Contents Online
Mathematical Research Letters
Volume 12 (2005)
Number 2
Ergodic Potentials With a Discontinuous Sampling Function Are Non-Deterministic
Pages: 187 – 192
DOI: https://dx.doi.org/10.4310/MRL.2005.v12.n2.a4
Authors
Abstract
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.
Published 1 January 2005