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.