Anderson localization for multi-frequency quasi-periodic operators on $$\pmb {\mathbb {Z}}^{d}$$

Abstract

We establish Anderson localization for general analytic k-frequency quasi-periodic operators on Z d for arbitrary k, d.

Analytic solutions of nonlinear elliptic equations on rectangular tori

“…The following result was indeed proved by Bourgain [10] in case d = 3 and it can be easily extended to any d ≥ 1 (see [26] for details). Suppose that for ∅ = I ⊂ {1, · · · , d}, θ ∈ R d and N τ2 ≤ L ≤ N τ1 , there is no sequence…”

“…Finally, we introduce the powerful matrix-valued Cartan estimate with 1-dimensional parameters. For a generation to several variables case, we refer to [26]. Lemma 4.6 (Matrix-valued Cartan estimate, [10,14]).…”

“…As in [10], on each unit cube in B(0, 3N ), using Lemma 4.6, Lemma 4.5 and the resolvent identity, one can find such X N for ν ∈ Ω N . We refer to [26] (see also [10]) for details. We fix a large integer A satisfying (i) If ν ∈ I ∈ Γ r , then there exists some set X A r such that, for 1 ≤ j ≤ d,…”

On the Spectra of Separable 2D Almost Mathieu Operators

“…Similar but distinct quasiperiodic (not necessarily separable) operators have been studied by S. Jitomirskaya, W. Liu, and Y. Shi [25]. The [10,25] results pertain to the spectral type but not the topological structure of the spectrum as a set. In Sect.…”

“…(f) The separable operator considered in Theorem 1.6 but with added background potential (so the resulting operator is not necessarily separable) has been studied by J. Bourgain and I. Kachkovskiy [10]. Similar but distinct quasiperiodic (not necessarily separable) operators have been studied by S. Jitomirskaya, W. Liu, and Y. Shi [25]. The [10,25] results pertain to the spectral type but not the topological structure of the spectrum as a set.…”