Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Eigenvalues of the Laplacian on a compact manifold with density   In this paper, we study the spectrum of the weighted Laplacian (also called
Bakry-Emery or Witten Laplacian) $L_\sigma$ on a compact, connected, smooth
Riemannian manifold $(M,g)$ endowed with a measure $\sigma dv_g$. First, we
obtain upper bounds for the $k-$th eigenvalue of $L_{\sigma}$ which are
consistent with the power of $k$ in Weyl's formula. These bounds depend on
integral norms of the density $\sigma$, and in the second part of the article,
we give examples showing that this dependence is, in some sense, sharp. As a
corollary, we get bounds for the eigenvalues of Laplace type operators, such as
the Schr\"{o}dinger operator or the Hodge Laplacian on $p-$forms. In the
special case of the weighted Laplacian on the sphere, we get a sharp inequality
for the first nonzero eigenvalue which extends Hersch's inequality.

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The categories of this paper are: nlin
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The categories of this paper are: supr-con
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The categories of this paper are: chao-dyn
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The categories of this paper are: hep-ex
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The categories of this paper are: hep-th
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The categories of this paper are: comp-gas
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The categories of this paper are: cs
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The categories of this paper are: acc-phys
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The categories of this paper are: nucl-th
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The categories of this paper are: cmp-lg
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The categories of this paper are: nucl-ex
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The categories of this paper are: adap-org
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The categories of this paper are: gr-qc
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The categories of this paper are: q-fin
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The categories of this paper are: atom-ph
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The categories of this paper are: cond-mat