Perturbative diagonalization and spectral gaps of quasiperiodic operators on $\ell^2(\mathbb Z^d)$ with monotone potentials

TL;DR

This paper presents a perturbative approach to prove localization and spectral gap properties of quasiperiodic operators on multi-dimensional integer lattices with monotone potentials, focusing on small hopping regimes.

Contribution

It introduces a local, convergent KAM-type diagonalization method for quasiperiodic operators with monotone potentials, establishing localization and spectral gap results.

Findings

Proves localization for small hopping in quasiperiodic operators.

Shows spectra contain infinitely many gaps.

Develops a convergent KAM-type diagonalization scheme.

Abstract

We obtain a perturbative proof of localization for quasiperiodic operators on $\ell^2(\Z^d)$ 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 proof that the spectra of such operators contain infinitely many gaps.