Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The perturbational stability of the Schr$\ddot{o}$dinger equation   By using the Wigner transform, it is shown that the nonlinear
Schr$\ddot{\textmd{o}}$dinger equation can be described, in phase space, by a
kinetic theory similar to the Vlasov equation which is used for describing a
classical collisionless plasma. In this paper we mainly show Landau damping in
the quantum sense, namely,quantum Landau damping exists for the Wigner-Poisson
system. At the same time, we also prove the existence and the stability of the
nonlinear Schr$\ddot{\textmd{o}}$dinger equation under the quantum stability
assumption.

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The categories of this paper are: nucl-th
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The categories of this paper are: chem-ph
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The categories of this paper are: patt-sol
27
The categories of this paper are: comp-gas
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The categories of this paper are: quant-ph
33
The categories of this paper are: plasm-ph
29
The categories of this paper are: funct-an
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The categories of this paper are: atom-ph
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The categories of this paper are: acc-phys
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The categories of this paper are: math-ph
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The categories of this paper are: ao-sci
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The categories of this paper are: eess
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The categories of this paper are: hep-th
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The categories of this paper are: alg-geom
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The categories of this paper are: adap-org
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The categories of this paper are: q-alg