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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow><mml:msup><mml:mi>ℓ</mml:mi> <mml:mn>2</mml:mn></mml:msup> <mml:mrow><mml:mo>(</mml:mo> <mml:msup><mml:mi>Z</mml:mi> <mml:mi>d</mml:mi></mml:msup> <mml:mo>)</mml:mo></mml:mrow> </mml:mrow> </mml:math> 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.