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Julien Sabin

Publications and source records attributed to Julien Sabin.

17 recordsLinked to original sources
arXiv2025

Scattering for the positive density Hartree equation

We study the asymptotic stability for large times of homogeneous stationary states for the nonlinear Hartree equation for density matrices in Rd for d\geq3. We can reach both the optimal Sobolev and Schatten exponents for the initial data, with a wide class of interaction potentials w (under the sole assumption that w is bounded, including in particular delta potentials). Our method relies on fractional Leibniz rules for density matrices to deal with the fractional critical Sobolev regularity s = d/2 -1 for odd d, as well as Christ-Kiselev lemmas in Schatten spaces.

math.AP↗
arXiv2023

Dynamics of mean-field Fermi systems with nonzero pairing

We study the dynamics of many-body Fermi systems, for a class of initial data which are close to quasi-free states exhibiting a nonvanishing pairing matrix. We focus on the mean-field scaling, which for fermionic systems is naturally coupled with a semiclassical scaling. Under the assumption that the initial datum enjoys a suitable semiclassical structure, we give a rigorous derivation of the time-dependent Hartree-Fock-Bogoliubov equation, a nonlinear effective evolution equation for the one-particle density matrix of the system, as the number of particles goes to infinity. Our result holds for all macroscopic times, and provides bounds for the rate of convergence.

math-ph↗
arXiv2022

Compactness methods in Lieb's work

We review some compactness methods appearing in the work of Lieb, with an emphasis on the techniques developed around his 1983 article on the optimizers for the Hardy-Littlewood-Sobolev inequality.

math.AP↗
arXiv2021

The Dirac-Klein-Gordon system in the strong coupling limit

We study the Dirac equation coupled to scalar and vector Klein-Gordon fields in the limit of strong coupling and large masses of the fields. We prove convergence of the solutions to those of a cubic non-linear Dirac equation, given that the initial spinors coincide. This shows that in this parameter regime, which is relevant to the relativistic mean-field theory of nuclei, the retarded interaction is well approximated by an instantaneous, local self-interaction. We generalize this result to a many-body Dirac-Fock equation on the space of Hilbert-Schmidt operators.

math.AP↗
arXiv2020

Sharp Weyl laws with singular potentials

We consider the Laplace--Beltrami operator on a three-dimensional Riemannian manifold perturbed by a potential from the Kato class and study whether various forms of Weyl's law remain valid under this perturbation. We show that a pointwise Weyl law holds, modified by an additional term, for any Kato class potential with the standard sharp remainder term. The additional term is always of lower order than the leading term, but it may or may not be of lower order than the sharp remainder term. In particular, we provide examples of singular potentials for which this additional term violates the sharp pointwise Weyl law of the standard Laplace-Beltrami operator. For the proof we extend the method of Avakumović to the case of Schrödinger operators with singular potentials.

math-ph↗
arXiv2019

The Hartree and Vlasov equations at positive density

We consider the nonlinear Hartree and Vlasov equations around a translation-invariant (homogeneous) stationary state in infinite volume, for a short range interaction potential. For both models, we consider time-dependent solutions which have a finite relative energy with respect to the reference translation-invariant state. We prove the convergence of the Hartree solutions to the Vlasov ones in a semi-classical limit and obtain as a by-product global well-posedness of the Vlasov equation in the (relative) energy space.

math-ph↗
arXiv2018

Extremizers for the Airy-Strichartz inequality

We identify the compactness threshold for optimizing sequences of the Airy-- Strichartz inequality as an explicit multiple of the sharp constant in the Strichartz inequality. In particular, if the sharp constant in the Airy--Strichartz inequality is strictly smaller than this multiple of the sharp constant in the Strichartz inequality, then there is an optimizer for the former inequality. Our result is valid for the full range of Airy--Strichartz inequalities (except the endpoints) both in the diagonal and off-diagonal cases.

math.AP↗
arXiv2016

The Stein-Tomas inequality in trace ideals

The goal of this review is to explain some recent results regarding generalizations of the Stein-Tomas (and Strichartz) inequalities to the context of trace ideals (Schatten spaces).

math.AP↗
arXiv2016

Maximizers for the Stein-Tomas inequality

We give a necessary and sufficient condition for the precompactness of all optimizing sequences for the Stein-Tomas inequality. In particular, if a well-known conjecture about the optimal constant in the Strichartz inequality is true, we obtain the existence of an optimizer in the Stein-Tomas inequality. Our result is valid in any dimension.

math.CA↗
arXiv2014

Restriction theorems for orthonormal functions, Strichartz inequalities, and uniform Sobolev estimates

