Study of fine spectral properties of quasiperiodic and similar discrete Schrödinger operators involves dealing with problems caused by small denominators, and until recently was only possible using perturbative methods, requiring certain small parameters and complicated KAM-type schemes. We review the recently developed nonperturbative methods for such study which lead to stronger results and are significantly simpler. Numerous applications mainly due to J. Bourgain, M. Goldstein, W. Schlag, and the author are also discussed.
We consider continuous one-dimensional multifrequency Schrödinger operators, with analytic potential, and prove Anderson localization in the regime of positive Lyapunov exponent for almost all phases and almost all Diophantine frequencies.
We propose a new method to prove Anderson localization for quasiperiodic Schrödinger operators and apply it to the quasiperiodic model considered by Sinai and Fröhlich-Spencer-Wittwer. More concretely, we prove Anderson localization for even $C^2$ cosine type quasiperiodic Schrödinger operators with large coupling constants, Diophantine frequencies and Diophantine phases.
We establish non-perturbative Anderson localization for a wide class of 1D quasiperiodic operators with unbounded monotone potentials, extending the classical results on Maryland model and perturbative results for analytic potentials by Béllissard, Lima, and Scoppola.
We prove sharp spectral transition in the arithmetics of phase between localization and singular continuous spectrum for Diophantine almost Mathieu operators. We also determine exact exponential asymptotics of eigenfunctions and of corresponding transfer matrices throughout the localization region. This uncovers a universal structure in their behavior governed by the exponential phase resonances. The structure features a new type of hierarchy, where self-similarity holds upon alternating reflections.
We consider a system of two discrete quasiperiodic 1D particles as an operator on $\ell^2(\mathbb Z^2)$ and establish Anderson localization at large disorder, assuming the potential has no cosine-type symmetries. In the presence of symmetries, we show localization outside of a neighborhood of finitely many energies. One can also add a deterministic background potential of low complexity, which includes periodic backgrounds and finite range interaction potentials. Such background potentials can only take finitely many values, and the excluded energies in the symmetric case are associated to tho…
We establish localization type dynamical bounds as a corollary of positive Lyapunov exponents for general operators with quasiperiodic potentials defined by piecewise Holder functions.
We prove Cantor spectrum and almost-sure Anderson localization for quasiperiodic discrete Schrödinger operators $H = \varepsilonΔ+ V$ with potential $V$ sampled with Diophantine frequency $α$ from an asymmetric, smooth, cosine-like function $v \in C^2(\mathbb{T},[-1,1])$ for sufficiently small interaction $\varepsilon \leq \varepsilon_0(v,α)$. We prove this result via an inductive analysis on scales, whereby we show that locally the Rellich functions of Dirichlet restrictions of $H$ inherit the cosine-like structure of $v$ and are uniformly well-separated.
We study quasiperiodicity-induced localization of waves in strongly precompressed granular chains. We propose three different setups, inspired by the Aubry--André (AA) model, of quasiperiodic chains; and we use these models to compare the effects of on-site and off-site quasiperiodicity in nonlinear lattices. When there is purely on-site quasiperiodicity, which we implement in two different ways, we show for a chain of spherical particles that there is a localization transition (as in the original AA model). However, we observe no localization transition in a chain of cylindrical particles in …
Sharp Logarithmic Quantum Dynamics for Quasiperiodic Schrödinger Operators
[W.\ Liu] Department of Mathematics, Texas A\&M University, College Station, TX 77843, USA；[X.\ Wang] Department of Mathematics, Texas A\&M University, College Station, TX 77843, USA
