UC Irvine

We consider discrete quasiperiodic Schrödinger operators with analytic sampling

functions. The thesis has two main themes: rst, to provide a sharp arithmetic criterion

of full spectral dimensionality for analytic quasiperiodic Schrödinger operators

in the positive Lyapunov exponent regime. Second, to provide a concrete example of

Schrödinger operator with mixed spectral types.

For the first theme, we introduce a notion of beta-almost periodicity and prove quantitative

lower spectral/quantum dynamical bounds for general bounded beta-almost periodic

potentials. Applications include the sharp arithmetic criterion in the positive

Lyapunov exponent regime and arithmetic criteria for families with zero Lyapunov

exponents, with applications to Sturmian potentials and the critical almost Mathieu

operator.

For the second part, we consider a family of one frequency discrete analytic quasiperiodic

Schrödinger operators which appear in [18]. We show that this family provides

an example of coexistence of absolutely continuous and point spectrum for some

parameters as well as coexistence of absolutely continuous and singular continuous

spectrum for some other parameters.