Long time behaviour of quantum and classical systems

25 mars 2024, 09:00 → 29 mars 2024, 14:00 Europe/Paris

IRMAR

Julien Sabin

The goal of the meeting is to bring together people working on the three areas:

* many-body (linear) scattering (quantum and classical)

* non-linear quantum scattering

* non-linear classical scattering

and share techniques/ideas between communities. In particular, there will be three mini-courses given by Christian Gérard, Nicola Visciglia, and Daniel Han Kwan.

Organisation: Julien Sabin

Gestion des participants et des repas: Florian Rogowski et Aude Guiny

Supports financiers: ANR via la CPJ de Julien Sabin, Université de Rennes, IRMAR

Coming to IRMAR: 

The math department (IRMAR) is situated on the Beaulieu campus of the Université de Rennes. You can reach it by bus, there are two possible bus stops near the department:

You can take either the bus C4 (direction Rennes/ZA Saint Sulpice) and stop at "Les préales" or "Beaulieu Université), or the bus C6 (direction Cesson-Sévigné) and stop at "Les préales". All the informations on public transportation in Rennes may be found here: https://www.star.fr/

For those staying at "Cité Internationale Paul Ricoeur", the closest bus stop is "Musée Beaux Arts".

All talks will be held at the ground floor of the IRMAR building, next to the coffee room. To enter the building, a badge a needed but since there are many people coming and going from the building all the time, you will be able to enter after waiting a bit.

 

 

lundi 25 mars

1 Welcome of participants

12:00 Lunch

2 Scattering for nonlinear Vlasov equations I

The minicourse will be about the long-time behaviour of solutions to the Vlasov--Poisson or Vlasov--screened Poisson equations, set in the whole space.
We will mostly try to review: (1) global existence and (modified) scattering for small initial data; (2) scattering in the screened case for initial data close to stable homogeneous equilibria. The approach we will follow for both problems is based on the study of the characteristics associated with Vlasov equations.

Orateur: Daniel Han Kwan (Université de Nantes)

16:00 Coffee break

3 On the asymptotic analysis of the Einstein-Vlasov system

The Einstein-Vlasov system is the natural generalization of the classical Vlasov-Poisson system in the context of the general theory of relativity. I will start by an introduction to the Einstein equations and relativistic (collisionless) kinetic theory. I will then present several results concerning the asymptotics and the global existence of small data solutions for the Einstein-Vlasov system with either massive or massless particules.

Orateur: Jacques Smulevici

mardi 26 mars

09:30 Welcome coffee

4 Scattering for nonlinear Vlasov equations II

The minicourse will be about the long-time behaviour of solutions to the Vlasov--Poisson or Vlasov--screened Poisson equations, set in the whole space.
We will mostly try to review: (1) global existence and (modified) scattering for small initial data; (2) scattering in the screened case for initial data close to stable homogeneous equilibria. The approach we will follow for both problems is based on the study of the characteristics associated with Vlasov equations.

Orateur: Daniel Han Kwan

12:00 Lunch

5 Scattering theory for NLS I

We study the long time behaviour of solutions to NLS. We shall first show that for long-range nonlinearities the nonlinear solutions cannot behave as free waves for large times.
Next we shall focus on the short range case and we shall describe several results proving that nonlinear solutions behave as free wave in a suitable sense. In particular we shall discuss the scattering theory in H^1 and in \Sigma spaces. We shall also present during the lectures some open problems.

Orateur: Nicola Visciglia

16:00 Coffee break

6 Stability results for resonant Schrödinger equations on Diophantine tori

This presentation is devoted to a stability result for cubic Schrödinger equations
(NLS) on Diophantine tori. We prove that the majority of small solutions in high
regularity Sobolev spaces do not exchange energy from low to high frequencies over
very long time scales. We first provide context on the Birkhoff normal form approach in the study of the long-time dynamics of the solutions to Hamiltonian partial differential equations. Then, we present the induction on scales normal form which is at the heart of
the proof. Throughout the iteration, we ensure appropriate non-resonance properties
while modulating the frequencies (of the linearized system) with the amplitude of the
Fourier coefficients of the initial data. Our main challenge is then to addressing very
small divisor problems, and describing the set of admissible initial data. The results are based on a joint work with Joackim Bernier, and an ongoing joint work with Gigliola Staffilani.

Orateur: Nicolas Camps

mercredi 27 mars

09:30 Welcome coffee

7 Scattering theory for NLS II

We study the long time behaviour of solutions to NLS. We shall first show that for long-range nonlinearities the nonlinear solutions cannot behave as free waves for large times.
Next we shall focus on the short range case and we shall describe several results proving that nonlinear solutions behave as free wave in a suitable sense. In particular we shall discuss the scattering theory in H^1 and in \Sigma spaces. We shall also present during the lectures some open problems.

