Perturbative diagonalization and spectral gaps of quasiperiodic operators on ℓ²(\mathbb Z^d) with monotone potentials

2024/08/10 by Ilya Kachkovskiy, Kachkovskiy, Ilya, Leonid Parnovski +3
Mathematics · #Spectral Theory in Mathematical Physics #Numerical methods in inverse problems #Differential Equations and Boundary Problems

paper · pdf · doi:10.48550/arxiv.2408.05650

Abstract

We obtain a perturbative proof of localization for quasiperiodic operators on ℓ²(\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.

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