Localization and Cantor spectrum for quasiperiodic discrete Schrödinger operators with asymmetric, smooth, cosine-like sampling functions

Abstract:

A discrete Schrödinger operator $H_V = \varepsilon\Delta + V$ on $\ell^2(\mathbb{Z})$ is called Anderson localized if it exhibits a basis of exponentially decaying eigenvectors. If $V_n$ is sampled from a potential function by Diophantine rotations on the one-dimensional torus, $H_V$ is known to be almost-surely Anderson localized for sufficiently small $\varepsilon$ if the potential is either analytic, or cosine-like and symmetric. In this talk, we discuss a new perturbative proof of almost-sure localization for Schrödinger operators with potential sampled from any $C^2$-smooth Morse function with two monotonicity intervals along a Diophantine rotation orbit on the circle. This is a joint work with Tom VandenBoom.