arXiv subjects

Sébastien Breteaux

Publications and source records attributed to Sébastien Breteaux.

17 recordsLinked to original sources

arXiv2026

Existence of Solutions for time-dependent fractional Kohn-Sham Equations

We consider time-dependent Kohn-Sham equations in dimension $3$ with a fractional dispersion relation $(1-Δ)^s$, $s\in(0,\frac32)$, and a class of interaction terms including, in particular, external potentials, internal potentials associated to Hartree-type non-linearities, and exchange terms described by energy subcritical pure-power non-linearities. We prove the local existence of weak solutions in $H^s$ using an approximation procedure regularizing the non-linearities. Assuming that the interaction energies can be controlled by the kinetic energy, we show that the solutions can be extended to global solutions using energy estimates. If $s\in[1,\frac32)$, we establish in addition the well-posedness of the time-dependent Kohn-Sham equations using Strichartz estimates.

math.AP↗

arXiv2025

Exponential Decay outside of the Light Cone for the Pseudo-Relativistic Non-Autonomous Schrödinger Equation

We establish a maximal velocity bound for a pseudo-relativistic quantum particle in an external time-dependent potential. Our estimate shows that the probability for the particle, starting in a convex set $X\subset\mathbb{R}^d$ at $t=0$, to reach a convex set $Y\subset\mathbb{R}^d$ at a time $t>0$, is bounded by $e^{-2δ}$ where $δ$ is the distance from $Y$ to the section at time $t$ of the light cone generated by $X$.

math-ph↗

arXiv2025

Propagation Estimates for the Boson Star Equation

We consider the boson star equation with a general two-body interaction potential $w$ and initial data $ψ_0$ in a Sobolev space. Under general assumptions on $w$, namely that $w$ decomposes as a sum of a finite, signed measure and an essentially bounded function, we prove that the (local in time) solution cannot propagate faster than the speed of light, up to a sharp exponentially small remainder term. If $w$ is short-range and $ψ_0$ is regular and small enough, we prove in addition asymptotic phase-space propagation estimates and minimal velocity estimates for the (global in time) solution, depending on the momentum of the scattering state associated to $ψ_0$.

math-ph↗

arXiv2024

On the number of bound states for fractional Schr{ö}dinger operators with critical and super-critical exponent

We study the number $N_{<0}(H_s)$ of negative eigenvalues, counting multiplicities, of the fractional Schrödinger operator $H_s=(-Δ)^s-V(x)$ on $L^2(\mathbb{R}^d)$, for any $d\ge1$ and $s\ge d/2$. We prove a bound on $N_{<0}(H_s)$ which depends on $s-d/2$ being either an integer or not, the critical case $s=d/2$ requiring a further analysis. Our proof relies on a splitting of the Birman-Schwinger operator associated to this spectral problem into low- and high-energies parts, a projection of the low-energies part onto a suitable subspace, and, in the critical case $s=d/2$, a Cwikel-type estimate in the weak trace ideal $\mathcal{L}^{2,\infty}$ to handle the high-energies part.

math.AP↗

arXiv2023

Light cones for open quantum systems

We consider Markovian open quantum dynamics (MOQD). We show that, up to small-probability tails, the supports of quantum states evolving under such dynamics propagate with finite speed in any finite-energy subspace. More precisely, we prove that if the initial quantum state is localized in space, then any finite-energy part of the solution of the von Neumann-Lindblad equation is approximately localized inside an energy-dependent light cone. We also obtain an explicit upper bound for the slope of this light cone.

quant-ph↗

arXiv2022

Quasi-classical Ground States. II. Standard Model of Non-relativistic QED

We consider a non-relativistic electron bound by an external potential and coupled to the quantized electromagnetic field in the standard model of non-relativistic QED. We compute the energy functional of product states of the form $u\otimes Ψ_f$, where $u$ is a normalized state for the electron and $Ψ_f$ is a coherent state in Fock space for the photon field. The minimization of this functional yields a Maxwell--Schr{ö}dinger system up to a trivial renormalization. We prove the existence of a ground state under general conditions on the external potential and the coupling. In particular, neither an ultraviolet cutoff nor an infrared cutoff needs to be imposed. Our results provide the convergence in the ultraviolet limit and the second-order asymptotic expansion in the coupling constant of the ground state energy of Maxwell--Schrödinger systems.

math.AP↗

arXiv2022

Quasi-classical Ground States. I. Linearly Coupled Pauli-Fierz Hamiltonians

We consider a spinless, non-relativistic particle bound by an external potential and linearly coupled to a quantized radiation field. The energy $\mathcal{E}(u,f)$ of product states of the form $u\otimes Ψ_f$, where $u$ is a normalized state for the particle and $Ψ_f$ is a coherent state in Fock space for the field, gives the energy of a Klein-Gordon--Schr\''odinger system. We minimize the functional $\mathcal{E}(u,f)$ on its natural energy space. We prove the existence and uniqueness of a ground state under general conditions on the coupling function. In particular, neither an ultraviolet cutoff nor an infrared cutoff is imposed. Our results establish the convergence in the ultraviolet limit of both the ground state and ground state energy of the Klein-Gordon--Schr\''odinger energy functional, and provide the second-order asymptotic expansion of the ground state energy at small coupling.

