Construction of quasiperiodic Schrödinger operators with Cantor spectrum (Q2330444)

The main result of the paper states that for each \(s \in (0, 1/2)\) and for any Diophantine vector \(\omega\in\mathbb{R}^d\), there exists a set \(K \in \mathbb{Z}^d\) such that the spectrum of the quasi-periodic Schrödinger operator generated by the expression \[ -u''(x) + \lambda v(\phi + x \omega) u(x) \] on \(L_2(\mathbb{R})\) with the potential \[ v(\theta) := \sum_{k\in K} e^{-|k|^s} \cos \langle k, \theta \rangle \] is a Cantor set for all \(\lambda \in [-1, 1] \setminus \{ 0 \}\). An explicit construction of such a set \(K\) is also given for some values of \(\omega\).