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, $V_\omega(n) = f(T^n(\omega))$, where $T$ is an ergodic transformation acting on a space $\Omega$ and $f: \Omega \to {\mathBB R}$. 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.

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Published 1 January 2005