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Although concrete operators with singular continuous spectrum have proliferated re-, we still don't really understand much about singular continuous spectrum. In part, this is because it is normally defined by what it isn't — neither pure point nor absolutely continuous. An important point of view, going back in part to Rodgers and Taylor [27,28], and… (More)

- S. Jitomirskaya, B. Simon
- 1994

We prove that one-dimensional Schrr odinger operators with even almost periodic potential have no point spectrum for a dense G in the hull. This implies purely singular continuous spectrum for the almost Mathieu equation for coupling larger than 2 and a dense G in even if the frequency is an irrational with good Diophantine properties. x1. Introduction This… (More)

A random polymer model is a one-dimensional Jacobi matrix randomly composed of two finite building blocks. If the two associated transfer matrices commute, the corresponding energy is called critical. Such critical energies appear in physical models, an example being the widely studied random dimer model. It is proven that the Lyapunov exponent vanishes… (More)

We prove the conjecture (known as the " Ten Martini Problem " after Kac and Simon) that the spectrum of the almost Mathieu operator is a Cantor set for all nonzero values of the coupling and all irrational frequencies.

- Mihai Stoiciu, David Damanik, +7 authors Sara Mena
- 2005

ii c 2005 Mihai Stoiciu All Rights Reserved iii Acknowledgements I would like to express my deepest gratitude to my advisor, Professor Barry Simon, for his help and guidance during my graduate studies at Caltech. During the time I worked under his supervision, Professor Simon provided a highly motivating and challenging scientific environment, which was… (More)

We study the almost Mathieu operator (h λ,α,θ u)(n) = u(n + 1) + u(n − 1) + λ cos(παn + θ)u(n) on 2 (Z), and prove that the dual of point spectrum is absolutely continuous spectrum. We use this to show that for λ = 2 it has purely singular continuous spectrum for a.e. pairs (α, θ). The α's for which we prove this are explicit. Our main goal in this paper is… (More)

- J Bourgain, S Jitomirskaya
- 2007

We examine various issues relevant to localization in the Anderson model. We show there is more to localization than exponentially localized states by presenting an example with such states but where x(t) 2 /t 2−δ is unbounded for any δ > 0. We show that the recently discovered instability of localization under rank one perturbations is only a weak… (More)

We prove that for any > 2 and a.e. !; the point spectrum of the almost Mathieu operator (H())) n = n?1 + n+1 + cos(2(+ n!))) n contains the essential closure ess of the spectrum. Corresponding eigenfunctions decay exponentially. The singular continuous component, if it exists, is concentrated on a zero measure set which is nowhere dense in ess :

We prove that for any diophantine rotation angle ! and a.e. phase the almost Mathieu operator (H())) n = n?1 + n+1 + cos(2(+ n!))) n has pure point spectrum with exponentially decaying eigenfunctions for 15: We also prove the existence of some pure point spectrum for any 5:4: