Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Regularity of the density of states of Random Schr\"odinger Operators   In this paper we solve a long standing open problem for Random Schr\"odinger
operators on $L^2(\mathbb{R}^d)$ with i.i.d single site random potentials. We
allow a large class of free operators, including magnetic potential, however
our method of proof works only for the case when the random potentials satisfy
a complete covering condition. We require that the supports of the random
potentials cover $\mathbb{R}^d$ and the bump functions that appear in the
random potentials form a partition of unity. For such models, we show that the
Density of States (DOS) is $m$ times differentiable in the part of the spectrum
where exponential localization is valid, if the single site distribution has
compact support and has H\"older continuous $m+1$ st derivative. The required
H\"older continuity depends on the fractional moment bounds satisfied by
appropriate operator kernels. Our proof of the Random Schr\"odinger operator
case is an extensions of our proof for Anderson type models on
$\ell^2(\mathbb{G})$, $\mathbb{G}$ a countable set, with the property that the
cardinality of the set of points at distance $N$ from any fixed point grows at
some rate in $N^\alpha, \alpha >0$. This condition rules out the Bethe lattice,
where our method of proof works but the degree of smoothness also depends on
the localization length, a result we do not present here. Even for these models
the random potentials need to satisfy a complete covering condition. The
Anderson model on the lattice for which regularity results were known earlier
also satisfies the complete covering condition.

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The categories of this paper are: hep-ex
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The categories of this paper are: eess
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The categories of this paper are: dg-ga
27
The categories of this paper are: comp-gas
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The categories of this paper are: supr-con
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The categories of this paper are: cmp-lg
15
The categories of this paper are: nucl-ex
30
The categories of this paper are: q-alg
34
The categories of this paper are: chem-ph
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The categories of this paper are: stat
33
The categories of this paper are: plasm-ph
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The categories of this paper are: adap-org
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The categories of this paper are: astro-ph
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The categories of this paper are: funct-an
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The categories of this paper are: nlin
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The categories of this paper are: nucl-th