Eigenvalues of perturbed Laplace operators on compact manifolds

TL;DR

This paper derives upper bounds for eigenvalues of perturbed Laplace operators on compact manifolds, relating them to integral quantities of potentials and conformal invariants, with applications to weighted Laplacians.

Contribution

It introduces new eigenvalue bounds for Schr"odinger and Bakry-Emery Laplacians using conformal invariants and variational methods, extending classical spectral estimates.

Findings

Upper bounds depend on integral quantities of potential q

Bounds are compatible with eigenvalue asymptotics

New bounds for weighted Laplacians with non-negative Bakry-Emery Ricci curvature

Abstract

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.