Mixed spectral regimes for square Fibonacci Hamiltonians

Abstract

For the square tridiagonal Fibonacci Hamiltonian, we prove existence of an open set of parameters which yield mixed interval-Cantor spectra (i.e. spectra containing an interval as well as a Cantor set), as well as mixed density of states measure (i.e. one whose absolutely continuous and singular continuous components are both nonzero). Using the methods developed in this paper, we also show existence of parameter regimes for the square continuum Fibonacci Schrödinger operator yielding mixed interval-Cantor spectra. These examples provide the first explicit examples of an interesting phenomenon that has not hitherto been observed in aperiodic Hamiltonians. Moreover, while we focus only on the Fibonacci model, our techniques are equally applicable to models based on any two-letter primitive invertible substitution.

Quantum and spectral properties of the Labyrinth model

“…The following theorem was proven in [14] for the Fibonacci Hamiltonian case, and recently it was extended to tridiagonal Fibonacci Hamiltonians in [18]. It follows by repeating the argument of [14].…”

“…By repeating the argument from [13], one can show that the analogous results of the square Fibonacci Hamiltonian hold for the square tiling. Recently, [18] considered the square tridiagonal Fibonacci Hamiltonians, which include the square Fibonacci Hamiltonian and the square tiling as special cases.…”

Spectral transitions for the square Fibonacci Hamiltonian

“…The study of the structure and the properties of sums of Cantor sets is motivated by applications in dynamical systems [38,39,40,42], number theory [7,27,33], harmonic analysis [3,4], and spectral theory [16,17,18,21,59]. In many cases dynamically defined Cantor sets are of special interest.…”

“…(d) For other work on the square Fibonacci Hamiltonian and related models, see [16,17,18,21,50,51,56,57,59].…”

Spectral Properties of Continuum Fibonacci Schrödinger Operators

“…Until somewhat recently, most of the effort was devoted to discrete Schrödinger operators arising in this manner. On the other hand, there have been several recent investigations concerning continuum Schrödinger operators [6,18,21], Jacobi matrices [14,30], and CMV matrices [7,8,10,11,12,13,22].…”