| 14h - 15h | Clément Tauber (Strasbourg) |
Approach to
equilibrium in translation-invariant quantum systems Abstract: In this talk I will formulate the problem of approach to equilibrium in algebraic quantum statistical mechanics and study some of its structural aspects, focusing on the relation between the zeroth law of thermodynamics (approach to equilibrium) and the second law (increase of entropy). The main result is that approach to equilibrium is necessarily accompanied by a strict increase of the specific (mean) entropy. In the course of our analysis, I will introduce the concept of quantum weak Gibbs state which is of independent interest. This talk is based on joint work with Vojkan Jaksic and Claude-Alain Pillet. |
| 15h15 - 16h15 | Dmitry Jakobson (McGill) |
Nodal sets and negative eigenvalues in conformal geometry Abstract: We study zero and negative eigenvalues of the conformal Laplacian, and some conformal invariants that arise from nodal sets of the corresponding eigenfunctions. We discuss applications to certain curvature prescription problems. |
| 14h - 15h | Silvia Pappalardi (ENS Paris) |
Low temperature quantum bounds on simple models Abstract: In the past few years, there has been considerable activity around a set of quantum bounds on transport coefficients (viscosity, conductivity) and chaos (Lyapunov exponents), relevant at low temperatures. The interest comes from the fact that black-hole models seem to saturate all of them. However, the relation between the different bounds and physical properties of the systems saturating the is still a matter of ongoing research. In this talk, I will discuss how one can gain physical intuition by studying classical and quantum free dynamics on curved manifolds. Thanks to the curvature, such models display chaotic dynamics up to low temperatures and - as I will show how- they violate the bounds in the classical limit. The talk aims to discuss three different ways in which quantum effects arise to enforce the bounds in practice. For instance, I will show how chaotic behaviour is limited by the quantum effects of the curvature itself. As an illustrative example, I will consider the simple case of a free particle on a two-dimensional manifold, constructed by joining the surface of constant negative curvature — a paradigmatic model of quantum chaos — to a cylinder. The resulting phenomenology can be generalized to the case of several (constant) curvatures. The presence of a hierarchy of length scales enforces the bound to chaos up to zero temperature. * Pappalardi, Kurchan, Low temperature quantum bounds on simple models, SciPost Phys. 13, 006 (2022) |
| 15h15 - 16h15 | Jean-Claude Cuenin (Loughborough) |
Effective bounds on scattering resonances Abstract: The celebrated Weyl law describes the asymptotic distribution of eigenvalues of the Laplacian on a compact manifold. Scattering resonances are analogues of eigenvalues when the underlying manifold is non-compact. The simplest case concerns Schrödinger operators -Δ+V on Euclidean space Rd with compactly supported potential V. The object of interest is the resonance counting function nV(r), that is, the number of resonances in a disk of radius r. In dimensions greater than one, asymptotics are known only in a few special cases. The topic of this talk are polynomial upper bounds on the resonance counting function. These have a long history, starting in the 80’s with work of Melrose. The sharp upper bound nV(r)< CV rd was proved by Zworski. In this talk I will present effective versions of this upper bound for non compactly supported potentials. Effective means that CV does not depend on V itself but only on some weighted norms. The proof of this result features a combination of harmonic, functional and complex analysis. |
| 14h - 15h | Viviana Grasselli (Toulouse) |
Dispersive equations on asymptotically conical manifolds Abstract: We consider solutions to three dispersive equations, such as Schrödinger, wave and Klein-Gordon equations, on a quite general class of non compact manifold which are asymptotically conical. We will be interested in L2 estimates of the associated evolution operators in the low frequency regime. With the aid of the spectral measure, we can write these operators in terms of limiting values of the resolvent of the Schrödinger operator, hence obtaining estimates for the resolvents can then be translated in time decay estimates for the evolution operators. We will see how we can use Mourre theory to obtain existence and estimates for such limiting values, provided we prove smallness of a suitable spectral localisation of the operator. |
| 15h15 - 16h15 | Angèle Niclas (Ecole Polytechnique) |
