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) $L_σ$ on a compact, connected, smooth Riemannian manifold $(M,g)$ endowed with a measure $σdv_g$. First, we obtain upper bounds for the $k-$th eigenvalue of $L_σ$ which are consistent with the power of $k$ in Weyl's formula. These bounds depend on integral norms of the density $σ$, 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 Schrödinger 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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