Eigenvalues of perturbed Laplace operators on compact manifolds

Abstract

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

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