Svetlana Jitomirskaya

Operators with singular continuous spectrum, IV. Hausdorff dimensions, rank one perturbations, and localization

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 and Taylor [27,28], and studied recently within spectral theory by Last [22] (also see references therein), is the idea of using Hausdorff measures and dimensions to classify measures. Our main goal in this paper is to look at the singular spectrum produced by rank one perturbations (and discussed in [7,11,33]) from this point of view. A Borel measure μ is said to have exact dimension α ∈ [0, 1] if and only if μ(S) = 0 if S has dimension β < α and if μ is supported by a set of dimension α. If 0 < α < 1, such a measure is, of necessity, singular continuous. But, there are also singular continuous measures of exact dimension 0 and 1 which are “particularly close” to point and a.c. measures, respectively. Indeed, as we’ll explain, we know of “explicit” Schrödinger operators with exact dimension 0 and 1, but, while they presumably exist, we don’t know of any with dimension α ∈ (0, 1). While we’re interested in the abstract theory of rank one perturbations, we’re especially interested in those rank one perturbations obtained by taking a random Jacobi matrix and making a Baire generic perturbation of the potential at a single point. It is a disturbing fact that the strict localization (dense point spectrum with ‖xe−itHδ0‖2 = (e−itHδ0, x2e−itHδ0) bounded in t), that holds a.e. for the random case, can be destroyed by arbitrarily small local perturbations [7,11]. We’ll ameliorate this discovery in the present paper in three ways: First, we’ll see that, in this case, the spectrum is always of dimension zero, albeit sometimes pure point and sometimes singular continuous. Second, we’ll show that not

Metal-insulator transition for the almost Mathieu operator

We prove that for Diophantine ! and almost every µ; the almost Mathieu operator, (H!;‚;µ“)(n )=“ (n +1 ) +“ (ni 1) +‚ cos 2…(!n+µ)“(n), exhibits localization for ‚> 2 and purely absolutely continuous spectrum for ‚< 2:

Continuity of the Lyapunov Exponent for Quasiperiodic Operators with Analytic Potential

We study regularity properties of the Lyapunov exponent L of one-frequency quasiperiodic operators with analytic potential, under no assumptions on the Diophantine class of the frequency. We prove joint continuity of L, in frequency and energy, at every irrational frequency.

Power-law subordinacy and singular spectra I. Half-line operators

We study Hausdorff-dimensional spectral properties of certain “whole-line” quasiperiodic discrete Schrodinger operators by using the extension of the Gilbert–Pearson subordinacy theory that we previously developed in [19].

Operators with singular continuous spectrum. III. Almost periodic Schrödinger operators

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

Delocalization in random polymer models

Abstract: 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 quadratically at a generic critical energy and that the density of states is positive there. Large deviation estimates around these asymptotics allow to prove optimal lower bounds on quantum transport, showing that it is almost surely overdiffusive even though the models are known to have pure-point spectrum with exponentially localized eigenstates for almost every configuration of the polymers. Furthermore, the level spacing is shown to be regular at the critical energy.

Power Law Subordinacy and Singular Spectra.¶II. Line Operators

Abstract:We study Hausdorff-dimensional spectral properties of certain “whole-line” quasiperiodic discrete Schrödinger operators by using the extension of the Gilbert–Pearson subordinacy theory that we previously developed in [19].

Erratum to: Analytic Quasi-Perodic Cocycles with Singularities and the Lyapunov Exponent of Extended Harper’s Model

We show how to extend (and with what limitations) Avila’s global theory of analytic SL(2,C) cocycles to families of cocycles with singularities. This allows us to develop a strategy to determine the Lyapunov exponent for extended Harper’s model, for all values of parameters and all irrational frequencies. In particular, this includes the self-dual regime for which even heuristic results did not previously exist in physics literature. The extension of Avila’s global theory is also shown to imply continuous behavior of the LE on the space of analytic M2(C)-cocycles. This includes rational approximation of the frequency, which so far has not been available.

Singular continuous spectrum is generic

In a variety of contexts, we prove that singular continuous spectrum is generic in the sense that for certain natural complete metric spaces of operators, those with singular spectrum are a dense G delta.

Complex one-frequency cocycles

We show that on a dense open set of analytic one-frequency complex valued cocycles in arbitrary dimension Oseledets filtration is either dominated or trivial. The underlying mechanism is different from that of the Bochi-Viana Theorem for continuous cocycles, which links non-domination with discontinuity of the Lyapunov exponent. Indeed, in our setting the Lyapunov exponents are shown to depend continuously on the cocycle, even if the initial irrational frequency is allowed to vary. On the other hand, this last property provides a good control of the periodic approximations of a cocycle, allowing us to show that domination can be characterized, in the presence of a gap in the Lyapunov spectrum, by additional regularity of the dependence of sums of Lyapunov exponents.