Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Scattering theory in homogeneous Sobolev spaces for Schr\"odinger and
  wave equations with rough potentials   We study the scattering theory for the Schr\"odinger and wave equations with
rough potentials in a scale of homogeneous Sobolev spaces. The first half of
the paper concerns with an inverse-square potential in both of subcritical and
critical constant cases, which is a particular model of scaling-critical
singular perturbations. In the subcritical case, the existence of the wave and
inverse wave operators defined on a range of homogeneous Sobolev spaces is
obtained. In particular, we have the scattering to a free solution in the
homogeneous energy space for both of the Schr\"odinger and wave equations. In
the critical case, it is shown that the solution is asymptotically a sum of a
$n$-dimensional free wave and a rescaled two-dimensional free wave. The second
half of the paper is concerned with a generalization to a class of strongly
singular decaying potentials. We provides a simple criterion in an abstract
framework to deduce the existence of wave operators defined on a homogeneous
Sobolev space from the existence of the standard ones defined on a base Hilbert
space.

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The categories of this paper are: plasm-ph
3
The categories of this paper are: physics
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The categories of this paper are: atom-ph
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The categories of this paper are: patt-sol
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The categories of this paper are: bayes-an
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The categories of this paper are: econ
5
The categories of this paper are: gr-qc
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The categories of this paper are: solv-int
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The categories of this paper are: q-fin
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The categories of this paper are: nucl-ex
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The categories of this paper are: hep-ex
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The categories of this paper are: stat
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The categories of this paper are: math-ph
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The categories of this paper are: acc-phys
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The categories of this paper are: chem-ph
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The categories of this paper are: cmp-lg