We study the eigenvalues or singular continuous spectrum of the free Laplacian embedded in the essential spectrum(absolutely continuous spectrum) on either asymptotically flat or asymptotically hyperbolic manifolds. The essential spectrum is $[/frac{c^2}{4},/infty]$ if $/Delta r /to c$ as $r$ goes to infinity, where $r(x)$ is the distance function. Kumura proved that there are no eigenvalues embedded in the essential spectrum $/sigma_{{/rm ess}}(-/Delta)=/left[/frac{1}{4}(n-1)^2,/infty/right)$ of Laplacians on asymptotically hyperbolic manifolds, where asymptotic  hyperbolicity is characterized by the radial curvature, i.e., $K_{/rm rad}=-1+o(r^{-1})$. He also constructed a manifold for which an eigenvalue $/frac{(n-1)^2}{4} + 1$ is embedded  into its essential spectrum $[ /frac{(n-1)^2}{4} , /infty )$ with the radial curvature $K_{/rm rad}(r)  = -1+O(r^{-1})$.

The first part of the talk, based on a joint work with S.Jitomirskaya, is devoted to construction of manifolds with embedded eigenvaluesand singular continous spectrum. Given any finite (countable)  positive energies $/{/lambda_n/}/in [/frac{K_0}{4}(n-1)^2,/infty)$, we construct     Riemannian manifolds  with the decay of order $K_{/rm rad}+K_0=O(r^{-1})$   with $K_0/geq 0$ ($K_{/rm rad}+K_0=/frac{C(r)}{r}$, where $C(r)/geq 0$ and $C(r)/to /infty $ arbitrarily slowly) such that  the eigenvalues $/{/lambda_n/}$  are embedded  in the essential spectrum  $/sigma_{{/rm ess}}(-/Delta)=/left[/frac{K_0}{4}(n-1)^2,/infty/right)$.

We also construct Riemannian manifolds with the decay of order $K{/rm rad}+K_0=/frac{C(r)}{r}$, where $C(r)/geq 0$ and $C(r)/to /infty $ arbitrarily slowly such that there is singular continous spectrum    embedded in the essential spectrum $/sigma_{{/rm ess}}(-/Delta)=/left[/frac{K_0}{4}(n-1)^2,/infty/right)$.

In the second part, I discuss criteria for the absence of eigenvalues embedded into essential spectrum in terms of the asymptotic behavior of $/Delta r$ . Under a weaker  convexity o.

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