Derivation of the 1d Gross-Pitaevskii equation for strongly confined 3d bosons
Lea Boßmann
Universität Tübingen
We consider the dynamics of $N$ interacting bosons initially exhibiting Bose-Einstein condensation. Due to an external trapping potential, the bosons are strongly confined in two spatial directions, with the transverse extension of the trap being of order $\varepsilon$. The non-negative interaction potential is scaled such that its scattering length is positive and of order $(N/\varepsilon^2)^{-1}$, the range of the interaction scales as $(N/\varepsilon^2)^{-\beta}$ for $\beta\in(0,1]$. We prove that in the simultaneous limit $N\rightarrow\infty$ and $\varepsilon\rightarrow 0$, the condensation is preserved by the dynamics and the time evolution is asymptotically described by a nonlinear Schr{\"o}dinger equation in one dimension. The strength of the nonlinearity depends on the interaction and on the shape of the confining potential. For $\beta=1$, the effective equation is a physically relevant one-dimensional Gross-Pitaevskii equation, where the coupling parameter contains the scattering length of the unscaled interaction. For our analysis, we adapt an approach by Pickl to the problem with dimensional reduction. Joint work with Stefan Teufel, based on \textit{arXiv:1803.11011} and \textit{arXiv:1803.11026}.
On the Hartree limit for the ground state of a bosonic atom
without Born-Oppenheimer approximation
Erik De Amorim
Rutgers University
The goal of this work is to establish the Hartree theory for the ground state of a large atom with ``bosonic electrons'' in the limit of an infinite number of these, without invoking the Born-Oppenheimer approximation of an infinitely massive nucleus. A permutation-symmetric system of relative coordinates in the center-of-mass frame is developed which facilitates the study of the quantum ground state energy of such many-electron atoms with finite-mass nucleus (which applies equally well to bosonic and fermionic electrons). In the new coordinates the intrinsic Hamiltonian has an equivalent many-body interpretation of N electrons electrically attracted to a multiple of the empirical mean of their own positions.
Asymptotic behaivor and stability problem for the Schrödinger-Lohe model
Dohyun Kim
Seoul National University
We present asymptotic behavior and stability problem for the Schr\"{o}dinger-Lohe(S-L) system which was first introduced as a possible phenomenological model exhibiting quantum synchronization. We present several sufficient frameworks leading to the emergent behavior of the S-L system. More precisely, we show that there are only two possible asymptotic states: completely synchronized state or bi-polar state. Furthermore, we provide the standing wave solutions for the S-L model with the harmonic potential and discuss the stability for standing wave solutions.
Many-body blow-up of bosons stars
Dinh-Thi Nguyen
LMU Munich
We study ground states of a system of $N$ identical bosons in $\mathbb{R}^3$, described by the Hamiltonian $$ H_N = \sum_{i=1}^{N}\left(\sqrt{-\Delta_{x_i}+m^2}+V(x_i)\right)-\frac{a}{N-1}\sum_{1\leq i<j \leq N}|x_i-x_j|^{-1}. $$ acting on Hilbert space $\bigotimes_{\rm sym}^N L^2(\mathbb{R}^3)$. Here the parameter $m > 0$ is the mass of particles, $a>0$ describes the strength of the attractive interaction, and $V\geq 0$ is an external potential. We are interested in the behavior of the ground state energy per particle of $H_N$ and the corresponding ground state when $N\to\infty$ and $a:=a_N$ tends to $a^*$ (Chandrasekhar limit) from below. We first study blow-up behavior of ground state energy as well as of ground states when $a$ tends to $a^*$ in the effective model: Hartree theory.
On a Dissipative Gross-Pitaevskii-Type Model for Exciton-Polariton Condensates
Ryan Obermeyer
University of Illinois at Chicago
We study a generalized dissipative Gross-Pitaevskii-type model arising in the description of exciton-polariton condensates. We derive rigorous existence and uniqueness results for this model posed on the one dimensional torus and derive various a-priori bounds on its solution. Then, we analyze in detail the long time behavior of spatially homogenous solutions and their respective steady states. In addition, we will present numerical simulations in the case of more general initial data. We also study the corresponding adiabatic regime which results in a single damped-driven Gross-Pitaveskii equation and compare its dynamics to the one of the full coupled system. This is joint work with my advisor, Christof Sparber, as well as Paolo Antonelli (GSSI), Peter Markowich (KAUST), and Jesus Sierra (KAUST), to be submitted in the near future.
Microscopic derivation for time-dependent point interactions in ionization models
Marco Olivieri
Sapienza, Università di Roma
We show how ionization models, consisting of a moving particle influenced by a point interaction modulated by a time-evolving coefficient, can be derived as effective models obtained in quasi-classical limit from the polaron model. The approximation by the scale limit has a physical justification considering time-dependent, squeezed coherent states with high intensity of the field, and it can be proved that particle’s observables, evolving w.r.t. microscopic dynamics, weak converge to time-evolved observables according to the effective dynamics. From a joint work with R. Carlone, M. Correggi and M. Falconi.
Pointwise NLS in dimension two
Lorenzo Tentarelli
Sapienza, Università di Roma
We discuss some recent results on the twodimensional nonlinear Schr\"odinger equation with pointwise nonlinearity. We start by presenting local well-posedness of the associated Cauchy problem, as well as mass and energy conservation along the flow. Then, we show that in the repulsive case solutions are global-in-time, whereas in the attractive case one can exhibit a class of initial data that give rise to blow-up phenomena. Finally, we exhibit the family of the standing waves of the problem. The talk is based on three works in collaboration with R. Adami, R. Carlone, M. Correggi and A. Fiorenza.
Global existence, stability and scattering of solutions
to one-dimensional quantum hydrodynamic equations
Hao Zheng
GSSI
In this talk we consider the Cauchy problem for the one-dimensional quantum hydrodynamic (QHD) system, namely the compressible Euler equations with a quantum correction term. We consider finite energy initial data with certain higher order bounds, which imply the square integrability of the chemical potential. These type of models have been extensively used to investigate Superfluidity, Superconductivity and recently in the modelling of semiconductor devices. The main result is the global existence of weak solutions to the Cauchy problem. Our approach does not require the initial data to be given through a wave function as in previous results. Nevertheless the higher order bounds allow to construct a related wave function. We prove such higher order quantities are bounded for any finite time. Furthermore, those a priori estimates allow us to prove the stability of weak solutions to the QHD system. Last, we present some scattering properties of solutions to the QHD system, which can be proved in purely hydrodynamic way.
