Perturbative Diagonalization and Spectral Gaps of Quasiperiodic Operators on \(\ell ^2(\mathbb Z^d)\) with Monotone Potentials

We obtain a perturbative proof of localization for quasiperiodic operators on \(\ell ^2(\mathbb Z^d)\) with one-dimensional phase space and monotone sampling functions, in the regime of small hopping. The proof is based on an iterative scheme which can be considered as a local (in the energy and the phase) and convergent version of KAM-type diagonalization, whose result is a covariant family of uniformly localized eigenvalues and eigenvectors. We also prove that the spectra of such operators contain infinitely many gaps.