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

Abstract

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 proof that the spectra of such operators contain infinitely many gaps.

33 pages. Final accepted version

References in corpus (3)

• Localization for Almost-Periodic Operators with Power-law Long-range Hopping: A Nash-Moser Iteration Type Reducibility Approach
• Stability of the gapless pure point spectrum of self-adjoint operators
• On gaps in the spectra of quasiperiodic Schrödinger operators with discontinuous monotone potentials