Title and Abstracts

Chong, Jacky

• Title:  Aspects of Semiclassical Limit from the Hartree–Fock Equation to the Vlasov–Poisson Equation

• We investigate the semiclassical limit from the Hartree–Fock equation with singular interaction (with emphasis on the Coulomb interaction potential) to the Vlasov (or Vlasov–Poisson) equation. The objective of the talk is to present the semiclassical propagation of moments for the Hartree–Fock equation with singular interaction potential, as well as some novel semi-classical stability estimates for the difference of the square roots of solutions to Hartree-type equations, which will help improve the rate of convergence from the Hartree–Fock equation to the Vlasov–Poisson equation in the L² norm when ℏ approaches 0. If time permits, we will discuss their connections to the derivation of Vlasov equation from many-body quantum dynamics. The talk is based on the recent joint works with Laurent Lafleche and Chiara Saffirio.

Dietze, Charlotte

• Title: Semiclassical 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.

Giacomelli, Emanuela

• Title: A 3D-Schrödinger operator under magnetic steps with semiclassical applications

• Abstract: In this talk we study the spectrum of a Schrödinger operator on the half-space with a discontinuous magnetic field having a piecewise-constant strength and a uniform direction. Motivated by applications in the theory of superconductivity, we use the Schrödinger operator on the half-space to study a new semiclassical problem in bounded domains of the space, considering a magnetic Neumann Laplacian with a piecewise-constant magnetic field. We then make precise the localization of the semiclassical ground state near specific points at the discontinuity jump of the magnetic field.

Junge, Lukas

• Title: Can you determine the domain from the lowest eigenvalue?

• Abstract: In 1966 Mac posed the famous question "can you hear the shape of a drum", which means; can you determine the domain from the spectrum of the Dirichlet Laplaican. This sparked wide interest and set fire to the field of spectral geometry. Although the question was answered negatively in 1992 the interest persisted and many spectral geometric properties of the Laplacian has since been derived. In this talk we shall see how some of these properties even extends to the more complicated magnetic Laplacian. More precisely we will prove for a shape Ω of unit volume and homogenous magnetic field B that

λ₁(B,Ω)-λ₁(B,D) ≥ c exp(-η B) A(Ω)

here D is the unit disk and A is a metric measuring the roundness of Ω. 

Ko, Hyerim

• Title: Sobolev regularity estimates for restricted X-ray transforms

• Abstract: In this talk, we consider the regularity property of a restricted X-ray transform. We prove sharp Lᵖ Sobolev regularity estimates for restricted X-ray transforms in all dimen- sions, which extends the result of Pramanik–Seeger in R³. This is achieved by constructing inductive argument and making use of the decoupling inequality for curves. This is joint work with Sanghyuk Lee and Sewook Oh.

Lee, Kiyeon

• Title: Modified scattering results for semi-relativistic equations and their nonrelativistic limit

• Abstract: In this talk, we give a survey of asymptotic behavior for the nonlinear semi-relativistic Hartree type equations with Coulomb potential by a space-time resonant argument. This argument is first introduced by Germain-Masmoudi-Shatah in 2009. In the 2 spatial dimensions, since the difficulty stems come from the lack of time decay and Strichartz estimates do not work for our nonlinearity, we observe space, time, and space-time resonant sets which observe the oscillation in time or frequencies. Especially, we give the asymptotic behavior of our main equations which behaves unlike the standard linear scattering. This work is based on the joint work with S. Kwon and C. Yang. Regarding the profile of modified scattering, we can observe the nonrelativistic limit to the profile of modified scattering for Schrödinger equations. In the last of this talk, I will describe our main statement and argument to show that. This work is in preparation with H. Yoon.

Lee, Myeong-Su

• Title: Derivation of the relativistic BGK-type model for gas mixtures.

• Abstract: In this talk, we provide a brief overview of the relativistic Boltzmann equation for gas mixtures. We discuss the fundamental properties of the Boltzmann equation, such as the conservation laws, the state in equilibrium, the indifferentiability principle, and the H-theorem. Then, we study the rigorous derivation of a BGK-type relaxation model that satisfies the aforementioned properties. We further see that our model can recover the classical BGK model in the Newtonian limit.

Lee, Yoonjung

• Title : Regularity estimates for nondivergence elliptic equations with certain potential.

• Abstract : To the scope of the Schrödinger operator, we are interested in nondivergence elliptic operators with small BMO coefficients and a potential satisfying a reverse Hölder condition. Motivated by the Calderón-Zygmund  estimates, Lᵖ type of the Hessian estimates for the Schrodinger equation was studied by Shen in 1995. In this talk, we overview the known results for such Lᵖ estimates of the nondivergence elliptic equations and present our recent result for an interior estimate on the weighted space based on the Maximal function method and modified Vitali covering lemma developed by Caffarelli. This work is a joint work with Mikyoung Lee.

Lill, Sascha

• Title: Momentum Distribution of a Fermi Gas in the Random Phase Approximation

• Abstract: We consider a 3d fermionic quantum gas at high density in the mean-field regime. For this model, a recent work by Benedikter, Nam, Porta, Schlein and Seiringer introduced a particularly convenient trial state, which is energetically close to the ground state. Within this trial state, we prove a formula for the occupation densities of the momentum space modes (also called momentum distribution). As expected, they converge towards a Fermi ball profile at vanishing coupling, while for low coupling, the Fermi surface persists. Further, our leading-order correction to the Fermi ball profile agrees with the formulas predicted by Daniel and Vosko in 1960. The talk is based on joint work with Niels Benedikter.

Nguyen, Dinh-Thi

• Title: Asymptotic analysis for 2D rotating Bose-Einstein condensate at the critical rotation speed.

• Abstract: We study the minimizers of a magnetic 2D non-linear Schrödinger energy functional in a quadratic trapping potential, describing a rotating Bose-Einstein condensate. In the first part, we consider the case of a repulsive interaction potential. We derive an effective Thomas-Fermi-like model in the rapidly rotating limit where the centrifugal force compensates the confinement. The available states are restricted to the lowest Landau level. The coupling constant of the Thomas-Fermi functional is to link the emergence of vortex lattices (the Abrikosov problem). In the second part, we consider the case an attractive interaction potential. When the strength of the interaction approaches a critical value from below, the system collapses to a profile obtained from the (unique) optimizer of a Gagliardo-Nirenberg interpolation inequality. This was established before in the case of fixed rotation frequency. We extend the result to rotation frequencies approaching, or even equal to, the critical frequency at which the centrifugal force compensates the trap. We prove that the blow-up scenario is to leading order unaffected by such a strong deconfinement mechanism. In particular the blow-up profile remains independent of the rotation frequency.

Nguyen, Ngoc Nhi

• Title: Spectral cluster bounds for orthonormal functions on compact manifolds with non-smooth metrics

• Abstract: There has been substantial recent interest in functional inequalities for systems of orthonormal functions. The game is to prove an optimal dependence on the number of functions involved. In this talk we focus on a family of inequalities called “spectral cluster bounds”, which concern Lᵖ norms of (linear combinations of) eigenfunctions of the Laplace-Beltrami operator on a compact closed manifold. Since the seminal work of Sogge in the 1980’s, these bounds have been generalized in various directions. Frank and Sabin recently established a version of Sogge’s bounds for systems of orthonormal functions. The result is valid for smooth metrics. We will show that the same result holds for C¹'¹ metrics. The analogue in the one-function case was proved by Smith. The talk is based on joint ongoing work with Jean-Claude Cuenin.

Olivieri, Marco

• Title: The energy of the dilute Bose gases in thermodynamic regime

• Abstract: We consider a system of bosons interacting via a repulsive, pairwise potential and study it in thermodynamic regime, i.e., when the particle's number and the volume are huge but still the density of particles remains constant. In the dilute limit, it is possible to derive an asymptotic expansion for the energy density of the aforementioned system given by the so-called Lee-Huang-Yang formula (LHY). The LHY formula shows how the first terms of the expansion are independent of the shape of the potential and depend on it only via its scattering length, giving a universal formula. In this talk we recall the historical development that has brought to the rigorous proof of the LHY formula and we present the most recent results in 3D and 2D.

Ryu, Jaehyun

•  Title: Almost everywhere convergence of Bochner-Riesz means for the twisted Laplacian

• Abstract: We study almost everywhere convergence of the Bochner-Riesz means associated with the twisted Laplacian. In the talk, we present the result stating that, for 2 ≤ p ≤ ∞, the sharp summability index for the convergence is half of the index appearing in the result for the classical Bochner-Riesz mean. An interesting aspect of the proof is that it does not require the use of the Aₚ weight theory, unlike the case of the classical Bochner-Riesz mean. This is a joint work with Eunhee Jeong and Sanghyuk Lee.