Absolutely continuous convolutions of singular measures and an application to the square Fibonacci Hamiltonian

Abstract

Abstract. We prove for the square Fibonacci Hamiltonian that the density of states measure is absolutely continuous for almost all pairs of small coupling constants. This is obtained from a new result we establish about the absolute continuity of convolutions of measures arising in hyperbolic dynamics with exact-dimensional measures.

Spectral continuity for aperiodic quantum systems I. General theory

“…As discussed before, the methods of transfer matrices and trace maps are powerful tools. On the other hand, these techniques are limited to Hamiltonians with nearest neighbor interaction and to one-dimensional systems or models that can be decomposed in one-dimensional systems [38]. But higher dimensional systems like the Penrose tiling or the Octagonal lattice cannot be treated with this methods [20].…”

Sums of two self-similar Cantor sets

“…Sums of two Cantor sets. Sums of two Cantor sets arise naturally in dynamical systems (e.g., [6], [7]), in number theory (e.g., [3], [4]) and also in spectral theory (e.g., [1], [2]). In 1970's, Palis conjectured that for generic pairs of dynamically defined Cantor sets their sumset contains an interval if the sum of their Hausdorff dimensions is greater than 1 (see, e.g., [7]).…”

Quantum and spectral properties of the Labyrinth model

“…proof of (1.5). The proof is essentially the repetition of the proof of Proposition A.3 of [13]. For the reader's convenience, we will repeat the argument.…”

“…To get an idea of spectral properties of higher dimensional quasicrystals, simpler models have been considered. In two dimensional case, we have, for example, the square Fibonacci Hamiltonian [13], the square tiling, and the Labyrinth model. The Labyrinth model is the main subject of this paper.…”