伪随机性与衍射
Pseudorandomness and Diffraction
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中文总结 AI 辅助
该研究从衍射性质角度验证了一类余弦势场的伪随机性，证明其衍射测度为纯绝对连续，为薛定谔算子相关的伪随机猜想提供了新证据，并揭示了谱测度与衍射测度的对偶性。
中文摘要 AI 辅助
伪随机结构是指表现得如同随机结构、但其本身未必具有随机性的结构。有一个备受关注且显然难度极高的猜想认为，从薛定谔算子的角度来看，$$ V(n) = λ\big( 2 π\big( x_1 + n x_2 + \frac{n(n-1)}{2} α\big) \big) $$ 具有伪随机性：即带势场 $V$ 的 $\boldsymbol{\rm \textellipsis^2(\bbold{Z})}$ 上的薛定谔算子会呈现安德森局域化，也就是对几乎所有参数值，算子具有纯点谱，且本征函数呈指数衰减——这与随机势场产生的谱特征完全一致。我们证明，$V$ 在衍射性质层面具有伪随机性：对所有 $λ≠0$、所有无理数 $α$ 以及所有 $x_1,x_2$，其关联的衍射测度是纯绝对连续的——这一性质同样与随机情形相符。我们的结果为薛定谔情形下的该猜想提供了进一步证据，同时阐明了薛定谔谱测度与衍射测度之间明显的对偶行为。
英文摘要
Pseudorandom structures are structures that behave like random ones, without necessarily being random themselves. It is a prominent, and evidently very difficult, conjecture that $$ V(n) = λ\cos \Big( 2 π\Big( x_1 + n x_2 + \frac{n(n-1)}{2} α\Big) \Big) $$ is pseudorandom from a Schrödinger operator perspective in that the Schrödinger operator in $\ell^2(\mathbb{Z})$ with potential $V$ displays Anderson localization, that is, pure point spectrum with exponentially decaying eigenfunctions for almost all parameter values --- the same spectral features as those produced by random potentials. We show that $V$ is pseudorandom in terms of its diffraction properties, that is, the associated diffraction measure is purely absolutely continuous for all $λ\not= 0$, all irrational $α$, and all $x_1,x_2$ --- which is the case as well for the random case. Our result gives further evidence for the conjecture in the Schrödinger case and it elucidates the apparent dual behavior of Schrödinger spectral measures and diffraction measures.
发表机构
- Rice University（莱斯大学）
- MacEwan University（麦克尤恩大学）
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