Eigenvalues of perturbed Laplace operators on compact manifolds

We obtain upper bounds for the eigenvalues of the Schr\"odinger operator $L=\Delta_g+q$ depending on integral quantities of the potential $q$ and a conformal invariant called the min-conformal volume. Moreover, when the Schr\"odinger operator $L$ is positive, integral quantities of $q$ which appear in upper bounds, can be replaced by the mean value of the potential $q$. The upper bounds we obtain are compatible with the asymptotic behavior of the eigenvalues. We also obtain upper bounds for the eigenvalues of the weighted Laplacian or the Bakry-Emery Laplacian $\Delta_\phi=\Delta_g+\nabla_g\phi\cdot\nabla_g$ using two approaches: First, we use the fact that $\Delta_\phi$ is unitarily equivalent to a Schr\"odinger operator and we get an upper bound in terms of the $L^2$-norm of $\nabla_g\phi$ and the min-conformal volume. Second, we use its variational characterization and we obtain upper bounds in terms of the $L^\infty$-norm of $\nabla_g\phi$ and a new conformal invariant. The second approach leads to a Buser type upper bound and also gives upper bounds which do not depend on $\phi$ when the Bakry-Emery Ricci curvature is non-negative.