Method of variations of potential of quasi-periodic Schrodinger equation

TL;DR

This paper investigates the spectral properties of the one-dimensional discrete quasi-periodic Schrödinger equation, introducing a new concept of potential variations to analyze the positivity of the Lyapunov exponent for typical smooth potentials at large coupling.

Contribution

It introduces the notion of potential variations and demonstrates that for typical smooth potentials with large coupling, the Lyapunov exponent is positive for most frequencies and all energies.

Findings

Lyapunov exponent is positive for most frequencies at large coupling.

Introduction of potential variations to define typical potentials.

Results apply to typical C^3 smooth potentials.

Abstract

We study the one-dimensional discrete quasi-periodic Schrodinger equation. We introduce the notion of variations of potential and use it to define "typical" potential. We show that for typical C^3 potential, if the coupling constant is large, then for most frequencies, the Lyapunov exponent is positive for all energies.