Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Unique continuation principle for spectral projections of Schr\" odinger
  operators and optimal Wegner estimates for non-ergodic random Schr\" odinger
  operators   We prove a unique continuation principle for spectral projections of Schr\"
odinger operators. We consider a Schr\" odinger operator $H= -\Delta + V$ on
$\mathrm{L}^2(\mathbb{R}^d)$, and let $H_{\Lambda}$ denote its restriction to a
finite box $\Lambda$ with either Dirichlet or periodic boundary condition. We
prove unique continuation estimates of the type $\chi_I (H_\Lambda) W \chi_I
(H_\Lambda) \ge \kappa\, \chi_I (H_\Lambda) $ with $\kappa >0$ for appropriate
potentials $W\ge 0$ and intervals $I$. As an application, we obtain optimal
Wegner estimates at all energies for a class of non-ergodic random Schr\"
odinger operators with alloy{-type random potentials (`crooked' Anderson
Hamiltonians). We also prove optimal Wegner estimates at the bottom of the
spectrum with the expected dependence on the disorder (the Wegner estimate
improves as the disorder increases), a new result even for the usual (ergodic)
Anderson Hamiltonian. These estimates are applied to prove localization at high
disorder for Anderson Hamiltonians in a fixed interval at the bottom of the
spectrum.

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The categories of this paper are: plasm-ph
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The categories of this paper are: hep-ph
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The categories of this paper are: ao-sci
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The categories of this paper are: patt-sol
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The categories of this paper are: econ
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The categories of this paper are: q-fin
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The categories of this paper are: chem-ph
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The categories of this paper are: solv-int
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The categories of this paper are: q-alg
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The categories of this paper are: q-bio
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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: atom-ph
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The categories of this paper are: dg-ga
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The categories of this paper are: chao-dyn
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The categories of this paper are: stat