Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Maximum Decay Rate for the Nonlinear Schr\"odinger Equation   In this paper, we consider global solutions for the following nonlinear
Schr\"odinger equation $iu_t+\Delta u+\lambda|u|^\alpha u=0,$ in $\R^N,$ with
$\lambda\in\R$ and $0\le\alpha<\frac{4}{N-2}$ $(0\le\alpha<\infty$ if $N=1).$
We show that no nontrivial solution can decay faster than the solutions of the
free Schr\"odinger equation, provided that $u(0)$ lies in the weighted Sobolev
space $H^1(\R^N)\cap L^2(|x|^2;dx),$ in the energy space, namely $H^1(\R^N),$
or in $L^2(\R^N),$ according to the different cases.

13
The categories of this paper are: quant-ph
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The categories of this paper are: alg-geom
37
The categories of this paper are: supr-con
32
The categories of this paper are: atom-ph
5
The categories of this paper are: gr-qc
25
The categories of this paper are: patt-sol
35
The categories of this paper are: mtrl-th
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The categories of this paper are: hep-lat
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The categories of this paper are: dg-ga
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The categories of this paper are: physics
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The categories of this paper are: nlin
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The categories of this paper are: solv-int
0
The categories of this paper are: hep-ph
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The categories of this paper are: bayes-an
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The categories of this paper are: adap-org
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The categories of this paper are: ao-sci