math.AP↗

arXiv2022

Maximal Speed of Propagation in Open Quantum Systems

We prove a maximal velocity bound for the dynamics of Markovian open quantum systems. The dynamics are described by one-parameter semi-groups of quantum channels satisfying the von Neumann-Lindblad equation. Our result says that dynamically evolving states are contained inside a suitable light cone up to polynomial errors. We also give a bound on the slope of the light cone, i.e., the maximal propagation speed. The result implies an upper bound on the speed of propagation of local perturbations of stationary states in open quantum systems.

quant-ph↗

arXiv2018

The time-dependent Hartree-Fock-Bogoliubov equations for Bosons

In this article, we use quasifree reduction to derive the time-dependent Hartree-Fock-Bogoliubov (HFB) equations describing the dynamics of quantum fluctuations around a Bose-Einstein condensate in $\mathbb R^d$. We prove global well-posedness for the HFB equations for sufficiently regular pair interaction potentials, and establish key conservation laws. Moreover, we show that the solutions to the HFB equations exhibit a symplectic structure, and have a form reminiscent of a Hamiltonian system. In particular, this is used to relate the HFB equations to the HFB eigenvalue equations encountered in the physics literature. Furthermore, we construct the Gibbs states at positive temperature associated with the HFB equations, and establish criteria for the emergence of Bose-Einstein condensation.

math-ph↗

arXiv2018

On the Hartree-Fock-Bogoliubov equations

We review some results of our paper arXiv:1602.05171v2 on the "nonlinear quasifree approximation" to the many-body Schrödinger dynamics of Bose gases. In that paper, we derive, with the help of this approximation, the time-dependent Hartree-Fock-Bogoliubov (HFB) equations, providing an approximate description of the dynamics of quantum fluctuations around a Bose-Einstein condensate and study properties of these equations.

math-ph↗

arXiv2017

Quantum mean field asymptotics and multiscale analysis

We study, via multiscale analysis, some defect of compactness phenomena which occur in bosonic and fermionic quantum mean-field problems. The approach relies on a combination of mean-field asymptotics and second microlocalized semiclassical measures. The phase space geometric description is illustrated by various examples.

math-ph↗

arXiv2015

Kinetic Energy Estimates for the Accuracy of the Time-Dependent Hartree-Fock Approximation with Coulomb Interaction

We study the time evolution of a system of $N$ spinless fermions in $\mathbb{R}^3$ which interact through a pair potential, e.g., the Coulomb potential. We compare the dynamics given by the solution to Schr{ö}dinger's equation with the time-dependent Hartree-Fock approximation, and we give an estimate for the accuracy of this approximation in terms of the kinetic energy of the system. This leads, in turn, to bounds in terms of the initial total energy of the system.

math-ph↗

arXiv2013

Generalized One-Particle Density Matrices and Quasifree States

In the spirit of the generalized one-particle density matrix for fermions, we introduce generalized one- and two-particle density matrices to state representability conditions up to second order for boson systems without assuming particle number-conservation. Furthermore, we show for both particle species that, for a semibounded Hamiltonian, the in?mum of the variation of the energy functional w.r.t quasifree states coincides with the one of a variation over pure quasifree states. Moreover, it is proven for fermions that only pure quasifree states have a generalized 1-pdm that is a projection, and a similar statement for bosons.

math-ph↗

arXiv2013

Minimization of the Energy of the Non-Relativistic One-Electron Pauli-Fierz Model over Quasifree States

In this article is proved the existence and uniqueness of a minimizer of the energy for the non-relativistic one electron Pauli-Fierz model, within the class of pure quasifree states. The minimum of the energy on pure quasifree states coincides with the minimum of the energy on quasifree states. Infrared and ultraviolet cutoffs are assumed, along with sufficiently small coupling constant and momentum of the dressed electron. A perturbative expression of the minimum of the energy on quasifree states for a small momentum of the dressed electron and small coupling constant is then given. We also express the Lagrange equation for the minimizer, in terms of the generalized one particle density matrix of the pure quasifree state.

math-ph↗

arXiv2012

Asymptotic Analysis of a Non-Linear Non-Local Integro-Differential Equation Arising from Bosonic Quantum Field Dynamics

We introduce a one parameter family of non-linear, non-local integro-differential equations and its limit equation. These equations originate from a derivation of the linear Boltzmann equation using the framework of bosonic quantum field theory. We show the existence and uniqueness of strong global solutions for these equations, and a result of uniform convergence on every compact interval of the solutions of the one parameter family towards the solution of the limit equation.

math.AP↗

arXiv2012

A geometric derivation of the linear Boltzmann equation for a particle interacting with a Gaussian random field

In this article the linear Boltzmann equation is derived for a particle interacting with a Gaussian random field, in the weak coupling limit, with renewal in time of the random field. The initial data can be chosen arbitrarily. The proof is geometric and involves coherent states and semi-classical calculus.

math-ph↗

arXiv2011

Higher order terms for the quantum evolution of a Wick observable within the Hepp method

The Hepp method is the coherent state approach to the mean field dynamics for bosons or to the semiclassical propagation. A key point is the asymptotic evolution of Wick observables under the evolution given by a time-dependent quadratic Hamiltonian. This article provides a complete expansion with respect to the small parameter ε> 0 which makes sense within the infinite-dimensional setting and fits with finite-dimensional formulae.

math-ph↗