eigenvector for- some parabolic equations in a bounded domain and in the whole space;- some transport equations in a bounded or unbounded domain, including some growth-fragmentationmodels and some kinetic models;- the kinetic Fokker-Planck equation in the torus and in the whole space;- some mutation-selection models.

We construct the infinitesimal generator of the Brox diffusion on a line with a periodic Brownian environment. This gives a new construction of the process and allows to solve the singular martingale problem. We prove that the associated semigroup is strong Feller with Gaussian lower and upper bounds. This also yields a construction of the Brox diffusion on a segment with periodic or Dirichlet boundary conditions. In this bounded space, we prove that there exists a unique measure and the existence of a spectral gap giving exponential ergodicity of the diffusion.

On a complete weighted Riemannian manifold $(M^n,g,\mu)$ satisfying the doubling condition and the Poincar{\'e} inequalities, we characterize the class of function $V$ such that the Schr{\"o}dinger operator $\Delta-V$ maps the homogeneous Sobolev space $W_o^{1,2} (M)$ to its dual space. On Euclidean space, this result is due to Maz'ya and Verbitsky. In the proof of our result, we investigate the weighted $L^2$-boundedness of the Hodge projector.

This work is concerned with operators of the type A = −c∆ acting in domains Ω := Ω ′ × (0, H) ⊆ R^d × R ^+. The diffusion coefficient c > 0 depends on one coordinate y ∈ (0, H) and is bounded but may be discontinuous. This corresponds to the physical model of "layered media", appearing in acoustics, elasticity, optical fibers... Dirichlet boundary conditions are assumed. In general, for each ε > 0, the set of eigenfunctions is divided into a disjoint union of three subsets : Fng (non-guided), Fg (guided) and Fres (residual). The residual set shrinks as ε → 0. The customary physical terminology of guided/non-guided is often replaced in the mathematical literature by concentrating/non-concentrating solutions, respectively. For guided waves, the assumption of "layered media" enables us to obtain rigorous estimates of their exponential decay away from concentration zones. The case of non-guided waves has attracted less attention in the literature. While it is not so closely connected to physical models, it leads to some very interesting questions concerning oscillatory solutions and their asymptotic properties. Classical asymptotic methods are available for c(y) ∈ C 2 but a lesser degree of regularity excludes such methods. The associated eigenfunctions (in Fng) are oscillatory. However, this fact by itself does not exclude the possibility of "flattening out" of the solution between two consecutive zeros, leading to concentration in the complementary segment. Here we show it cannot happen when c(y) is of bounded variation, by proving a "minimal amplitude hypothesis". However the validity of such results when c(y) is not of bounded variation (even if it is continuous) remains an open problem.; Dans ce papier nous considérons des opérateurs du type A = −c∆ agissant sur des ouverts Ω := Ω ′ × (0, H) ⊆ R^d × R^+, le coefficient de diffusion c > 0 dépendant de la seule coordonnée y ∈ (0, H). Il est borné mais peut être discontinu. Cette situation correspond au modèle physique des "milieux stratifiés" qui intervient en acoustique, en élasticité, dans les fibres optiques, ... Nous supposons la condition de Dirichlet homogène au bord de Ω. Pour chaque ε > 0, nous créons une partition de l'ensemble des fonctions propres en trois sous-ensembles : Fng (fonctions non guidées), Fg (fonctions guidées) et Fres (ensemble résiduel), ce dernier diminuant lorsque ε → 0. La terminologie classique guidée/non guidée est souvent remplacée en mathématiques par les expressions : solutions se concentrant, respectivement ne se concentrant pas. Pour les ondes guidées, le cadre des "milieux stratifiés" nous permet d'obtenir des estimations rigoureuses sur leur décroissance exponentielle loin des zones de concentration. La littérature s'est moins penchée sur le cas des ondes non guidées. Bien que semblant moins intéressante pour les modèles physiques, elle conduit à de très intéressantes questions concernant les solutions oscillantes et leurs propriétés asymptotiques. Des méthodes asymptotiques classiques sont disponibles lorsque (y) ∈ C^2 mais ne marchent pas avec une régularité moindre. Les fonctions de Fng sont oscillantes mais ceci n'exclue pas qu'elles puissent s'aplatir entre deux zéros consécutifs, conduisant à une concentration dans le segment complémentaire. Nous montrons ici que cela ne peut pas arriver quand c(y) est à variation bornée en vérifiant une "hypothèse d'amplitude minimale". Cependant la validité de tels résultats reste un problème ouvert quand c(y) n'est plus à variation bornée, même si ce coefficient est continu.

