Ergodic Potentials With a Discontinuous Sampling Function Are Non-Deterministic

Abstract

We prove absence of absolutely continuous spectrum for discrete one-dimensional Schrödinger operators on the whole line with certain ergodic potentials, $V_ω(n) = f(T^n(ω))$, where $T$ is an ergodic transformation acting on a space $Ω$ and $f: Ω\to \R$. The key hypothesis, however, is that $f$ is discontinuous. In particular, we are able to settle a conjecture of Aubry and Jitomirskaya--Mandel'shtam regarding potentials generated by irrational rotations on the torus. The proof relies on a theorem of Kotani, which shows that non-deterministic potentials give rise to operators that have no absolutely continuous spectrum.

5 pages