Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Spectral and scattering theory for perturbations of the Carleman
  operator   We study spectral properties of the Carleman operator (the Hankel operator
with kernel $h_{0}(t)=t^{-1}$) and, in particular, find an explicit formula for
its resolvent. Then we consider perturbations of the Carleman operator $H_{0}$
by Hankel operators $V$ with kernels $v(t)$ decaying sufficiently rapidly as
$t\to\infty$ and not too singular at t=0. Our goal is to develop scattering
theory for the pair $H_{0}$, $H=H_{0} +V $ and to construct an expansion in
eigenfunctions of the continuous spectrum of the Hankel operator $H$. We also
prove that under general assumptions the singular continuous spectrum of the
operator $H$ is empty and that its eigenvalues may accumulate only to the edge
points 0 and $\pi$ in the spectrum of $H_{0}$. We find simple conditions for
the finiteness of the total number of eigenvalues of the operator $H$ lying
above the (continuous) spectrum of the Carleman operator $H_{0}$ and obtain an
explicit estimate of this number. The theory constructed is somewhat analogous
to the theory of one-dimensional differential operators.

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The categories of this paper are: q-bio
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The categories of this paper are: chao-dyn
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The categories of this paper are: comp-gas
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The categories of this paper are: eess
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The categories of this paper are: funct-an
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The categories of this paper are: alg-geom
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The categories of this paper are: bayes-an
0
The categories of this paper are: hep-ph
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The categories of this paper are: cs
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The categories of this paper are: q-alg
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The categories of this paper are: gr-qc
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The categories of this paper are: nlin
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The categories of this paper are: plasm-ph
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The categories of this paper are: hep-th
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The categories of this paper are: hep-ex
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The categories of this paper are: cond-mat