International audience; 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.

We study spectral properties of sub-Riemannian Laplacians, which are hypoelliptic operators. The main objective is to obtain quantum ergodicity results, what we have achieved in the 3D contact case. In the general case we study the small-time asymptotics of sub-Riemannian heat kernels. We prove that they are given by the nilpotentized heat kernel. In the equiregular case, we infer the local and microlocal Weyl law, putting in light the Weyl measure in sR geometry. This measure coincides with the Popp measure in low dimension but differs from it in general. We prove that spectral concentration occurs on the shief generated by Lie brackets of length r-1, where r is the degree of nonholonomy. In the singular case, like Martinet or Grushin, the situation is more involved but we obtain small-time asymptotic expansions of the heat kernel and the Weyl law in some cases. Finally, we give the Weyl law in the general singular case, under the assumption that the singular set is stratifiable.

We present new methods to construct a Ruelle-Pollicott spectrum for the geodesic flow on manifolds with strictly negative curvature and a finite number of hyperbolic cusps.

We numerically compute the lowest Laplacian eigenvalues of several two-dimensional shapes with dihedral symmetry at arbitrary precision arithmetic. Our approach is based on the method of particular solutions with domain decomposition. We are particularly interested in asymptotic expansions of the eigenvalues $\lambda(n)$ of shapes with $n$ edges that are of the form $\lambda(n) \sim x\sum_{k=0}^{\infty} \frac{C_k(x)}{n^k}$ where $x$ is the limiting eigenvalue for $n\rightarrow \infty$. Expansions of this form have previously only been known for regular polygons with Dirichlet boundary condition and (quite surprisingly) involve Riemann zeta values and single-valued multiple zeta values, which makes them interesting to study. We provide numerical evidence for closed-form expressions of higher order $C_k(x)$ and give more examples of shapes for which such expansions are possible (including regular polygons with Neumann boundary condition, regular star polygons and star shapes with sinusoidal boundary)., Comment: 20 pages

We investigate spectral features of the Dirac operator with infinite mass boundary conditions in a smooth bounded domain of $\mathbb{R}^2$. Motivated by spectral geometric inequalities, we prove a non-linear variational formulation to characterize its principal eigenvalue. This characterization turns out to be very robust and allows for a simple proof of a Szeg\"o type inequality as well as a new reformulation of a Faber-Krahn type inequality for this operator. The paper is complemented with strong numerical evidences supporting the existence of a Faber-Krahn type inequality., Comment: 34 pages, 4 figures

Combining the methods of Cuenin [2019] and Borichev-Golinskii-Kupin [2009, 2018], we obtain the so-called Lieb-Thirring inequalities for non-selfadjoint perturbations of an effective Hamiltonian for bilayer graphene., 22 pages; few typos corrected

The question of determining for which eigenvalues there exists an eigenfunction which has the same number of nodal domains as the label of the associated eigenvalue (Courant-sharp property) was motivated by the analysis of minimal spectral partitions. In previous works, many examples have been analyzed corresponding to squares, rectangles, disks, triangles, tori, \ldots . A natural toy model for further investigations is the M\"obius strip, a non-orientable surface with Euler characteristic $0$, and particularly the "square" M\"obius strip whose eigenvalues have higher multiplicities. In this case, we prove that the only Courant-sharp Dirichlet eigenvalues are the first and the second, and we exhibit peculiar nodal patterns., Comment: Revised version prior to publication. Accepted for publication in Portugaliae Mathematica

We prove that the results in scattering theory that involve resonances are still valid for non-analytic potentials, even if the notion of resonance is not defined in this setting. More precisely, we show that if the potential of a semiclassical Schr\"odinger operator is supposed to be smooth and to decrease at infinity, the usual formulas relating scattering quantities and resonances still hold. The main ingredient for the proofs is a resolvent estimate of a new type, relating the resolvent of an operator with the resolvent of its cut-off counterpart., Comment: 49 pages, 12 figures

We control the behavior of the Poincar{\'e} constant along the Polchinski renormalization flow using a dynamic version of $\Gamma$-calculus. We also treat the case of higher order eigenvalues. Our method generalizes a method introduced by B. Klartag and E. Putterman to analyze the evolution of log-concave distributions along the heat flow. Furthermore, we apply it to general $\Phi$ 4-measures and discuss the interpretation in terms of transport maps.

