# Oberseminar: Calculus of Variations and Applications

## Summer Semester 2022

The seminar takes place on Wednesday, starting from 4:15 pm, at room B 134, unless indicated otherwise.### Organizers: Phan ThÃ nh Nam, Arnaud Triay

### Speakers

- 18.5.2022:
**Jinyeop Lee**(LMU Munich).

**Title:**A mixed-norm estimate of two-particle reduced density matrix of many-body SchrÃ¶dinger dynamics

**Abstract:**We provide a mixed-norm estimate of two-particle reduced density matrix of the solution of $N$-body SchrÃ¶dinger equation. Using that we present a new approach to obtain the Vlasov dynamics from the SchrÃ¶dinger equation through Hartree-Fock dynamics with $\hbar = N^{-1/3}$ as the re-scaled Plank constant. Furthermore, we provide that, in the sense of distribution, the mean-field residue term $\mathcal{R}_\mathrm{m}$ has higher rate than the semi-classical residue $\mathcal{R}_\mathrm{s}$, namely, $\mathcal{R}_\mathrm{s} \sim \hbar^{\frac{1}{2}-}$ and $\mathbb{R}_\mathrm{m} \sim N^{-1} \hbar^{-\frac{1}{2}-}$ $\sim N^{-\frac{1}{2}}\hbar^{1-}$.

- 01.6.2022:
**Toan T. Nguyen**(Penn State University).

**Title:**Landau damping

**Abstract:**Consider the classical meanfield Vlasov-Poisson system on the whole space, there are three basic damping mechanisms: dispersion due to velocity averaging, dispersion due to plasma oscillations (Schrodinger's type), and Landau's damping. This talk is to give a complete linear theory of these phenomena and present recent works on the nonlinear problem.

- 22.6.2022:
**Alessandro Olgiati**(UniversitÃ¤t ZÃ¼rich).

**Title:**Hard-core bosons in the Gross-Pitaevskii regime: a second order energy upper bound

**Abstract:**I will present an upper bound to the ground state energy of $N$ bosons interacting through a hard-core potential in the Gross-Pitaevskii regime. Our result matches the known expression for the energy in the case of integrable potentials, which depends universally on the scattering length of the interaction, thus confirming the prediction of Bogoliubov's theory. The trial state providing the right upper bound is a suitable perturbation of the Jastrow wave-function. The key observation is then that optimizing such a perturbation can be reduced to finding an upper bound for a bosonic Hamiltonian with an integrable potential in a less singular scaling. Joint work with G. Basti (GSSI), S. Cenatiempo (GSSI), G. Pasqualetti (University of Z\"urich), and B. Schlein (University of Z\"urich).

- 29.6.2022:
**Morris Brooks**(IST Austria).

**Title:**Validity of Bogoliubovâ€™s approximation for translation-invariant Bose gases

**Abstract:**We will discuss Bogoliubovâ€™s approximation for translation-invaraint Bose gases in the mean-field limit. The result is divided into two parts: First one needs to verify the existence of approximate ground states satisfying complete BEC, a result which we believe to be of independent interest, and in the second step one uses BEC together with a rigorous correspondence principle in order to verify Bogoliubovâ€™s prediction. In this talk we will mainly focus on the second part of the result.

- 13.7.2022:
**Jonas Lampart**(UniversitÃ© Bourgogne Franche-ComtÃ© & CNRS).

**Title:**The ultra-violet problem for polaron Hamiltonians

**Abstract:**Many models involving quantum fields that appear naturally in physics are ill-defined mathematically due to ultra-violet singularities. I will discuss the construction and renormalisation of Hamiltonians for such models generalising the well-known FrÃ¶hlich polaron model. For a class of models with a sub-critical behaviour, roughly meaning that the interaction looks weak at short distances, I will explain a construction of the Hamiltonian by an iterative procedure. This Hamiltonian is the limit of the cut-off Hamiltonians up to a divergent sequence of numbers related to the perturbation series of the ground state energy.

- 20.7.2022:
**Dinh-Thi Nguyen**(ENS Lyon & CNRS).

**Title:**Asymptotic analysis for 2D rotating Bose gases

**Abstract:**We study the minimizers of a magnetic 2D non-linear Schr\"odinger energy functional in a quadratic trapping potential, describing a rotating Bose-Einstein condensate. First, we consider the case of the repulsive interaction potential. We derive an effective Thomas-Fermi-like model in the rapidly rotating limit where the centrifugal force compensates the confinement. The available states are restricted to the lowest Landau level. The coupling constant of the Thomas--Fermi functional is to linked the emergence of vortex lattices (the Abrikosov problem). Second, we consider the case where the interaction potential is attractive. The system is unstable when either the rotating frequency is larger than the trapping frequency or the interaction strength goes beyond a_*, the critical strength of the focusing cubic nonlinear Schr\"odinger equation. We study the behavior of the energy and its minimizers when the coupling constant approaches a_*. We prove that blow-up profile given by the Gagliardo-Nirenberg solution is independent of the rotation speed, even in the rapidly rotating limit.