Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The Rigorous Derivation of the 2D Cubic Focusing NLS from Quantum
  Many-body Evolution   We consider a 2D time-dependent quantum system of $N$-bosons with harmonic
external confining and \emph{attractive} interparticle interaction in the
Gross-Pitaevskii scaling. We derive stability of matter type estimates showing
that the $k$-th power of the energy controls the $H^{1}$ Sobolev norm of the
solution over $k$-particles. This estimate is new and more difficult for
attractive interactions than repulsive interactions. For the proof, we use a
version of the finite-dimensional quantum di Finetti theorem from [49]. A high
particle-number averaging effect is at play in the proof, which is not needed
for the corresponding estimate in the repulsive case. This a priori bound
allows us to prove that the corresponding BBGKY hierarchy converges to the GP
limit as was done in many previous works treating the case of repulsive
interactions. As a result, we obtain that the \emph{focusing} nonlinear
Schr\"{o}dinger equation is the mean-field limit of the 2D time-dependent
quantum many-body system with attractive interatomic interaction and
asymptotically factorized initial data. An assumption on the size of the
$L^{1}$-norm of the interatomic interaction potential is needed that
corresponds to the sharp constant in the 2D Gagliardo-Nirenberg inequality
though the inequality is not directly relevant because we are dealing with a
trace instead of a power.

17
The categories of this paper are: q-fin
4
The categories of this paper are: cond-mat
37
The categories of this paper are: supr-con
24
The categories of this paper are: dg-ga
6
The categories of this paper are: astro-ph
23
The categories of this paper are: solv-int
5
The categories of this paper are: gr-qc
33
The categories of this paper are: plasm-ph
19
The categories of this paper are: eess
31
The categories of this paper are: ao-sci
26
The categories of this paper are: bayes-an
7
The categories of this paper are: hep-th
10
The categories of this paper are: q-bio
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The categories of this paper are: adap-org
9
The categories of this paper are: nlin
11
The categories of this paper are: hep-lat