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We prove Cantor spectrum and almost-sure Anderson localization forquasiperiodic discrete Schr\"odinger operators $H = \varepsilon\Delta + V$ withpotential $V$ sampled with Diophantine frequency $\alpha$ from an asymmetric,smooth, cosine-like function $v \in C^2(\mathbb{T},[-1,1])$ for sufficientlysmall interaction $\varepsilon \leq \varepsilon_0(v,\alpha)$. We prove thisresult via an inductive analysis on scales, whereby we show that locally theRellich functions of Dirichlet restrictions of $H$ inherit the cosine-likestructure of $v$ and are uniformly well-separated.

Localization and Cantor spectrum for quasiperiodic discrete Schr\"odinger operators with asymmetric, smooth, cosine-like sampling functions