Spectral transitions for Schr\"odinger operators with decaying potentials and Laplacians on asymptotically flat (hyperbolic) manifolds
We apply piecewise constructions and gluing technics 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, including perturbed periodic
operators, periodic Jacobi operators, and Stark operators.
We establish the asymptotic behavior (WKB for example) of eigensolutions
under small perturbations, which implies certain
rules for the absence of singular spectra. As a result, several sharp
spectral transitions (even criteria) for a single (finitely many or
countably many) embedded eigenvalues, singular continuous spectra and
essential supports of spectral measures are obtained.
The talk is based on several papers, some joint with Jitomirskaya and Ong.