Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Statistics of anomalously localized states at the center of band E=0 in
  the one-dimensional Anderson localization model   We consider the distribution function $P(|\psi|^{2})$ of the eigenfunction
amplitude at the center-of-band (E=0) anomaly in the one-dimensional
tight-binding chain with weak uncorrelated on-site disorder (the
one-dimensional Anderson model). The special emphasis is on the probability of
the anomalously localized states (ALS) with $|\psi|^{2}$ much larger than the
inverse typical localization length $\ell_{0}$. Using the solution to the
generating function $\Phi_{an}(u,\phi)$ found recently in our works we find the
ALS probability distribution $P(|\psi|^{2})$ at $|\psi|^{2}\ell_{0} >> 1$. As
an auxiliary preliminary step we found the asymptotic form of the generating
function $\Phi_{an}(u,\phi)$ at $u >> 1$ which can be used to compute other
statistical properties at the center-of-band anomaly. We show that at
moderately large values of $|\psi|^{2}\ell_{0}$, the probability of ALS at E=0
is smaller than at energies away from the anomaly. However, at very large
values of $|\psi|^{2}\ell_{0}$, the tendency is inverted: it is exponentially
easier to create a very strongly localized state at E=0 than at energies away
from the anomaly. We also found the leading term in the behavior of
$P(|\psi|^{2})$ at small $|\psi|^{2}<< \ell_{0}^{-1}$ and show that it is
consistent with the exponential localization corresponding to the Lyapunov
exponent found earlier by Kappus and Wegner and Derrida and Gardner.

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The categories of this paper are: cs
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The categories of this paper are: adap-org
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The categories of this paper are: nucl-th
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The categories of this paper are: ao-sci
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The categories of this paper are: nlin
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The categories of this paper are: alg-geom
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The categories of this paper are: acc-phys
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The categories of this paper are: funct-an
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The categories of this paper are: bayes-an
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The categories of this paper are: quant-ph
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The categories of this paper are: plasm-ph
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The categories of this paper are: astro-ph
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The categories of this paper are: econ
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The categories of this paper are: stat
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The categories of this paper are: q-fin
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The categories of this paper are: nucl-ex