Conference details

LIST OF ABSTRACTS

Jean-Marie BARBAROUX : Quantitative estimates of the binding energy
for Hydrogen atom in Nonrelativistic QED.

Abstract : Explicit computation of the binding energy for Hydrogen whose Hamiltonian is given by the Schrödinger-Coulomb operator is a very simple exercise.
However, a refined computation including the interactions with the quantized radiation field, seems to be an impossible task. Therefore a perturbative approach (in the size of the coupling, i.e., where the fine structure constant $\alpha$ is considered as a small parameter) ought to be the right approach.
Unfortunately, as we shall see, the standard perturbation theory failed to work in this case.
We will show that it is nevertheless possible to devise alternative method to compute the binding energy in powers of the fine structure constant, up to the order $\alpha4$, with rigorous error bounds.
Moreover, we will prove that deriving higher order estimates will exhibit the non analytic property of
the binding energy in the variable $\alpha$.

Laurent BRUNEAU : Thermal relaxation in a quantum cavity

Abstract : We study repeated interactions of the quantized electromagnetic field in a cavity with a sequence of two-level atoms, so-called "One-atom masers" experiments. We study the large time
behaviour of the system. We show that whenever the atoms are initially in thermal equilibrium at temperature T>0, and provided some non-resonant condition is satisfied, the cavity field relaxes towards thermal equilibrium at some renormalized temperature T*. Our result is non-perturbative in the strength of the atom-field coupling.

Jan DEREZINSKI : Scattering in non relativistic quantum field theory

Abstract: I will discuss the set-up and some rigorous results about scattering in nonrelativistic quantum field theory with a localized interaction. In particular, I will consider the (exactly solvable) van Hove model, the Pauli-Fierz model and a general class of perturbative models.

Christian GERARD : Sur le modèle de Nelson à coefficients variables

Résumé : Nous décrirons dans cet exposé quelques résultats préliminaires sur le modèle de Nel-
son à coefficients variables. Nous discuterons le problème ultraviolet, en montrant que
la troncature ultraviolette peut être éliminé comme dans le modèle de Nelson habituel.
Nous discuterons aussi le problème infrarouge (existence d'un état fondamental dans
l'espace de Hilbert), en montrant que si la masse décroit vers 0 assez lentement, il
existe un état fondamental. Enfin nous donnerons quelques problèmes ouverts et
quelques conjectures.

Alain JOYE : Leaky Repeated Interaction Quantum Systems

Abstract : We consider a small reference system S interacting with two large quantum systems of a different nature. On the one hand the system S interacts for a fixed duration with the successive elements E of an infinite chain C of identical independent quantum subsystems E. And, on the other hand, it interacts continuously with a heat reservoir R at a some inverse temperature given by an infinitely extended Fermi gas. The reservoir and the chain are not coupled. When the reservoir is absent, the state of the repeated interaction quantum system defined by S and the chain C approches a non-equilibrium asymptotic state for large times. When the chain is absent, the system S and the reservoir R reach an equilibrium state at large times. Our goal is to describe the large time behaviour of the fully coupled system S+R+C and to describe the asymptotic state of this system in the large times limit. This is joint work with Laurent Bruneau and Marco Merkli.

Mathieu LEWIN: The Hartree-Fock-Bogoliubov model for neutron stars and white dwarfs

Abstract : We present some recent results on the Hartree-Fock-Bogoliubov model for
attractive Fermi gases, as seen for instance in neutron stars and white dwarfs. We show
the existence of a minimizer when the energy is bounded from below and derive some of its
properties. We also address the question of the blow-up in finite time in the corresponding time-dependent equation. This is a joint work with Enno Lenzmann (MIT).

Jacob Schach MOLLER : Instability of a many-electron Fröhlich Hamiltonian

Abstract : The Fröhlich Hamiltonian describes the motion of $N$ electrons coupled to a second quantized phonon mode. The phonons are massive and have a constant dispersion relation. The model depends on $2$ parameters, an electron-electron repulsion strength $\lambda$, and the electron-phonon Fröhlich coupling constant $\alpha$. The physical regime corresponds
to constants in the halfplane $\lambda\geq \sqrt{2} \alpha$. In the talk it will be demonstrated that exactly in the unphysical parameter regime the model has an effective attractive electron-electron interaction resulting in a non-extensive groundstate energy diverging as $-N^{7/3}$. Inside the physical regime we prove a lower bound for the groundstate energy of the order $-N^2$, which may not be optimal.

Annalisa PANATI : Spectral and scattering theory for abstract QFT Hamiltonians

Abstract : We introduce an abstract class of bosonic QFT Hamiltonians and study their spectral and scattering theories. An example belonging to this class is the space-cutoff $P(\varphi)_{2}$ model with a variable metric of the form
\[H= \textrm{d}\Gamma(\omega)+ \int_{\mathbb{R}}g(x):\!P(x,\varphi(x))\!:\textrm{d} x,\]
on the bosonic Fock space $L^{2}(\mathbb{R})$, where the kinetic energy $\omega= h^{\frac{1}{2}}$ is the square root of a real second order differential operator $h= D_{x}a(x)D_{x}+ c(x)$, where the coefficients $a(x), c(x)$ tend respectively to $1$ and
$m_{\infty}^{2}$ at $\infty$ for some $m_{\infty}>0$. This is joint work with Christian Gérard.

Eric SERE : Ground state and charge renormalization in the reduced Bogoliubov -Dirac-Fock model

Abstract : This is joint work with Philippe GRAVEJAT and Mathieu LEWIN. We study the minimization of the reduced Bogoliubov-Dirac-Fock energy of a relativistic atom or molecule. We prove the existence of minimizers for a large range of values of the charge, and any positive value of the coupling constant. Our result covers neutral and positively charged molecules, provided that the positive charge is not large enough to create electron-positron pairs. We also prove that the density of any minimizer is an L 1 function and compute the effective charge of the system, recovering a
well-known formula for the renormalization of charge.

Friday 3 April	Saturday 4 April
9:00-9:30 Opening, Coffee
9:30-10:20 Jan DEREZINSZKI Scattering in non relativistic quantum field theory 1	9:30-10:20 Jan DEREZINSKI Scattering in non relativistic quantum field theory 2
10:20-10:45 COFFEE BREAK	10:20-10:45 COFFEE BREAK
10:45-11:30 Christian GERARD Sur le modèle de Nelson à coefficients variables	10:45-11:30 Jean-Marie BARBAROUX Quantitative estimates of the binding energy for Hydrogen atom in Nonrelativistic QED.
11:35-12:20 Eric SERE Ground state and charge renormalization in the reduced Bogoliubov-Dirac-Fock model	11:35-12:20 Mathieu LEWIN The Hartree-Fock-Bogoliubov model for neutron stars and white dwarfs
LUNCH	LUNCH
14:15-15:00 Annalisa PANATI Spectral and scattering theory for abstract QFT Hamiltonians
15:05-15:50 Alain JOYE Leaky repeated Interacting Quantum Systems
15:50-16:15 COFFEE BREAK
16:15-17:00 Laurent BRUNEAU Thermal relaxation in a quantum cavity
17:05-17:50 Jacob Schach MOLLER Instability of a many-electron Fröhlich Hamiltonian

PROGRAMME