We generalize the theorems of Stein--Tomas and Strichartz about surface restrictions of Fourier transforms to systems of orthonormal functions with an optimal dependence on the number of functions. We deduce the corresponding Strichartz bounds for solutions to Schrödinger equations up to the endpoint, thereby solving an open problem of Frank, Lewin, Lieb and Seiringer. We also prove uniform Sobolev estimates in Schatten spaces, extending the results of Kenig, Ruiz, and Sogge. We finally provide applications of these results to a Limiting Absorption Principle in Schatten spaces, to the well-posedness of the Hartree equation in Schatten spaces, to Lieb--Thirring bounds for eigenvalues of Schrödinger operators with complex potentials, and to Schatten properties of the scattering matrix.

math-ph↗
arXiv2014

The Hartree equation for infinitely many particles. I. Well-posedness theory

We show local and global well-posedness results for the Hartree equation $$i\partial_tγ=[-Δ+w*ρ_γ,γ],$$ where $γ$ is a bounded self-adjoint operator on $L^2(\R^d)$, $ρ_γ(x)=γ(x,x)$ and $w$ is a smooth short-range interaction potential. The initial datum $γ(0)$ is assumed to be a perturbation of a translation-invariant state $γ_f=f(-Δ)$ which describes a quantum system with an infinite number of particles, such as the Fermi sea at zero temperature, or the Fermi-Dirac and Bose-Einstein gases at positive temperature. Global well-posedness follows from the conservation of the relative (free) energy of the state $γ(t)$, counted relatively to the stationary state $γ_f$. We indeed use a general notion of relative entropy, which allows to treat a wide class of stationary states $f(-Δ)$. Our results are based on a Lieb-Thirring inequality at positive density and on a recent Strichartz inequality for orthonormal functions, which are both due to Frank, Lieb, Seiringer and the first author of this article.

math-ph↗
arXiv2013

The Hartree equation for infinitely many particles. II. Dispersion and scattering in 2D

We consider the nonlinear Hartree equation for an interacting gas containing infinitely many particles and we investigate the large-time stability of the stationary states of the form $f(-Δ)$, describing an homogeneous Fermi gas. Under suitable assumptions on the interaction potential and on the momentum distribution $f$, we prove that the stationary state is asymptotically stable in dimension 2. More precisely, for any initial datum which is a small perturbation of $f(-Δ)$ in a Schatten space, the system weakly converges to the stationary state for large times.

math-ph↗
arXiv2013

A family of monotone quantum relative entropies

We study here the elementary properties of the relative entropy $\cH(A,B)=\tr[ϕ(A)-ϕ(B)-ϕ'(B)(A-B)]$ for $ϕ$ a convex function and $A,B$ bounded self-adjoint operators. In particular, we prove that this relative entropy is monotone if and only if $ϕ'$ is operator monotone. We use this to appropriately define $\cH(A,B)$ in infinite dimension.

math-ph↗
arXiv2012

Charge renormalization and static electron/positron pair production for a nonlinear Dirac model with weak interactions

The Hartree-Fock approximation of Quantum Electrodynamics provides a rigorous framework for the description of relativistic electrons in external fields. This nonlinear model takes into account the infinitely many virtual electrons of Dirac's vacuum as well as the Coulomb interactions between all the particles. The state of the system is an infinite-rank projection satisfying a nonlinear equation. In this paper, we construct solutions to this equation, in the regime of weak interactions (that is, small coupling constant $α$), and strong external fields (that is, large atomic charge $Z$ such that $αZ:=κ$ stays fixed). In this regime, we are able to remove the ultraviolet cut-off $Λ$ as soon as $α\logΛ$ stays fixed. As an application of this result, we compare the critical strength $κ_c(α)$ of the external potential needed to produce an additional particle in the vacuum, when $α=0$ or $α>0$. We prove that $\lim_{α\to0}κ_c(α)/κ_c(0)> 1$, and we identify the limit exactly. Because of the dielectric behavior of Dirac's vacuum, static electron/positron pair production occurs in the interacting case for a stronger field that in the non-interacting case, which is a mere consequence of charge renormalization.

math-ph↗
arXiv2011

Static Electron-Positron Pair Creation in Strong Fields for a Nonlinear Dirac model

We consider the Hartree-Fock approximation of Quantum Electrodynamics, with the exchange term neglected. We prove that the probability of static electron-positron pair creation for the Dirac vacuum polarized by an external field of strength $Z$ behaves as $1-\exp(-κZ^{2/3})$ for $Z$ large enough. Our method involves two steps. First we estimate the vacuum expectation of general quasi-free states in terms of their total number of particles, which can be of general interest. Then we study the asymptotics of the Hartree-Fock energy when $Z\to+\infty$ which gives the expected bounds.

math-ph↗