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.

arXiv comment33 pages. Final accepted version

References in corpus (7)

• Bounds on the density of states for Schr\" odinger operators
• Arithmetic Spectral Transitions for the Maryland Model
• Singular continuous spectrum for singular potential
• Localization for Almost-Periodic Operators with Power-law Long-range Hopping: A Nash-Moser Iteration Type Reducibility Approach
• Perturbative diagonalisation for Maryland-type quasiperiodic operators with flat pieces
• Stability of the gapless pure point spectrum of self-adjoint operators
• On gaps in the spectra of quasiperiodic Schrödinger operators with discontinuous monotone potentials