Advisor: Svetlana Jitomirskaya
Thesis Abstract:
We consider quasi-periodic Jacobi operators with analytic sampling functions. Recent applications in physics of such operators are Graphene or the Quantum Hall effect. Even though much is known for their prototype, the almost Mathieu operator (AMO), not much can be said for models beyond that. The thesis has two main themes: Firstly, to provide a complete understanding of extended Harper's model (EHM), a natural generalization of the AMO proposed by D. J. Thouless, which so far has presented an open problem even from the physics point of view. Secondly, to address aspects of the spectral theory of general quasi-periodic, analytic Jacobi operators: continuity of the Lyapunov exponent, and continuity of spectral properties upon rational frequency approximation. As a result of our investigations, we provide a complete description of the model's ``metal-insulator phase diagram,'' as given by the Lyapunov exponent (LE). The main achievement was to develop a non-duality based approach, able to tackle the self-dual regime in parameter space. The latter is accomplished by developing Avila's ``global theory'' for analytic Jacobi-cocycles. Based on phase-complexification, the spectrum is partitioned into super-critical, sub-critical and critical regime. This provides a refined description of the set of zero LE, going beyond what is known from classical Kotani theory. For EHM, we identify the three regimes, for all values of the coupling and all irrational frequencies. Referring to the second theme of the thesis, we prove continuous dependence of the LE on the cocycle within the analytic category. The main achievement, is that our result allows for cocycles which are not everywhere invertible, a situation that naturally arises for Jacobi operators. Finally, we show how to recover the spectral properties of a quasi-periodic operator from rational approximation of the frequency. Up to sets of zero Lebesgue measure, the absolutely continuous spectrum can be obtained asymptotically from the ``intersection spectrum'' of the periodic operators associated with the continued fraction expansion of the frequency. This proves a conjecture of Y. Last and has not even been known for the AMO. Similarly, from the asymptotics of the ``union spectrum'', one recovers the spectrum.
