David Damanik

Multi-scale analysis implies strong dynamical localization

Abstract. We prove that a strong form of dynamical localization follows from a variable energy multi-scale analysis. This abstract result is applied to a number of models for wave propagation in disordered media.

Uniform Spectral Properties of One-Dimensional Quasicrystals, III. α-Continuity

Abstract: We study the spectral properties of one-dimensional whole-line Schrödinger operators, especially those with Sturmian potentials. Building upon the Jitomirskaya–Last extension of the Gilbert–Pearson theory of subordinacy, we demonstrate how to establish α-continuity of a whole-line operator from power-law bounds on the solutions on a half-line. However, we require that these bounds hold uniformly in the boundary condition.We are able to prove these bounds for Sturmian potentials with rotation numbers of bounded density and arbitrary coupling constant. From this we establish purely α-continuous spectrum uniformly for all phases.Our analysis also permits us to prove that the point spectrum is empty for all Sturmian potentials.

Uniform Spectral Properties of One-Dimensional Quasicrystals, I. Absence of Eigenvalues

Abstract:We consider discrete one-dimensional Schrödinger operators with Sturmian potentials. For a full-measure set of rotation numbers including the Fibonacci case, we prove absence of eigenvalues for all elements in the hull.

The Fractal Dimension of the Spectrum of the Fibonacci Hamiltonian

We study the spectrum of the Fibonacci Hamiltonian and prove upper and lower bounds for its fractal dimension in the large coupling regime. These bounds show that as

The Index of Sturmian Sequences

\lambda \to \infty, {\rm dim} (\sigma(H_\lambda)) \cdot {\rm log} \lambda

Power-law bounds on transfer matrices and quantum dynamics in one dimension

converges to an explicit constant,

Upper bounds in quantum dynamics

{\rm log}(1+\sqrt{2})\approx 0.88137