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

We prove Cantor spectrum and almost-sure Anderson localization for quasiperiodic discrete Schr"odinger operators $H=\epsilon \mathrm{\Delta }+V$ with potential $V$ sampled with Diophantine frequency $\alpha$ from an asymmetric, smooth, cosine-like function $v\in C^2(\mathbb{T},[-1,1])$ for sufficiently small interaction $\epsilon \le {\epsilon }_0(v,\alpha )$. We prove this result via an inductive analysis on scales, whereby we show that locally the Rellich functions of Dirichlet restrictions of $H$ inherit the cosine-like structure of $v$ and are uniformly well-separated.