# Program

All talks are 25 minutes + 5 minutes for questions.

Note that all times are stated in Central European Summer Time (UTC +2).

## Thursday, October 1, 2020

 13:45 - 14:00 Sören Petrat Welcome 14:00 - 14:30 Peter Pickl A Light Tracer Particle Interacting with a Bose Gas 14:30 - 15:00 Nikolai Leopold Derivation of the Landau-Pekar Equations in a Many-Body Mean-Field Limit 15:00 - 15:30 Marcin Napiórkowski Optimal Rate of Condensation for Trapped Bosons in the Gross-Pitaevskii Regime 15:30 - 16:00 Virtual coffee break 16:00 - 16:30 Lea Bossmann Asymptotic Expansion of the Low-energy Excitation Spectrum for Weakly Interacting Bosons 16:30 - 17:00 Sébastien Breteaux Accuracy of the Time-Dependent Hartree-Fock Approximation 17:00 - 17:30 Marco Falconi Quasi-Classical Dynamics 17:30 - 18:30 Open discussion

## Friday, October 2, 2020

 13:55 - 14:00 Sören Petrat Welcome 14:00 - 14:30 Jean-Bernard Bru Macroscopic Long-Range Dynamics of Fermions and Quantum Spins on the Lattice 14:30 - 15:00 David Mitrouskas Polaron Dynamics in the Strong Coupling Limit 15:00 - 15:30 Volker Bach On the Ultraviolet Limit of the Pauli-Fierz Hamiltonian in the Lieb-Loss Model 15:30 - 16:00 Virtual coffee break 16:00 - 16:30 Phan Thành Nam Emergence of Nonlinear Gibbs Measure from Bose Gases at Positive Temperature 16:30 - 17:00 Simone Rademacher Central Limit Theorem for Bose-Einstein Condensates 17:00 - 17:30 Alessandro Pizzo Local Lie-Schwinger Conjugations and Gapped Quantum Chains 17:30 - 18:30 Open discussion

# Abstracts

Peter Pickl, "A Light Tracer Particle Interacting with a Bose Gas"

We consider a particle entering a Bose gas of large density. We assume interaction among the gas particles in a weak coupling regime and between gas particles and the tracer particle. In contrast to other approaches we assume, that the mass of the tracer particle and the gas particles are the same. We assume that the density of the gas is roughly constant and use a coupling for the interaction which is such, that the influence on the tracer particles is of leading order. We will show that, of leading order, the tracer interacts with the excitations in the gas and that these excitations are subject to Bogoliubov time evolution.
This is joint work with Jonas Lampart.

Nikolai Leopold, "Derivation of the Landau-Pekar Equations in a Many-Body Mean-Field Limit"

We consider the Fröhlich Hamiltonian in a mean-field limit where many bosonic particles weakly couple to the quantized phonon field. For large particle number and suitably small coupling, we show that the dynamics of the system is approximately described by the Landau-Pekar equations. These describe a Bose-Einstein condensate interacting with a classical polarization field, whose dynamics is effected by the condensate, i.e., the back-reaction of the phonons that are created by the particles during the time evolution is of leading order. The talk is based on work in collaboration with David Mitrouskas and Robert Seiringer.

Marcin Napiórkowski, "Optimal Rate of Condensation for Trapped Bosons in the Gross-Pitaevskii Regime"

We study the Bose-Einstein condensates of trapped Bose gases in the Gross-Pitaevskii regime. We show that the ground state energy and ground states of the many-body quantum system are correctly described by the Gross-Pitaevskii equation in the large particle number limit, and provide the optimal convergence rate. Based on joint work with P. T. Nam, J. Ricaud and A. Triay.

Lea Bossmann, "Asymptotic Expansion of the Low-energy Excitation Spectrum for Weakly Interacting Bosons"

We consider a system of N bosons in the mean-field scaling regime for a class of interactions including the repulsive Coulomb potential. We derive an asymptotic expansion of the low-energy eigenstates and the corresponding energies, which provides corrections to Bogoliubov theory to any order in 1/N. Joint work with Robert Seiringer and Sören Petrat.

Sébastien Breteaux, "Accuracy of the Time-Dependent Hartree-Fock Approximation"

We study the time evolution of a system of N spinless fermions which interact through a pair potential, e.g., the Coulomb potential. We compare the dynamics given by the solution to Schrödinger'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.

Marco Falconi, "Quasi-Classical Dynamics"

In this talk we study the dynamics of quantum particles interacting with a semiclassical radiation field. A typical example of such models, omnipresent in physics, is that of electrons in an atom, subjected to electromagnetic radiation.
We prove that the quantum field can be effectively approximated by a classical one, provided that the number of excitations of the field is very large. Such classical field acts as an environment for the particles' subsystem, driving their evolution.
Our discussion is based on newly developed techniques of semiclassical analysis for partially classical composite systems.

