| 14h - 15h | Salma Lahbabi (Paris-Dauphine & Université Hassan II de Casablanca & UM6P) |
Density functional theory for two dimensional homogeneous materials Abstract: We study Density Functional Theory models for 2D materials. We first show that a homogeneous material can be seen as a limit of crystals. Next, we derive reduced models in the orthogonal direction. We show how the different terms of the energy are modified and prove some properties of the ground state. In Kohn-Sham models, we prove that the Pauli principle is replaced by a penalization term in the energy. |
| 15h15 - 16h15 | Laurent Laflèche (Univ. Lyon 1) |
Quantum Optimal Transport and Sobolev Spaces Abstract: In the context of proving the semiclassical mean-field limit from the N-body Schrödinger equation to the Hartree-Fock and Vlasov equations, a crucial component is obtaining inequalities uniform in the Planck constant and the number of particles. These inequalities are the analogue of the estimates obtained in the corresponding kinetic models of classical statistical mechanics. Hence, in this presentation, I will introduce analogous tools and inequalities within the realm of quantum mechanics, such as operator versions of optimal transport and Sobolev spaces on the phase space, and the corresponding classical inequalities. We will in particular see that the quantum analogue of Sobolev inequalities yield uncertainty inequalities concerning the Wigner–Yanase skew information, and that the latter also plays a significant role in controlling a quantum Wasserstein "self-distance". |
| 14h - 15h | Søren Mikkelsen (Bath) |
Sharp semiclassical spectral asymptotics Abstract: In this talk we will discuss sharp semiclassical spectral asymptotics for differential operators, where the coefficients are not smooth. Usually, such results are proven using the theory of pseudo-differential operators. This theory is unfortunately not directly applicable, when we do not assume smoothness of the coefficients. How do we then obtain sharp semiclassical spectral asymptotics? The main focus of this talk will be to present the ideas in the proof of sharp semiclassical spectral asymptotics. |
| 15h15 - 16h15 | Ruoyu Wang (Univ. College London) |
Unbounded damped waves: backward uniqueness and polynomial stability Abstract: In this talk, we discuss the wave semigroup with an unbounded damping. In such a setting, there are explicit examples where the linear damped waves would go into finite-time extinction. We will then find an optimal condition explicitly on the unboundedness to guarantee that the finite-time extinction cannot happen. We will also develop powerful yet flexible control-theoretic tools to establish novel polynomial stability and energy decay results for a variety of damped wave-like systems, including the singular damped waves, the linearised gravity water waves and Euler-Bernoulli beams. |
| 14h - 15h | Jinyeop Lee
(LMU, Munich) |
Derivation of
the Vlasov equation from the fermionic many-body Schrödinger system
using the Husimi measure Abstract: This seminar presents a derivation of the Vlasov equation from the fermionic many-body Schrödinger system, employing the Husimi measure as a liking tool. Our exploration begins with a heuristic understanding of the Vlasov equation's derivation. This is followed by a short review of the many-body Schrödinger equation. Then we will talk about the methodology of linking the solutions of the many-body Schrödinger equation with the Vlasov equation, namely the Wigner measure and the Husimi measure. Attendees will gain insights into the formalism of this approach and explore strategies for controlling residual terms appearing in the derivations. |
| 15h15 - 16h15 | Jacob
Schach Møller (Aarhus) |
Weighted
estimates for the Laplacian on the cubic metric lattice Abstract: We review how recent weighted resolvent estimates for the (discrete) Laplacian on Zd, together with a new embedding estimate, can be used to derive new weighted resolvent estimates for the Laplacian on the cubic metric lattice Ld = {x∈R d , for all but one j, x j ∈ Z}. In particular, some consequences for the spectrum of the (metric) Laplacian perturbed by suitable potentials will be discussed. The talk is based on joint work with E. Korotyaev and M. G. Rasmussen. |
| 14h - 15h | Léo Morin (U. de Copenhague) |
Quantum tunnelling between radial magnetic wells Abstract: This talk is devoted to the spectral analysis of 2D Schrödinger operators with inhomogeneous magnetic field. When the magnetic field has two symmetric minima, each well generates an eigenvalue of same order. Then, we expect tunnelling to happen: the two smallest eigenvalues are exponentially close to each other in the semiclassical limit. We prove this result in the case of radially symmetric wells. Based on the Helffer-Sjöstrand theory, we obtain an elegant and short proof. Even though non-magnetic tunnelling is already very well understood, magnetic fields come with specific issues. In particular, this is the first tunnelling result between purely magnetic wells, and the non-radial situation is still challenging. Finally, we observe some new purely magnetic effect on the interaction between the wells. |
| 15h15 - 16h15 | Olivier Bourget (UC Chile) |
On localization regimes for kicked quantum systems Abstract: We will discuss the spectral properties of some periodically kicked quantum systems defined on the lattice. We focus our analysis on the existence of (dynamical) localization regimes for a class of random perturbations. |
| 14h - 15h | Bérangère
Delourme (Paris-Nord) |
Guided waves in
honeycomb periodic structures Abstract: In this talk, we investigate the propagation of waves in a particular honeycomb structure made of thin tubes. Based on an asymptotic analysis, we prove that the dispersion surfaces associated with this system have conical points located at the vertices of the Brillouin zone: this is a well-known property for systems having hexagonal symmetry. Then, introducing so-called zig-zag perturbations in the structure generates guided waves propagating along the defect. More specifically, we show that the frequency of those guided modes may be independent of the quasi-periodicity parameter, leading to almost flat dispersion curves. We present numerical results to illustrate our results. |
| 15h15 - 16h15 | Asbjorn
Bækgaard Lauritsen (IST Austria) |
Energies of
dilute spin-polarized Fermi gasses Abstract: Recently the study of dilute quantum gases have received much interest, in particular regarding their ground state energies and pressures/free energies at positive temperature. I will present recent work on such problems concerning that of the ground state energy of a spin-polarized Fermi gas and the extension to the pressure at positive temperature. Compared to the free gas, the energy density/pressure of the interacting gas differs by a term of order a3 ρ8/3, with a the p-wave scattering length of the interaction. This talk is based on joint work with Robert Seiringer. |