On the singular spectrum for adiabatic quasi-periodic Schrodinger operators on the real line

TL;DR

This paper investigates the spectral characteristics of adiabatic quasi-periodic Schrödinger operators on the real line, deriving asymptotic formulas for the Lyapunov exponent and demonstrating the spectrum's purely singular nature in certain energy intervals.

Contribution

It provides new asymptotic formulas for the Lyapunov exponent and establishes the purely singular spectrum for adiabatic quasi-periodic Schrödinger operators under specific conditions.

Findings

Asymptotic formula for Lyapunov exponent derived.

Spectrum shown to be purely singular in certain energy intervals.

Extended iso-energetic curves along momentum are key to results.

Abstract

In this paper, we study spectral properties of a family of quasi-periodic Schrodinger operators on the real line in the adiabatic limit. We assume that the adiabatic iso-energetic curves are extended along the momentum direction. In the energy intervals where this happens, we obtain an asymptotic formula for the Lyapunov exponent, and show that the spectrum is purely singular.