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Although concrete operators with singular continuous spectrum have proliferated recently [7,11,13,17,34,35,37,39], 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… (More)

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

We consider polynomials on the unit circle defined by the recurrence relation Φk+1(z) = zΦk(z)− αkΦk(z) k ≥ 0, Φ0 = 1 For each n we take α0, α1, . . . , αn−2 to be independent identically distributed random variables uniformly distributed in a disk of radius r < 1 and αn−1 to be another random variable independent of the previous ones and distributed… (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)

q2 a P--E~ I --+ 0 (1.2) qn I as n ~ c ~ . Then for a.e. 8, hA=2,a,o has purely singular continuous spectrum. Remarks. (1) (1.2) is used because for such a, Last [22] has proven that the spectrum, a~,~, of h~,~,e (which is 0-independent [5]) has la2,al=0 (where I" I denotes Lebesgue measure). Our proof is such that for any other a with la~=2,~ I=0… (More)

- S. Jitomirskaya, B. Simon
- 1994

We prove that one-dimensional Schrödinger operators with even almost periodic potential have no point spectrum for a dense Gδ in the hull. This implies purely singular continuous spectrum for the almost Mathieu equation for coupling larger than 2 and a dense Gδ in θ even if the frequency is an irrational with good Diophantine properties. §

- 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〉/t2−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:

- S. Jitomirskaya
- 2003

Study of fine spectral properties of quasiperiodic and similar discrete Schrodinger operators involves dealing with problems caused by small denominators, and until recently was only possible using perturbative methods, requiring certain small parameters and complicated KAM-type schemes. We review the recently developed nonperturbative methods for such… (More)