Spectral Analysis Of Quasiperiodic Schr(?)dinger Operators

We consider discrete quasiperiodic Schrodinger operators with analytic sampling functions.The thesis has three main themes:first,to provide a sharp arithmetic cri-terion of full spectral dimensionality for analytic quasiperiodic Schrodinger operators in the positive Lyapunov exponent regime.Second,to study the Holder continuity of the Lyapunov exponent for weakly Liouville frequencies.Last,to provide a concrete example of Schodinger operator with mixed spectral types.For the first theme,we introduce a notion of β-almost periodicity and prove quan-titative lower spectral/quantum dynamical bounds for general bounded β-almost pe-riodic 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 obtain Holder continuity of the Lyapunov exponent for weakly Liouville frequencies,a result previously available only under the strong Dio-phantine condition in Goldstein-Schlag,[52].The extension to all irrationals is impos-sible according to[6],so the result is in some sense close to what one should expect in terms of the requirements on the arithmetics of the frequency.For the last part,we consider a family of one frequency discrete analytic quasi-periodic Schrodinger operators which appear in[23].We show that this family pro-vides 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.