International audience; We consider the propagation of Transverse Electric (TE) waves in a 2D photonic crystal made of an homogeneous bulk containing high permittivity identical inclusions localized at the vertices of a honeycomb lattice [1, 2]. In this high contrast regime, we analyse the band structure of the underlying elliptic operator. We prove global results on the evolution of the band structure with respect to the contrast as well as detailed local information such as the existence of Dirac points, which are conical crossings between consecutive dispersion surfaces. Our study is illustrated by numerical simulations. Moreover, essential differences between this electromagnetic setting and the quantum model of graphene [3] will be discussed during the talk.

This thesis is devoted to the mathematical study of electronic properties of matter. The systems, both molecular and crystalline, are described by nonlinear models coming from quantum mechanics. Then, we consider the dissociation regime, that is when the distances between the nuclei are large. In Chapter 1, we study the diatomic Hartree model, both in dimension two and three, and we precisely estimate the quantum tunneling between the first two eigenfunctions. In Chapter 2, we show that if a non-degeneracy condition is satisfied then the reduced Hartree-Fock model of graphene presents conical singularities, called Dirac points. In addition, we show that the Fermi level coincides with the energy of these cones. In this direction, we derive conditions under which the dispersion relation of periodic Schrödinger operator is given, to leading order and in the dissociation regime, by the corresponding tight-binding model.; Cette thèse porte sur l’étude mathématique des propriétés électroniques de la matière. Les systèmes, moléculaires ou cristallins, sont décrits à l’aide de modèles non linéaires issus de la mécanique quantique. On considère alors le régime de dissociation, c’est-à-dire lorsque les distances entre les noyaux sont grandes. Dans le Chapitre 1, on étudie le modèle de Hartree diatomique, en dimension deux ou trois, et on quantifie précisément l’effet tunnel entre les deux premiers modes propres. Dans le Chapitre 2, on montre que si une condition de non-dégénérescence est vérifiée alors le modèle de Hartree-Fock réduit du graphène présente des singularités coniques, appelées points de Dirac. De plus, on prouve que le niveau de Fermi coïncide avec le niveau d’énergie de ces cônes. Pour cela, on dérive certaines conditions sous lesquelles les relations de dispersion d’un opérateur de Schrödinger périodique sont données, au premier ordre et dans le régime de dissociation, par le modèle de liaison forte correspondant.

We propose a systematic construction of signed harmonic functions for discrete Laplacian operators with Dirichlet conditions in the quarter plane. In particular, we prove that the set of harmonic functions is an algebra generated by a single element, which conjecturally corresponds to the unique positive harmonic function., 42 pages, 8 figures

This paper is devoted to a general min-max characterization of the eigenvalues in a gap of the essential spectrum of a self-adjoint unbounded operator. We prove an abstract theorem, then we apply it to the case of Dirac operators with a Coulomb-like potential. The result is optimal for the Coulomb potential. An erratum is appended at the end of the tex., The main goal of this submission is to add an erratum to the old paper published in 2000

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 study the long-time behavior of the Schrodinger flow in a heterogeneous potential $$\lambda V$$ with small intensity $$00$$ . More precisely, the approximate normal form leads to an accurate long-time description of the Schrodinger flow as an effective unitary correction of the free flow. The approach is robust and generically applies to linear waves. For classical waves, for instance, this allows to extend diffractive geometric optics to quasiperiodically perturbed media.

