Eigenvalues of perturbed Laplace operators on compact manifolds

We obtain upper bounds for the eigenvalues of the Schr"odinger operator $L={\mathrm{\Delta }}_g+q$ 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 ${\mathrm{\Delta }}_{\varphi }={\mathrm{\Delta }}_g+{\mathrm{\nabla }}_g\varphi \cdot {\mathrm{\nabla }}_g$ using two approaches: First, we use the fact that ${\mathrm{\Delta }}_{\varphi }$ is unitarily equivalent to a Schr"odinger operator and we get an upper bound in terms of the $L^2$-norm of ${\mathrm{\nabla }}_g\varphi$ and the min-conformal volume. Second, we use its variational characterization and we obtain upper bounds in terms of the $L^{\mathrm{\infty }}$-norm of ${\mathrm{\nabla }}_g\varphi$ 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 $\varphi$ when the Bakry-Emery Ricci curvature is non-negative.