Defect reconstruction in waveguides using resonant frequencies Abstract: This talk aims at introducing a new multi-frequency method to reconstruct width defects in waveguides. Different inverse methods already exist. However, those methods are not using some frequencies, called resonant frequencies, where propagation equations are known to be ill-conditioned. Since waves seem very sensible to defects at these particular frequencies, we exploit them instead. After studying the forward problem at these resonant frequencies, we approximate the wavefield using a Schrödinger equation satisfied by each mode. Then, we use this approximation to solve the inverse problem: given partial wavefield measurements, we reconstruct width defects in a stable and precise way and provide numerical validations and comparisons with existing methods. |
| 14h - 15h | Charlotte Dietze (LMU Munich & IHES) |
Spectral estimates for Schrödinger operators with Neumann boundary conditions on Hölder domains Abstract: We prove a universal bound for the number of negative eigenvalues of Schrödinger operators with Neumann boundary conditions on bounded Hölder domains, under suitable assumptions on the Hölder exponent and the external potential. Our bound yields the same semiclassical behaviour as the Weyl asymptotics for smooth domains. We also discuss different cases where Weyl's law holds and fails. |
| 15h15 - 16h15 | Marouane Assal (Bordeaux) |
Quelques résultats d'analyse spectrale des systèmes d'opérateurs de Schrödinger avec des croisements de trajectoires classiques Résumé: Je présenterai deux résultats récents sur la distribution asymptotique, dans le régime semi-classique, des valeurs propres et des résonances d'un système d'opérateurs de Schrödinger unidimensionnel avec des croisements de trajectoires (hamiltoniennes) classiques. Le premier résultat concerne le "splitting" des valeurs propres pour un modèle captif-captif au dessus et près du niveau de croisement. Dans le deuxième résultat on donne l'asymptotique des résonances pour un modèle captif-non captif en présence de croisements dégénérés des trajectoires classiques. Dans les deux résultats, le coefficient principal de l'asymptotique est calculé explicitement en termes de quantités classiques, et reflète la géométrie des croisements. Ces résultats sont issus de travaux en collaboration avec S. Fujiié (Kyoto) et K. Higuchi (Ehime). |
| 14h - 15h | Francis White (Paris 13) |
Quadratic Evolution Equations and Fourier Integral Operators in the Complex Domain Abstract: In mathematical physics, non self-adjoint operators and their associated evolution equations are used to model dissipative phenomena. In this talk, I will present some new results concerning the propagation of global analytic singularities and Lp-bounds for solutions of Schrödinger equations on Rn with nonselfadjoint quadratic Hamiltonians. The main idea in the proofs is that, after conjugation by a metaplectic Fourier-Bros-Iagolnitzer (FBI) transform, the solution operator for such a quadratic evolution equation is a Fourier integral operator (FIO) associated to a complex canonical transformation acting on a suitable exponentially weighted space of entire functions. When this FIO is written in its so-called `Bergman form', previously out of reach questions can be answered. |
| 15h15 - 16h15 | Lucas Vacossin (Orsay) |
Influence of the trapped set of the billiard flow on the scattering resonances Abstract: In this presentation, we will focus on the scattering resonances for finitely many strictly convex obstacles in R^d. The distribution of such resonances is strongly influenced by the billiard flow associated with these obstacles, more particularly by the trapped set of this flow. This trapped set exhibits a fractal structure which plays a crucial role in different problems related with the distribution of the resonances. In this talk, we will discuss these relations and present a new result which establishes the existence of a spectral gap in the 2D problem, as conjectured by Zworski in 2017 in any dimension. This result relies on a recent tool of harmonic analysis, a Fractal Uncertainty Principle, developped in particular by Bourgain and Dyatlov. |
| 14h - 15h | Annalaura Stingo (Ecole Polytechnique) |
Global stability of the Kaluza-Klein theories Abstract: The Kaluza-Klein theories represent the classical mathematical approach to the unification of general relativity with electromagnetism and more generally with gauge fields. In these theories, general relativity is considered in 1+3+d dimensions and in the simplest case d=1 dimensional gravity is compactified on a circle to obtain at low energies a (3+1)-dimensional Einstein-Maxwell-Scalar systems. In this talk I will discuss the problem of the classical global stability of Kaluza-Klein theories when d=1. This is a joint work with C. Huneau and Z. Wyatt. |
| 15h15 - 16h15 | Julien Ricaud (Ecole Polytechnique) |
