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Wencai Liu : Spectral transitions for Schr\odinger operators with decaying potentials and Laplacians on asymptotically flat (hyperbolic) manifolds (Oct 2, 2018 11:55 AM)

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.

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