Abstract of Proposed Research Svetlana Jitomirskaya
This project is to study spectra and localization type effects for ergodic Schroedinger operators and the study of anomalous absolutely continuous spectrum and anomalous quantum transport in the quasiperiodic and Anderson model-type settings. It is also planned to study several models related to Bloch electrons in constant and/or random magnetic fields. The project involves the continuing development of non-perturbative methods both for the proofs of localization and other related properties as well as for the study of absolutely continuous spectrum. Other important objectives are the study of issues related to Cantor spectra of quasiperiodic operators, it's occurrence, prevalence, and scaling properties, and the development of smooth (rather than analytic) methods.
The proposed research investigates the anomalous spectral and diffusive properties of quasiperiodic and other deterministic and random structures. This is basic research on the fundamental properties of disordered systems that serve as models of systems with impurities. Quasiperiodic operators provide central or important models for integer quantum Hall effect, experimental quasicrystals, and quantum chaos theory. The development of the rigorous theory is expected to contribute to the understanding of all three phenomena, and in particular, may lead to finding new materials with desired physical properties. Disordered systems are also used in modeling many other micro and macro effects: from quantum localization to earthquakes. The proposed topics include studying properties of both highly and weakly disordered systems of Quantum Mechanics that demonstrate certain anomalous behavior.
The intellectual merit of this project consists in solving several open problems in analysis motivated by questions in solid state physics and development of corresponding theories. Our work on quasiperiodic operators included the final solution of the Ten Martini problem, the solution of Dry Ten Martini problem for the Diophantine case, proof of phase stability of absence of singular continuous spectrum, proof of optimal H Ì~Holder continuity of the density of states in the non-perturbative regime, full analysis of the Lyapunov exponents of the extended Harperâ~@~Ys model (including the self-dual region), and the first (and optimal) results on the continuity modulus of individual absolutely continuous spectral measures. It also includes the full non-perturbative version of Eliassonâ~@~Ys reducibility theory in the one- frequency case, and results on continuity of spectra and Lyapunov exponents. The work on quantum transport included a new method to prove upper bounds on quantum transport and exact expressions for diffusion exponents in the random polymer models, where dynamical delocalization (superdiffusive transport) coexists with pure point spectrum. The work on mul- tidimensional lattice models included a construction of a family of counterexamples showing that no form of unique continuation principle can hold in the discrete setting. Further results on spectral theory include work on various aspects of singular spectra. Another line of research by members of our group included development of numerical methods to circumvent the problem of broken ergodicity in Monte Carlo simulations, and study of nonlinear resonant interactions between entangled photons and matter leading to quantum pathway selection. Our focus has been both on developing new methods and theories in this branch of analy- sis/dynamical systems and on applications to models that are of interest to the physics community. As such, we obtained new results and proved open conjectures on the almost Mathieu (also called Harperâ~@~Ys) operator, on the operators describing tight-binding model of a two dimensional electron in a magnetic field, with general lattice geometry, and on the random polymer models. The project contributed to training of two postdoctoral fellows and four graduate students.
