具有不连续采样函数的遍历势是不确定的
Ergodic Potentials With a Discontinuous Sampling Function Are Non-Deterministic
摘要：
本文证明了具有一定遍历势$V_ω(n) = f(T^n(ω))$的离散一维薛定谔算子在整条直线上不存在绝对连续谱，其中$T$是作用在空间$Ω$和$f: Ω\to \R$上的遍历变换.然而，关键假设是$f$是不连续的。特别是，我们能够解决一个猜想的奥布里和Jitomirskaya-Mandel'shtam关于潜力所产生的不合理的旋转环面。证明依赖于Kotani定理，该定理表明，非确定性的潜力产生的运营商，没有绝对连续谱。
英文摘要：
We prove absence of absolutely continuous spectrum for discrete one-dimensional Schrödinger operators on the whole line with certain ergodic potentials, $V_ω(n) = f(T^n(ω))$, where $T$ is an ergodic transformation acting on a space $Ω$ and $f: Ω\to \R$. 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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