Wencai Liu

Arithmetic Spectral Transitions for the Maryland Model

We give a precise description of spectra of the Maryland model

Anderson localization for the almost Mathieu operator in the exponential regime

(h_{\lambda,\alpha,\theta}u)_n=u_{n+1}+u_{n-1}+ \lambda \tan \pi(\theta+n\alpha)u_n

Spectral gaps of almost Mathieu operators in the exponential regime

for all values of parameters. We introduce an arithmetically defined index

Anderson localization for the completely resonant phases

\delta (\alpha, \theta)

Continuous quasiperiodic Schrödinger operators with Gordon type potentials

and show that for

A Lower Bound on the Lyapunov Exponent for the Generalized Harper’s Model

\alpha\notin\mathbb{Q},\,

Hölder continuity of the spectral measures for one-dimensional Schrödinger operator in exponential regime

\sigma_{sc}(h_{\lambda,\alpha,\theta})=\overline{\{e:\gamma_{\lambda}(e) <\delta (\alpha, \theta) \}}

Universal hierarchical structure of quasiperiodic eigenfunctions

and

Noncompact complete Riemannian manifolds with dense eigenvalues embedded in the essential spectrum of the Laplacian

\sigma_{pp}(h_{\lambda,\alpha,\theta})=\{e:\gamma_{\lambda}(e) \geq \delta (\alpha, \theta) \}