# 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.

### 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.