Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Weyl sums and the Lyapunov exponent for the skew-shift Schr\"odinger
  cocycle   We study the one-dimensional discrete Schr\"odinger operator with the
skew-shift potential $2\lambda\cos\left(2\pi \left(\binom{j}{2}
\omega+jy+x\right)\right)$. This potential is long conjectured to behave like a
random one, i.e., it is expected to produce Anderson localization for
arbitrarily small coupling constants $\lambda>0$. In this paper, we introduce a
novel perturbative approach for studying the zero-energy Lyapunov exponent
$L(\lambda)$ at small $\lambda$. Our main results establish that, to second
order in perturbation theory, a natural upper bound on $L(\lambda)$ is fully
consistent with $L(\lambda)$ being positive and satisfying the usual
Figotin-Pastur type asymptotics $L(\lambda)\sim C\lambda^2$ as $\lambda\to 0$.
The analogous quantity behaves completely differently in the Almost-Mathieu
model, whose zero-energy Lyapunov exponent vanishes for $\lambda<1$. The main
technical work consists in establishing good lower bounds on the exponential
sums (quadratic Weyl sums) that appear in our perturbation series.

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The categories of this paper are: dg-ga
33
The categories of this paper are: plasm-ph
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The categories of this paper are: nucl-th
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The categories of this paper are: hep-lat
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The categories of this paper are: nlin
10
The categories of this paper are: q-bio
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The categories of this paper are: patt-sol
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The categories of this paper are: chao-dyn
1
The categories of this paper are: cs
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The categories of this paper are: alg-geom
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The categories of this paper are: supr-con
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The categories of this paper are: quant-ph
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The categories of this paper are: acc-phys
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The categories of this paper are: physics
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The categories of this paper are: stat
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The categories of this paper are: comp-gas