Stability of Spectral Types of Quasi-Periodic Schr\"odinger Operators With Respect to Perturbations by Decaying Potentials

TL;DR

This paper investigates how spectral properties of quasi-periodic Schrödinger operators are affected by decaying potentials, showing stability of the spectrum and localization under certain decay conditions.

Contribution

It establishes decay conditions under which spectral types are preserved, including absolute continuity and Anderson localization, in the context of quasi-periodic Schrödinger operators.

Findings

Essential spectrum remains purely absolutely continuous with finite discrete spectrum in each gap.

Anderson localization is preserved under exponentially decaying perturbations.

Spectral properties are stable in the almost reducibility regime with finite first moment decay.

Abstract

We consider perturbations of quasi-periodic Schr\"odinger operators on the integer lattice with analytic sampling functions by decaying potentials and seek decay conditions under which various spectral properties are preserved. In the (almost) reducibility regime we prove that for perturbations with finite first moment, the essential spectrum remains purely absolutely continuous and the newly created discrete spectrum must be finite in each gap of the unperturbed spectrum. We also prove that for fixed phase, Anderson localization occurring for almost all frequencies in the regime of positive Lyapunov exponents is preserved under exponentially decaying perturbations.