Orateur: Nicola Visciglia

12:00 Lunch

8 Large-time behavior for the Landau equation

I will present recent results for the Landau equation that prove the converge of solutions to the unique Maxwellian equilibrium (with quantitative rates).

Orateur: Kléber Carrapatoso

9 Stability Analysis of Quantum Dissipative Systems

In this talk I will present the stability properties of plane wave solutions for a system describing quantum particles interacting with a complex environment. From a mathematical point of view, this amounts to studying a system of PDEs coupled in a non-local (in time and space) way, which complicates the analysis considerably compared to the usual nonlinear Schrödinger equations. The strategy adopted is based on the identification of suitable Hamiltonian structures and Lyapunov functionals. Work in collaboration with T. Goudon.

Orateur: Simona Rota Nodari

16:00 Coffee break

10 Focusing dynamics of 2D Bose gases in the instability regime

We consider the dynamics of a 2D Bose gas with an interaction
potential of the form $N^{2\beta-1}w(N^\beta\cdot)$ for $\beta\in (0,3/2)$.
The interaction may be chosen to be negative and large, leading to the
instability regime where the corresponding focusing cubic nonlinear
Schrödinger equation (NLS) may blow up in finite time. We show that to
leading order, the $N$-body quantum dynamics can be effectively described by
the NLS prior to the blow-up time. Moreover, we prove the validity of the
Bogoliubov approximation, where the excitations from the condensate are
captured in a norm approximation of the many-body dynamics. This is joint work with Lea Boßmann and Phan Thành Nam.

Orateur: Charlotte Dietze

jeudi 28 mars

11 Semiclassical analysis of quantum scattering in the Yukawa interaction

In this talk, I will consider the quantum scattering theory of a Yukawa particle field
model in the semi-classical regime. The transition and diffusion amplitudes of such a model will be linked to those of the Schrödinger--Klein--Gordon equation. The phase space properties of asymptotic vacuum states will be highlighted and the concentration around the radiation-free solutions of the SKG equation will be established. I will also discuss the uniqueness of SKG energy minimizers as well as some open questions related to asymptotic completeness. This is a joint work with Marco Falconi and Marco Olivieri.

Orateur: Zied Ammari

12 Free time (PDE seminar) / coffee break

13 Scattering theory of classical particles

I will speak about classical scattering theory, both 2-body and N-body. It is much less developed than its quantum counterpart. Nevertheless, it has some interesting constructions and results, which are later useful in the quantum theory. Therefore, apart from its independent interest, my talk can serve as an introduction to Christian Gerard's lectures on quantum N-body scattering.

Orateur: Jan Derenzinski

rennes-slides.pdf

12:15 Lunch

14 Scattering for the quantum $N$-body problem I

We will review the scattering theory for the quantum $N$-body problem, focusing on asymptotic completeness. We will explain the methods developped in the eighties and nineties to tackle this problem, which led to a complete solution of the asymptotic completeness problem.

Orateur: Christian Gérard

Jacket.png

transparents-rennes-2024.pdf

16:00 Coffee break

15 A many-body RAGE theorem

The RAGE theorem is a general statement on the relation of spectrum and dynamics for self-adjoint operators.
It states that the large-time dynamics converge, in a certain weak sense, to the projection of the initial condition onto the bound states of the generator. The contributions of the continuous spectrum disappear in this limit.
The many-body version of the theorem, which is joint work with Mathieu Lewin, refines this statement in the case of N-body systems.
In an appropriate sense, these systems converge for large times to a combination of bound states of systems with n=0,..,N particles.
Including the decomposition into subsystems with strictly fewer particles resolves the contribution of the continuous spectrum on the N body generator, which is due to freely moving clusters of particles.

Orateur: Jonas Lampart

20:30 Conference diner (Crêperie Bretone, 7 rue Joseph Sauveur)

vendredi 29 mars

16 Scattering for the quantum $N$-body problem II

We will review the scattering theory for the quantum $N$-body problem, focusing on asymptotic completeness. We will explain the methods developped in the eighties and nineties to tackle this problem, which led to a complete solution of the asymptotic completeness problem.

Orateur: Christian Gérard

11:00 Coffee break

17 Some dynamical properties of the scattering operator for NLS

For the nonlinear Schrodinger equation with short range nonlinearity, the construction of this operator often implies sone analyticity properties of this operator, as we will explain in a first part. We will also show how, in the special case of the mass-critical nonlinearity, it is rather straightforward to construct final states, or initial data, for which the action of the scattering operator, or inverted wave operator, is explicit, consisting in the multiplication by an arbitrary complex number of modulus one.

Orateur: Rémi Carles

12:45 Lunch