Multidimensional Schrödinger operators whose spectrum features a half-line and a Cantor set

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