Titel: “The dipolar Gross-Pitaevskii functional with Lee-Huang-Yang corrections”
Startdatum: 15 Juli
Startzeit: 15:15
Stoppzeit: 16:45Uhr
Veranstalter: Institut für Mathematik
Herr Arnaud Triay (Ceremade Université Paris-Dauphine)  
Ort: HPS40, Raum 0450

We study the functional

$\mathcal{E} (u) = \int_{R^3} |\nabla u|^2 + a \int_{R^3} |u|^4 + b \int_{R^3} (K \ast |u|^2) |u|^2 + c \int_{R^3} |u|^5$

on $\int_{R^3} |u|^2 = d$ and where $K(x) = (1-3 x_3^2/|x|^2)/|x|^3$.


The Gross-Pitaevskii (GP) theory is an effective model describing the common state of all (or most of) the particles in a Bose-Einstein condensate. The GP energy is given by a non-linear Schrödinger functional with a local quartic term accounting for the short range interactions among the particles of the dilute gas. To deal with dipolar particles, a (second) long-range term need to be added and gives rise to an instability in the regime where the dipolar interaction dominates the repulsive short range one. Nevertheless, the existence of a metastable state in this unstable regime has recently been discovered in experiments. The stabilization mechanism is believed to be based on quantum fluctuations and is accounted for in the GP functional by a (third) quintic non-linearity called the Lee-Huang-Yang correction.


In this talk, we will give a brief overview of the link with the many-body framework and discuss the existence of minimizers for this functional.