- 04-47 David Damanik and Rowan Killip
- Ergodic Potentials With a Discontinuous Sampling
Function Are Non-Deterministic
Feb 25, 04
(auto. generated ps),
of related papers
Abstract. We prove absence of absolutely continuous spectrum for discrete
one-dimensional Schr\"odinger 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 $f: \Omega \to \R$.
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.