Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The Initial Value Problem, Scattering and Inverse Scattering, for
  Non-Linear Schr\"odinger Equations with a Potential and a Non-Local
  Non-Linearity   We consider non-linear Schr\"odinger equations with a potential, and
non-local non-linearities, that are models in mesoscopic physics, for example
of a quantum capacitor, and that also are models of molecular structure. We
study in detail the initial value problem for these equations. In particular,
existence and uniqueness of local and global solutions, continuous dependence
on the initial data and regularity. We allow for a large class of unbounded
potentials. We have no restriction on the growth at infinity of the positive
part of the potential. We also construct the scattering operator in the case of
potentials that go to zero at infinity. Furthermore, we give a method for the
unique reconstruction of the potential from the small amplitude limit of the
scattering operator. In the case of the quantum capacitor, our method allows us
to uniquely reconstruct all the physical parameters from the small amplitude
limit of the scattering operator.

20
The categories of this paper are: acc-phys
16
The categories of this paper are: stat
6
The categories of this paper are: astro-ph
11
The categories of this paper are: hep-lat
26
The categories of this paper are: bayes-an
23
The categories of this paper are: solv-int
31
The categories of this paper are: ao-sci
13
The categories of this paper are: quant-ph
4
The categories of this paper are: cond-mat
3
The categories of this paper are: physics
0
The categories of this paper are: hep-ph
17
The categories of this paper are: q-fin
28
The categories of this paper are: alg-geom
9
The categories of this paper are: nlin
22
The categories of this paper are: chao-dyn
34
The categories of this paper are: chem-ph