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Ergodic Potentials With a Discontinuous Sampling Function Are Non-Deterministic

David Damanik·Rowan Killip
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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.

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David Damanik, Rowan Killip. 2004-02-25. Ergodic Potentials With a Discontinuous Sampling Function Are Non-Deterministic. https://arxiv.org/abs/math-ph/0402070

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