We consider the continuous Anderson Hamiltonian with white noise potential on $(-L/2,L/2)^d$ in dimension $d\le 3$, and derive the asymptotic of the smallest eigenvalues when $L$ goes to infinity. We show that these eigenvalues go to $-\infty$ at speed $(\log L)^{1/(2-d/2)}$ and identify the prefactor in terms of the optimal constant of the Gagliardo-Nirenberg inequality. This result was already known in dimensions $1$ and $2$, but appears to be new in dimension $3$. We present some conjectures on the fluctuations of the eigenvalues and on the asymptotic shape of the corresponding eigenfunctions near their localisation centers., To appear in Stochastics and Partial Differential Equations: Analysis and Computations

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 consider the Steklov zeta function $\zeta$ $\Omega$ of a smooth bounded simply connected planar domain $\Omega$ $\subset$ R 2 of perimeter 2$\pi$. We provide a first variation formula for $\zeta$ $\Omega$ under a smooth deformation of the domain. On the base of the formula, we prove that, for every s $\in$ (--1, 0) $\cup$ (0, 1), the difference $\zeta$ $\Omega$ (s) -- 2$\zeta$ R (s) is non-negative and is equal to zero if and only if $\Omega$ is a round disk ($\zeta$ R is the classical Riemann zeta function). Our approach gives also an alternative proof of the inequality $\zeta$ $\Omega$ (s) -- 2$\zeta$ R (s) $\ge$ 0 for s $\in$ (--$\infty$, --1] $\cup$ (1, $\infty$); the latter fact was proved in our previous paper [2018] in a different way. We also provide an alternative proof of the equality $\zeta$ $\Omega$ (0) = 2$\zeta$ R (0) obtained by Edward and Wu [1991].

International audience; We consider discrete Schrödinger operators on ${\mathbb{Z}}^d$ for which the perturbation consists of the sum of a long-range type potential and a Wigner-von Neumann type potential. Still working in a framework of weighted Mourre theory, we improve the limiting absorption principle (LAP) that was obtained in [Ma1]. To our knowledge, this is a new result even in the one-dimensional case. The improvement consists in a weakening of the assumptions on the long-range potential and better LAP weights. The improvement relies only on the fact that the generator of dilations (which serves as conjugate operator) is bounded from above by the position operator. To exploit this, Loewner's theorem on operator monotone functions is invoked.

International audience; In this work we study the number of small eigenvalues and the convergence to the equilibrium of the Bouncy Particle Sampler process and the Zig-Zag process generators in the small temperature regime. Such processes, which fall in the class of Piecewise Deterministic Markov Processes, are non diffusive and non reversible.

We study the concentration properties of low-energy states for quantum systems in the semiclassical limit, in the setting of Toeplitz operators, which include quantum spin systems as a large class of examples. We establish tools proper to Toeplitz quantization to give a general subprincipal criterion for localisation. In addition, we build up symplectic normal forms in two particular settings, including a generalisation of Helffer–Sjostrand miniwells, in order to prove asymptotics for the ground state and estimates on the number of low-lying eigenvalues.

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.

In this paper we consider a Stokes system with Navier-slip boundary conditions. The main results concern the behaviour of the cost of null controllability with respect to the diffusion coefficient when the control acts in the interior. In particular, we prove in the square that for a sufficiently large time the cost decays exponentially as the diffusion coefficient vanishes, whereas in the cube we prove that for most of the control domains and for any time T > 0 the cost explodes exponentially as the diffusion coefficient vanishes.

We give an explicit description of all minimal self-adjoint extensions of a densely defined, closed symmetric operator in a Hilbert space with deficiency indices $(1, 1)$., Comment: 4 pages, no figures, submitted

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.

International audience

In this work, we give sharp asymptotic equivalents in the small temperature regime of the smallest eigenvalues of the generator of some piecewise deterministic Markov processes (including the ZigZag process and the Bouncy Particle Sampler process) with refreshment rate $\alpha$ on the one-dimensional torus T. These asymptotic equivalents are usually called Eyring-Kramers type formulas in the literature. The case when the refreshment rate $\alpha$ vanishes on T is also considered.

We estimate the bottom of the $L^2$ spectrum of the Laplacian on locally symmetric spaces in terms of the critical exponents of appropriate Poincar\'e series. Our main result is the higher rank analog of a characterization due to Elstrodt, Patterson, Sullivan and Corlette in rank one. It improves upon previous results obtained by Leuzinger and Weber in higher rank., Comment: To appear in Geometriae Dedicata

We consider Fokker-Planck type differential operators associated with general Langevin processes admitting a Gibbs stationary distribution. Under assumptions insuring suitable resolvent estimates, we prove Eyring-Kramers formulas for the bottom of the spectrum of these operators in the low temperature regime. Our approach is based on the construction of sharp Gaussian quasimodes which avoids supersymmetry or PT-symmetry assumptions., 45 pages, 3 figures

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.