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.

In this thesis we study discrete quasiperiodic Jacobi operators as well as ergodic operators driven by more general zero topological entropy dynamics. Such operators are deeply connected to physics (quantum Hall effect and graphene) and have enjoyed great attention from mathematics (e.g. several of Simon’s problems). The thesis has two main themes. First, to study spectral properties of quasiperiodic Jacobi matrices, in particular when off-diagonal sampling function has non-zero winding number or singularities. Second, to address the consequences of positive Lyapunov exponent for Schr\"odinger operators with a class of potentials of bounded discrepancy, prime example being those driven by shifts and skew-shifts on multi-dimensional tori.

Within the first theme, one of our results provides an if and only if topological criterion for obtaining localization from reducibility of the dual Jacobi cocycles. As an application of this result to the extended Harper’s model, we obtain sharp arithmetic spectral transition in the positive Lyapunov exponent regime. Two other results about the extended Harper’s model include a proof of non-degeneracy of all possible spectral gaps (known as Dry Ten Martini Problem) for the non-self-dual regions, and an arithmetic result on purely continuous spectrum for the self-dual region that is optimal and improves on a recent work by Avila-Jitomirskaya-Marx, who proved a measure-theoretic version.

The most important contribution among the second group is a general localization- type result for ergodic potentials of bounded discrepancy. As concrete applications of our general result, we build the first arithmetic localization-type results for potentials defined by shifts and (the first non-perturbative ones for) skew-shifts of higher- dimensional tori. Similar consideration also leads to the continuity of spectral data, for Schr\"odinger operators with such underlying dynamics in the positive Lyapunov exponent regime.

In this thesis we will prove various types of localization for some classes of one-dimensionalrandom Schrodinger operators. The central theme for all models considered will be singularity. Here, we use the term singularity mainly to refer to the possible lack of continuity in the probability distribution governing the randomness of the potential terms; although, we also deal with the other notion of singularity: that of Jacobi matrices and its counterpart, the unboundedness of the potential. In particular, we will prove spectral localization for unbounded one dimensional random Jacobi operators. Such operators are obtained by incorporating independent and identically distributed randomness into the off-diagonal terms of the standard Anderson model. The operators exhibit spectral localization if almost surely the spectrum is pure point and all of the eigenfunctions decay exponentially. We also consider so-called random word models. These generalizations of random Schrodinger operators have potential terms given by (row) vectors of bounded but random length which permits consideration of local correlations within the potential. These operators sometimes have a finite set of critical energies where the rate of localization tends to zero as one approaches these energies. Nevertheless, we will prove that these operators exhibit exponential dynamical localization in expectation on compact sets not containing any critical energies.

In this thesis, we will study the conductance of several models originating from condensed matter physics, including the Anderson model for random systems and the Bistritzer-MacDonald (BM) model for twisted bilayer graphene (TBG). In fact, in their time, both models exhibited unexpected conductance properties which bewildered mathematicians and physicists. The Anderson model, developed in 1958 by physicist P.W. Anderson, exhibited unexpected localization/insulating phenomena in the 1D and 2D cases, while TBG was discovered experimentally in 2018 \cite{C18} to have unconventional superconductance at certain ``magic angles'' with relatively flat bands. This thesis has two primary parts. In the first part, we prove different types of localization results in the Anderson model and other related models. In the second part, we study the BM model in various magnetic fields from spectral, semi-classical and physical perspectives; in particular, we focus on the existence and persistence of flat bands which, though mysterious, is believed to be related to the superconductance of TBG \cite{LKV21}. More specifically, for the first part, we initially provide a short non-perturbative proof of Anderson localization and dynamical localization for the 1D Anderson model with arbitrary disorder (e.g. including Bernoulli potential). After that, we derive the dynamical localization in expectation in a related random CMV model with arbitrary disorder. Finally, we work with 2D Anderson model with Bernoulli potential and prove strong dynamical localization in expectation in this setting.

We start the second part by first discussing the influence of different magnetic and electric potentials on the existence/persistence of flat bands for TBG. After the general discussion, we divert our attention the strong constant magnetic fields and provide the explicit asymptotic expansion of the density of states (DOS). In particular, we point out the intrinsically different roles that chiral and anti-chiral potentials play in the magnetic response of TBG. Finally, from the expansion of the DOS, we are able to study the physical phenomena, including magnetic oscillations and quantum Hall effect of the TBG. We find that the chiral potential enhances these phenomena, while the anti-chiral potential diminishes them.

We consider quasiperiodic Jacobi and Schr\"odinger operators of both a single- and multi-frequency. These operators appear very naturally in condensed matter physics, where they have seen applications in the study of Graphene and the Quantum Hall effect. The prototypical example of the single-frequency operator is the almost Mathieu operator (AMO). While much is known about the AMO, less may be said about the multifrequency analogues and even some perturbed single-frequency models. This thesis has one recurring theme: properties of the Lyapunov exponent (LE); moreover, this thesis may be split into two parts. First, we explore continuity of the LE for multifrequency analytic quasiperiodic cocycles and positivity of the LE for single-frequency analytic quasiperiodic Schr\"odinger operators with an additional background potential. We then derive upper bounds in quantum dynamics as a consequence of LE regularity, and explore the fractal properties of spectral measures. In addressing the first part, we prove joint continuity of the LE for non-identically singular multifrequency analytic quasiperiodic cocycles in both cocycle and frequency, and we prove that the (lower) LE for single frequency analytic quasiperiodic Schr\"odinger operators with added background potential can be made uniformly positive by taking a sufficiently large coupling constant independent of the background. The former is accomplished by adapting an inductive argument of Bourgain which was originally used to derive similar results for $SL(2,\C)$-cocycles. The latter involves complexification in phase of the associated transfer matrix and appealing to various properties of analytic and subharmonic functions. In the second part, we first derive a lite version of dynamical localization: under suitable assumptions on the frequency and LE, (time-averaged) moments of the position operator grow no faster than a power of the logarithm. The main achievement here is that our notion is stable under perturbations and holds for all values of the phase and an arithmetically defined set of frequencies of full measure. We then extend the Jitomirskaya-Last power-law subordinacy theory and Last theory of quantum dynamics to encompass a more general version of Hausdorff dimension. This allows us to study fine dimensional properties of spectral measures, particularly `zero-dimensional' measures.