  We construct multidimensional Schr\"odinger operators with a spectrum that
has no gaps at high energies and that is nowhere dense at low energies. This
gives the first example for which this widely expected topological structure of
the spectrum in the class of uniformly recurrent Schr\"odinger operators,
namely the coexistence of a half-line and a Cantor-type structure, can be
confirmed. Our construction uses Schr\"odinger operators with separable
potentials that decompose into one-dimensional potentials generated by the
Fibonacci sequence and relies on the study of such operators via the trace map
and the Fricke-Vogt invariant. To show that the spectrum contains a half-line,
we prove an abstract Bethe--Sommerfeld criterion for sums of Cantor sets which
may be of independent interest.
