Eigenvalues of the Laplacian on a compact manifold with density

In this paper, we study the spectrum of the weighted Laplacian (also called Bakry-Émery or Witten Laplacian) Lσon a compact, connected, smooth Riemannian manifold (M,g) endowed with a measure σdvg. First, we obtain upper bounds for the k-th eigenvalue of Lσ 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.