Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Quantum scattering at low energies   For a class of negative slowly decaying potentials, including
$V(x):=-\gamma|x|^{-\mu}$ with $0<\mu<2$, we study the quantum mechanical
scattering theory in the low-energy regime. Using modifiers of the
Isozaki-Kitada type we show that scattering theory is well behaved on the whole
continuous spectrum of the Hamiltonian, including the energy 0. We show that
the S-matrices are well-defined and strongly continuous down to the zero energy
threshold. Similarly, we prove that the wave matrices and generalized
eigenfunctions are norm continuous down to the zero energy if we use
appropriate weighted spaces. These results are used to derive (oscillatory)
asymptotics of the standard short-range and Dollard type S-matrices for the
subclasses of potentials where both kinds of S-matrices are defined. For
potentials whose leading part is $-\gamma|x|^{-\mu}$ we show that the location
of singularities of the kernel of $S(\lambda)$ experiences an abrupt change
from passing from positive energies $\lambda$ to the limiting energy
$\lambda=0$. This change corresponds to the behaviour of the classical orbits.
Under stronger conditions we extract the leading term of the asymptotics of the
kernel of $S(\lambda)$ at its singularities; this leading term defines a
Fourier integral operator in the sense of H\"ormander.

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The categories of this paper are: cond-mat
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The categories of this paper are: chao-dyn
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The categories of this paper are: patt-sol
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The categories of this paper are: q-fin
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The categories of this paper are: q-bio
32
The categories of this paper are: atom-ph
12
The categories of this paper are: nucl-th
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The categories of this paper are: nlin
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The categories of this paper are: hep-ex
29
The categories of this paper are: funct-an
18
The categories of this paper are: econ
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The categories of this paper are: cmp-lg
34
The categories of this paper are: chem-ph
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The categories of this paper are: nucl-ex
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The categories of this paper are: dg-ga
31
The categories of this paper are: ao-sci