Ergodic Potentials with a Discontinuous Sampling Function Are Non-deterministic

We prove absence of absolutely continuous spectrum for discrete one-dimensional Schrödinger operators on the whole line with certain ergodic potentials, Vω(n) = f(T n(ω)), where T is an ergodic transformation acting on a space Ω and f : Ω → 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.