Mathematical Research Letters

Volume 12 (2005)

Number 2

Ergodic Potentials With a Discontinuous Sampling Function Are Non-Deterministic

Pages: 187 – 192

DOI: http://dx.doi.org/10.4310/MRL.2005.v12.n2.a4

Authors

David Damanik

Rowan Killip

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.

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