Localization and Cantor spectrum for quasiperiodic discrete Schr\"odinger operators with asymmetric, smooth, cosine-like sampling functions

TL;DR

This paper establishes Cantor spectrum and Anderson localization for certain quasiperiodic Schrödinger operators with asymmetric, smooth cosine-like potentials, using an inductive scale analysis to demonstrate spectral properties.

Contribution

It proves Cantor spectrum and localization for quasiperiodic operators with asymmetric smooth cosine-like potentials, extending previous results to more general potential functions.

Findings

Proves Cantor spectrum for the operators.

Establishes almost-sure Anderson localization.

Shows local Rellich functions inherit cosine-like structure.

Abstract

We prove Cantor spectrum and almost-sure Anderson localization for quasiperiodic discrete Schr\"odinger operators $H = \varepsilon\Delta + V$ with potential $V$ sampled with Diophantine frequency $\alpha$ from an asymmetric, smooth, cosine-like function $v \in C^2(\mathbb{T},[-1,1])$ for sufficiently small interaction $\varepsilon \leq \varepsilon_0(v,\alpha)$. 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.