Spectral stability in the nonlinear Dirac equation with Soler-type nonlinearity Abstract: This talk concerns the (generalized) Soler model: a nonlinear (massive) Dirac equation with a nonlinearity taking the form of a space dependent mass. The equation admits standing wave solutions and they are generally expected to be stable (i.e., small perturbations in the initial conditions stay small) based on numerical simulations. However, contrarily to the nonlinear Schrödinger equation for example, there are very few results in this direction. The results that I will discuss concern the simpler question of spectral stability (and instability), i.e., the absence (or presence) of exponentially growing solutions to the linearized equation around a solitary wave. As in the case of the nonlinear Schrödinger equation, this is equivalent to the presence or absence of "unstable eigenvalues" of a non-selfadjoint operator with a particular block structure. I will highlight the differences and similarities with the Schrödinger case, present some results for the one-dimensional case, and discuss open problems. This is joint work with Danko Aldunate, Edgardo Stockmeyer, and Hanne Van Den Bosch. |
| 14h - 15h | Angeliki Menegaki (IHES) |
Spectral gap for harmonic and weakly anharmonic chain of oscillators Abstract: We consider one-dimensional chains and multi-dimensional networks of harmonic oscillators coupled to two Langevin heat reservoirs at different temperatures. Each particle interacts with its nearest neighbours by harmonic potentials and all individual particles are confined by harmonic potentials, too. In previous works we investigated the sharp N-particle dependence of the spectral gap of the associated generator in different physical scenarios and for different spatial dimensions. We also obtained estimates on the gap after perturbing weakly the quadratic potentials, through a Log-Sobolev Inequality. In this talk I will present new results on the behaviour of the spectral gap when considering longer-range interactions in the purely harmonic chain. In particular, depending on the strength of the longer-range interaction, there are different regimes appearing where the gap drastically changes behaviour but even the hypoellipticity of the operator breaks down. Parts of this talk are joint works with Simon Becker (ETH). |
| 15h15 - 16h15 | Xiaolong Han (CSUN & IHES) |
Fourier restriction estimates on hyperbolic manifolds Abstract: The Fourier restriction phenomenon asks whether one can meaningfully restrict the Fourier transform of a function onto a hypersurface (such as a sphere) in the frequency space. Stein's restriction conjectures state that the Fourier transform of an Lp function restricts to a well-defined Lq function on the sphere, for appropriate ranges of exponents p and q. While the full conjecture remains open, Tomas and Stein in the 1970s proved the case when q=2. Via the spectral measure, the Tomas-Stein restriction estimates have been proved in geometries other than the Euclidean spaces such as asymptotically conic or hyperbolic manifolds, all of which require that there is no geodesic trapping, i.e., all geodesics extend to infinity. In this talk, we study how the restriction estimates are influenced by this trapping condition. We present the first examples of manifolds with geodesic trapping for which the Tomas-Stein restriction estimates hold. |
| 14h - 15h | Jack Thomas (Orsay) |
Locality of interatomic interactions in electronic structure models Abstract: We survey some recent results on the locality and sparsity of the potential energy landscape (PEL) aimed at justifying and extending the theory of machine-learning for interatomic potentials. We decompose the PEL into the sum of exponentially localised site potentials and characterise the rate of decay. By reviewing classical results in approximation theory, which may be of broader interest, we are able to construct low body-order approximations of the PEL and describe the rate of approximation both in terms of vanishing Fermi-temperature and band gap. We discuss potential consequences of this observation for modelling the PEL, as well as for solving the electronic structure problem. Based on joint work with Huajie Chen and Christoph Ortner [1,2]. Time permitting, we will discuss the related problem of screening in periodic systems, ongoing joint work with Antoine Levitt. [1] Jack Thomas, Huajie Chen, and Christoph Ortner. Body-ordered approximations of atomic properties. Archive for Rational Mechanics and Analysis, 246(1): 1-60 (2022) [2] Christoph Ortner, Jack Thomas, and Huajie Chen. Locality of interatomic forces in tight binding models for insulators. ESAIM: Mathematical Modelling and Numerical Analysis, 54(6): 2295-2318 (2020) |
| 15h15 - 16h15 | Cyril Labbé (Paris-Dauphine) |
Autour de l'hamiltonien d'Anderson avec bruit blanc en dimension 1 Résumé: On s'intéresse à l'opérateur de Schrödinger aléatoire obtenu en perturbant le Laplacien par un bruit blanc, en dimension 1. Dans un premier temps, je présenterai un résultat de localisation d'Anderson pour cet opérateur : le spectre est purement ponctuel et les fonctions propres sont exponentiellement localisées. Dans un second temps, je me concentrerai sur la restriction de cet opérateur à un volume fini et je présenterai une étude détaillée des statistiques locales des valeurs propres et fonctions propres dans la limite de grand volume : nous verrons qu'en fonction du niveau d'énergie auquel on s’intéresse, des comportements très variés apparaissent. Cet exposé est basé sur une séries de travaux en collaboration avec Laure Dumaz. |