Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Derivation of the Time Dependent Two Dimensional Focusing NLS Equation   In this paper, we present a microscopic derivation of the two-dimensional
focusing cubic nonlinear Schr\"odinger equation starting from an interacting
$N$-particle system of Bosons. The interaction potential we consider is given
by $W_\beta(x)=N^{-1+2 \beta}W(N^\beta x)$ for some bounded and compactly
supported $W$. We assume the $N$-particle Hamiltonian fulfills stability of
second kind. The class of initial wave functions is chosen such that the
variance in energy is small. We then prove the convergence of the reduced
density matrix corresponding to the exact time evolution to the projector onto
the solution of the corresponding nonlinear Schr\"odinger equation in either
Sobolev trace norm, if the external potential is in some $L^p$ space, $p \in
]2, \infty]$, or in trace norm, for more general external potentials.

16
The categories of this paper are: stat
10
The categories of this paper are: q-bio
22
The categories of this paper are: chao-dyn
9
The categories of this paper are: nlin
24
The categories of this paper are: dg-ga
26
The categories of this paper are: bayes-an
31
The categories of this paper are: ao-sci
30
The categories of this paper are: q-alg
32
The categories of this paper are: atom-ph
23
The categories of this paper are: solv-int
5
The categories of this paper are: gr-qc
7
The categories of this paper are: hep-th
35
The categories of this paper are: mtrl-th
13
The categories of this paper are: quant-ph
12
The categories of this paper are: nucl-th
34
The categories of this paper are: chem-ph