Noncompact complete Riemannian manifolds with dense eigenvalues embedded in the essential spectrum of the Laplacian

Abstract

We prove sharp criteria on the behavior of radial curvature for the existence of asymptotically flat or hyperbolic Riemannian manifolds with prescribed sets of eigenvalues embedded in the spectrum of the Laplacian. In particular, we construct such manifolds with dense embedded point spectrum and sharp curvature bounds.

Growth of the Eigensolutions of Laplacians on Riemannian Manifolds I: Construction of Energy Function

“…Putting (13), (14), (15) together and using (11), we conclude that Let 0 < t < 1 be small enough, 2ρ ′ = a 4 + a5 r and q 1 = 0. Direct computation of (9) implies that Proof.…”

Growth of the Eigensolutions of Laplacians on Riemannian Manifolds II: Positivity of the Initial Energy

“…He excludes eigenvalues greater than (n−1) 2 4 under the assumption that K rad (r) = −1 + o(r −1 ), and also constructs a manifold for which exact an eigenvalue (n−1) 2 4 + 1 is embedded into its essential spectrum [ (n−1) 2 4 , ∞) with the radial curvature K rad (r) = −1 + O(r −1 ). Jitomirskaya and Liu constructed examples which show that dense eigenvalues and singular continuous spectrum can be embedded into the essential spectrum [14,15].…”

Criteria for eigenvalues embedded into the absolutely continuous spectrum of perturbed Stark type operators

“…Proof of Theorem 1.9. Once we have Proposition 8.1, we can prove Theorem 1.9 by the arguments in [17,33]. We omit the details here.…”

“…Recently, by the combination of Prüfer transformation (generalized Prüfer transformation) and piecewise potentials, Jitomirskaya-Liu and Liu-Ong constructed asymptotically flat (hyperbolic) manifolds, perturbed periodic operators and perturbed Jacobi operators with finitely or countable many embedded eigenvalues [17,30,33]. Here, we develop the piecewise potential technics in [17,30,33] in several aspects. Firstly, we gave the universal constructions in an effective way.…”