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.

arXiv commentReferences updated, GAFA to appear

References in corpus (7)

• Criteria for embedded eigenvalues for discrete Schrödinger operators
• Criteria for eigenvalues embedded into the absolutely continuous spectrum of perturbed Stark type operators
• Sharp spectral transition for eigenvalues embedded into the spectral bands of perturbed periodic operators
• Limiting absorption principle on manifolds having ends with various measure growth rate limits
• The radial curvature of an end that makes eigenvalues vanish in the essential spectrum II
• Growth of the eigensolutions of Laplacians on Riemannian manifolds I: construction of energy function
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Cited by in corpus (7)

• Criteria for embedded eigenvalues for discrete Schrödinger operators
• Topics on Fermi varieties of discrete periodic Schrödinger operators
• Sharp spectral transition for eigenvalues embedded into the spectral bands of perturbed periodic operators
• Noncompact complete Riemannian manifolds with singular continuous spectrum embedded into the essential spectrum of the Laplacian, I. The hyperbolic case
• Growth of the eigensolutions of Laplacians on Riemannian manifolds I: construction of energy function
• Sharp bounds for finitely many embedded eigenvalues of perturbed Stark type operators
• Manifolds with cylindrical ends having a finite and positive number of embedded eigenvalues