# Spectral transitions for Schr\"odinger operators with decaying potentials and Laplacians on asymptotically flat (hyperbolic) manifolds

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We apply piecewise constructions and gluing techniques to construct

asymptotically flat (hyperbolic) manifolds such that associated

Laplacians have dense embedded eigenvalues or singular continuous

spectra. The method also allows us to provide various examples of

operators with embedded singular spectra. With additional perturbation theory, several sharp

spectral transitions (even criteria) for singular spectra are obtained.

In this talk, I will focus on two models-Laplacians on manifolds and Stark operators.