We obtain upper bounds for
the eigenvalues of the Schrödinger operator L = Δg+ q depending on integral
quantities of the potential q and a conformal invariant called the min-conformalvolume. When the Schrödinger operator L is positive, integral quantities of q
appearing 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–Émery Laplacian Δϕ= Δg+ ∇gϕ ⋅∇g using two
approaches: first, we use the fact that Δϕ is unitarily equivalent to a Schrödinger
operator and we get an upper bound in terms of the L2-norm of ∇gϕ and the
min-conformal volume; second, we use its variational characterization and we obtain
upper bounds in terms of the L∞-norm of ∇gϕ and a new conformal invariant. The
second approach leads to a Buser type upper bound and also gives upper
bounds that do not depend on ϕ when the Bakry–Émery Ricci curvature is
nonnegative.
