  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
Schrodinger 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.
