 0447 David Damanik and Rowan Killip
 Ergodic Potentials With a Discontinuous Sampling
Function Are NonDeterministic
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Feb 25, 04

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Abstract. We prove absence of absolutely continuous spectrum for discrete
onedimensional 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 JitomirskayaMandel'shtam
regarding potentials generated by irrational rotations on the torus.
The proof relies on a theorem of Kotani, which shows that nondeterministic potentials
give rise to operators that have no absolutely continuous spectrum.
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