| 11h15 - 12h15 | Nicolas
Popoff
(Bordeaux) |
Spectrum of
the
Robin Laplacian with singular boundary conditions Abstract: In this talk, I will review various results about the spectrum of the Robin Laplacian with “attractive” boundary condition in a bounded domain. As the Robin parameter gets large, the first eigenvalues go to infinity. I will describe the mechanism giving the first orders of the asymptotics when the domain has corners, and give improvements when the domain is regular. This includes the description of the bottom of the spectrum in infinite cones. For 3d cones with regular cross section, we are able to count the number of discrete eigenvalues. Finally, if time allows, I will consider the case of a variable, vanishing, Robin coefficient. In that case the operator may not be self-adjoint and we are able to compute the indices of deficiency. Those are joint works with Vincent Bruneau, Konstantin Pankrashkin and Sergei Nazarov. |
| Déjeuner | ||
| 14h - 15h | Simona Rota
Nodari (Dijon) |
Uniqueness
and
non-degeneracy for a class of semilinear elliptic equations Abstract: In this talk, I will present a result on the uniqueness and the non-degeneracy of positive radial solutions for a class of semilinear elliptic equations. Next, I will illustrate this result with two examples : a nonlinear Schrödinger equation for a nucleon and a Schrödinger equation with a double power nonlinearity. This talk is based on joint works with Mathieu Lewin. |
| 15h15 - 16h15 | Dorian Le
Peutrec (Orsay) |
Sharp
spectral
asymptotics for non-reversible metastable diffusion processes Abstract: In this talk, we are interested in the overdamped Langevin dynamics More precisely, we are interested in the low spectral properties of the infinitesimal generator of the dynamics, that is L=-hΔ+∇∙U, and in its connections with the long-time behaviour of the dynamics in the regime h →0 . In particular, when the function V is a smooth Morse function admitting n local minima, we will show that there exists ε> 0 such that when h →0 , L admits precisely n eigenvalues in the strip {0 ≤ Re(z) < ε} which have moreover exponentially small moduli. In addition, under a generic assumption on the potential barriers of the function V , we will show that the asymptotic behaviors of these small eigenvalues are given by Eyring-Kramers type formulas. This generalizes previous results obtained by several authors in the gradient case, i.e. when U has the form U=∇∙V. Joint work with Laurent Michel. |
| 11h15 - 12h15 | Lisette
Jager (Reims) |
Three
pseudodifferential calculi in an infinite dimensional setting Abstract: Pseudodifferential analysis is a classical and powerful tool to treat problems in partial differential equations. Is there an analogue in an infinite dimensional framework ? Which spaces can be used to replace the phase and configuration spaces ? Which notions and results can be extended ? In this talk, we give a construction of a Weyl quantization on a probability space, the Wiener space, and specify its links with the Wick and Anti-Wick calculi, thanks to a version of the heat operator. Then we present an application to an evolution operator which appears in a model in nuclear magnetic resonance. This talk is based on articles in collaboration with Laurent Amour and Jean Nourrigat. |
| Déjeuner | ||
| 14h - 15h | Kouichi
Taira (Univ. Tokyo) |
Essential
self-adjointness of real principal type operators on Euclidean spaces Abstract: In this talk, we discuss the essential selfadjointness of variable coefficients Klein-Gordon operators in the setting where the corresponding Lorentzian metric is asymptotically flat. The main difficulty of the proof lies in the absence of ellipticity (previous similar results heavily depend on ellipticity or positivity of the operator). To overcome this problem, we use propagation of singularities and results from the geometric scattering theory, and analyze the singularities of solutions to stationary and time-dependent Schrödinger equations. This is a joint work with Shu Nakamura. |
| 15h15 - 16h15 | Frédéric
Naud (Jussieu) |
Résonances
et trous spectraux des surfaces hyperboliques aléatoires Résumé: On fera un court rappel sur la théorie spectrale des surfaces hyperboliques de volume infini et on introduira une notion de revêtements aléatoires de surfaces, baséé sur un modèle standard de graphe régulier aléatoire. On verra comment, dans le régime de grand degré, on peut obtenir des bornes explicites asymptotiquement presque sûres concernant le trou spectral. Travail commun avec Michael Magee. |
| 11h15 - 12h15 | Albert Werner
(Copenhague) |
Spectral
and dynamical properties of electric quantum walks Abstract: Quantum walks constitute a framework for both quantum search algorithms as well as for discrete time quantum simulators of single and few particles. In this talk, I will report on some recent results on the spectral and dynamical properties of one-dimensional quantum walks placed in homogenous electric fields, according to a discrete version of the minimal coupling principle. We relate the properties of these systems to the continued fraction expansion of the field and show in particular, that for all irrational fields the absolutely continuous spectrum of these systems is empty, and prove Anderson localization for almost all (irrational) fields. Making use of a connection between quantum walks and CMV matrices, our result also implies Anderson localization for CMV matrices with a particular choice of skew-shift Verblunsky coefficients, as well as for quasi-periodic unitary band matrices. |