Jean-Bernard Bru, "Macroscopic Long-Range Dynamics of Fermions and Quantum Spins on the Lattice"

The aim of the current paper is to illustrate our recent rigorous studies on the dynamical properties of fermions and quantum-spin systems with long-range, or mean-field, interactions. For pedagogical reasons, we will consider mainly the example of the strong-coupling BCS-Hubbard model, instead of considering an abstract and general formulation of the problem, as it is done in our recent papers.

David Mitrouskas, "Polaron Dynamics in the Strong Coupling Limit"

We discuss recent results about the dynamics generated by the Froehlich Hamiltonian which describes an electron interacting with a quantized phonon field. For suitable initial product states and for large coupling constant alpha, the quantum dynamics can be described by product states that are determined by the classical Landau-Pekar equations. In the talk, we show how to improve the approximation by adding quantum fluctuations to the effective dynamics of the phonons. This allows to extend the validity of the approximation to the relevant time scale which is of order alpha^2. The talk is based on joint work with N. Leopold, S. Rademacher, B. Schlein and R. Seiringer.

Volker Bach, "On the Ultraviolet Limit of the Pauli-Fierz Hamiltonian in the Lieb-Loss Model"

Two decades ago, Lieb and Loss approximated the ground state energy of a free, nonrelativistic electron coupled to the quantized radiation field by the infimum $E_{\alpha, \Lambda}$ of all expectation values $\langle \phi_{el} \otimes \psi_{ph} | H_{\alpha, \Lambda} (\phi_{el} \otimes \psi_{ph}) \rangle$, where $H_{\alpha, \Lambda}$ is the corresponding Hamiltonian with fine structure constant $\alpha >0$ and ultraviolet cutoff $\Lambda < \infty$, and $\phi_{el}$ and $\psi_{ph}$ are normalized electron and photon wave functions, respectively. Lieb and Loss showed that $c \alpha^{1/2} \Lambda^{3/2} \leq E_{\alpha, \Lambda} \leq c^{-1} \alpha^{2/7} \Lambda^{12/7}$ for some constant $c >0$.
In a joint work with Alexander Hach we prove the existence of a constant $C < \infty$, such that $\bigg| \frac{E_{\alpha, \Lambda}}{F[1] \, \alpha^{2/7} \, \Lambda^{12/7}} - 1 \bigg| \leq C \, \alpha^{4/105} \, \Lambda^{-4/105}$ holds true, where $F[1] >0$ is an explicit universal number. This result shows that Lieb and Loss' upper bound is actually sharp and gives the asymptotics of $E_{\alpha, \Lambda}$ uniformly in the limit $\alpha \to 0$ and in the ultraviolet limit $\Lambda \to \infty$.

Phan Thành Nam, "Emergence of Nonlinear Gibbs Measure from Bose Gases at Positive Temperature"

I will report a rigorous derivation of nonlinear Gibbs measures from many-body quantum Bose gases at just above the Bose-Einstein phase transition. The Gibbs measures are very singular, while the quantum problem is regular but involves non commutative operators. The derivation thus requires a Wick renormalization in a semiclassical limit. Our proof relies on a new entropy estimate and a novel method to control the quantum variance. This is joint work with Mathieu Lewin and Nicolas Rougerie.

Simone Rademacher, "Central Limit Theorem for Bose-Einstein Condensates"

We consider a Bose gas trapped in the unit torus in the Gross-Pitaevskii regime. We show that in the ground state fluctuations of bounded one-particle observables satisfy a central limit theorem. The correlations among the particles in this regime affect the variance of the limiting Gaussian distribution. The proof is based on a norm approximation of the ground state recently established. This is joint work with Benjamin Schlein.

Alessandro Pizzo, "Local Lie-Schwinger Conjugations and Gapped Quantum Chains"

We consider quantum chains whose Hamiltonians are perturbations, by interactions of short range, of a Hamiltonian that does not couple the degrees of freedom located at different sites of the chain and has a strictly positive energy gap above its ground-state energy. For interactions that are form-bounded w.r.t. the on-site Hamiltonian terms, we prove that the spectral gap of the perturbed Hamiltonian above its ground-state energy is bounded from below by a positive constant uniformly in the length of the chain, for small values of a coupling constant. Under the same hypothesis, we prove that the ground state energy is analytic for values of the coupling constant in a fixed interval, uniformly in the length of the chain. In our proof we use a novel method based on local Lie-Schwinger conjugations of the Hamiltonians associated with connected subsets of the chain, that can be also applied to complex Hamiltonians obtained by considering complex values of the coupling constant. We can treat fermions and bosons on the same footing, and our technique does not face a large field problem, even though bosons are involved, in contrast to most approaches.
(Based on joint work with J. Fröhlich, and with S. Del Vecchio, J. Fröhlich, and S. Rossi.)