Kumura showed that there are no eigenvalues embedded in the essential
spectrum of the Laplacian on $n$-dimensional noncompact
complete Riemannian manifold $(M_n, g)$, if the radial curvature $K_{\rm
rad}+1=o(r^{-1})$ as $r$ goes to infinity.

Given any finite/countable set of positive energies $\{\lambda_n\}$, we
can
construct a Riemannian manifold with the decay order
$K_{\rm rad}+1=O(r^{-1})$/$K_{\rm rad}+1=\frac{C(r)}{r}$, where $C(r)\geq
0$ and $C(r) $ goes to infinity arbitrarily slowly, such that the
eigenvalues $\{\frac{(n-1)^2}{4}+\lambda_n\}$ are embedded in the
essential
spectrum $\sigma_{{\rm ess}}(-\Delta_g)=\left[\frac{(n-1)^2}{4},\infty
\right)$.

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