| Déjeuner | ||
| 14h - 15h | Noema Nicolussi
(Vienne) |
Spectral
Theory of Infinite Quantum Graphs Abstract: Quantum graphs (Laplacians on metric graphs) play an important role as an intermediate setting between Laplacians on Riemannian manifolds and discrete Laplacians on graphs, and share properties with both. Whereas for finite metric graphs the Laplacian is selfadjoint and has discrete spectrum, Laplacians on graphs with infinitely many vertices and edges are much less understood. In this talk, we plan to discuss basic spectral properties of infinite quantum graphs and their relationship to (weighted) discrete Laplacians on infinite graphs. In particular, we focus on the selfadjointness problem and recently discovered connections to the notion of graph ends, a classical graph boundary introduced independently by Freudenthal and Halin. Based on joint works with Aleksey Kostenko, Mark Malamud and Delio Mugnolo. |
| 15h15 - 16h15 | Yannick
Bonthonneau (Rennes) |
Localisation
exponentielle dans un puits magnétique Résumé: Nous avons considéré le problème suivant : la localisation des fonctions propres de basse énergie du Laplacien magnétique semiclassique, en absence de champ électrique. Plus précisément, dans le cas bidimensionnel, quand le champ présente un puits non dégénéré en l’origine, et pour un champ analytique, nous obtenons une décroissance des fonctions propres hors du puits, à la vitesse e-c/h. J’expliquerai comment tout ceci fonctionne grâce à des outils assez classiques, en particulier une estimée d’Agmon microlocale. Collaboration avec Nicolas Raymond et San Vũ Ngọc. |
| 15h - 16h | Jean Lagacé (Univ. College London) |
Homegenisation
in geometric spectral theory Abstract: The question to find the best upper bound for the first nonzero Steklov eigenvalue of a surface goes back to Weinstock, who proved in 1954 that the first nonzero perimeter-normalised Steklov eigenvalue of a simply-connected planar domain is 2π, with equality iff the domain is a disk. In a series of recent works, we constructed families of surfaces, for which the perimeter-normalised first eigenvalue tends to 8π. In combination with Kokarev's bound from 2014, this solves the isoperimetric problem completely for the first nonzero eigenvalue. The surfaces are obtained by removing small geodesic balls that are asymptotically densely uniformly distributed as their radius tends to zero, in the spirit of homogenisation by obstacles. The main difficulty was in the construction of a homogenisation limit on manifolds, most of which lack any sort of periodic structure. A surprising consequence is the existence of free boundary minimal surfaces immersed in the unit ball by first Steklov eigenfunctions and with area strictly larger than 2π, which was previously claimed to be impossible. This is based on joint work with Alexandre Girouard (U.Laval), Antoine Henrot (Nancy) and Mikhail Karpukhin (UC Irvine) |
| 16h15 - 17h15 | Martin
Gebert (UC Davis) |
Lieb-Robinson
bounds for a class of continuum many-body fermion systems Abstract: We introduce a class of UV-regularized two-body interactions for fermions in Rd and prove a Lieb-Robinson estimate for the dynamics of this class of many-body systems. As a step towards this result, we also prove a propagation bound of Lieb-Robinson type for one-particle Schrödinger operators. We apply the propagation bound to prove the existence of a strongly continuous infinite-volume dynamics on the Canonical Anticanonical Relations algebra. |
| 14h-15h | Roland Bauerschmidt (Univ. of Cambridge) | Log-Sobolev
inequality for the continuum Sine-Gordon model Abstract: We derive a multiscale generalisation of the Bakry-Emery criterion for a measure to satisfy a Log-Sobolev inequality. Our criterion relies on the control of an associated PDE well known in renormalisation theory: the Polchinski equation. It implies the usual Bakry-Emery criterion, but we show that it remains effective for measures which are far from log-concave. Indeed, as an application, we prove that the massive continuum Sine-Gordon model with β<6π satisfies asymptotically optimal Log-Sobolev inequalities for Glauber and Kawasaki dynamics. This is joint work with Thierry Bodineau. |
| 15h15 - 16h15 | Jeffrey
Galkowski (Univ. College London) |
Viscosity
limits for 0th order operators Abstract: In recent work, Colin de Verdière-Saint-Raymond and Dyatlov-Zworski showed that a class of zeroth order pseudodifferential operators coming from experiments on forced waves in fluids satisfies a limiting absorption principle. Thus, these operators have absolutely continuous spectra, with possibly finitely many embedded eigenvalues. In this talk, we discuss the effect of a small viscosity on the spectra of these operators, showing that the spectrum of the operator with small viscosity converges to the poles of a certain meromorphic continuation of the resolvent through the continuous spectrum. In order to do this, we introduce functional spaces based on an FBI transform, which allows the testing of microlocal analyticity properties. This talk is based on joint work with M. Zworski. |