Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Scattering theory for 3d cubic inhomogeneous NLS with inverse square
  potential   In this paper, we study the scattering theory for the cubic inhomogeneous
Schr\"odinger equations with inverse square potential $iu_t+\Delta
u-\frac{a}{|x|^2}u=\lambda |x|^{-b}|u|^2u$ with $a>-\frac14$ and $0<b<1$ in
dimension three. In the defocusing case (i.e. $\lambda=1$), we establish the
global well-posedness and scattering for any initial data in the energy space
$H^1_a(\mathbb R^3)$. While for the focusing case(i.e. $\lambda=-1$), we obtain
the scattering for the initial data below the threshold of the ground state, by
making use of the virial/Morawetz argument as in Dodson-Murphy [Proc. Amer.
Math. Soc.,145(2017), 4859-4867.] and Campos-Cardoso [arXiv: 2101.08770v1.]
that avoids the use of interaction Morawetz estimate.

17
The categories of this paper are: q-fin
15
The categories of this paper are: nucl-ex
31
The categories of this paper are: ao-sci
5
The categories of this paper are: gr-qc
13
The categories of this paper are: quant-ph
30
The categories of this paper are: q-alg
14
The categories of this paper are: math-ph
32
The categories of this paper are: atom-ph
4
The categories of this paper are: cond-mat
3
The categories of this paper are: physics
35
The categories of this paper are: mtrl-th
23
The categories of this paper are: solv-int
7
The categories of this paper are: hep-th
20
The categories of this paper are: acc-phys
21
The categories of this paper are: adap-org
16
The categories of this paper are: stat