Localization for a class of one dimensional quasi-periodic Schrödinger operators

Abstract

We prove for small ɛ and α satisfying a certain Diophantine condition the operator

$$H = - \varepsilon ^2 \Delta + \frac{1}{{2\pi }}\cos 2\pi (j\alpha + \theta ) j \in \mathbb{Z}$$

has pure point spectrum for almost all θ. A similar result is established at low energy for$H = - \frac{{d^2 }}{{dx^2 }} - K^2 (\cos 2\pi x + \cos 2\pi (\alpha x + \theta ))$ providedK is sufficiently large.