论文标题
高度相关的安德森型模型的频谱和动态对比度
Spectral and Dynamical contrast on highly correlated Anderson-type models
论文作者
论文摘要
我们研究随机schrödinger运算符的光谱和动态属性$ h _ {\ mathrm {diag}} = - a _ {\ mathbb {g} _ {\ mathrm {diag}}}+v_Ω+v_Ω$在某些二维图上$ {\ mathbb {g} $ {\ mathbb {g} _ {\ mathrm {diag}}} $。与标准安德森模型不同，随机电位不是独立的，而是沿任何垂直线，即$v_Ω（n）=ω（n_1）$，对于$ n =（n_1，n_2）$。特别是，此处研究的电位表现出远距离相关性。我们列出了示例，其中几何变化对基础图的变化与高混乱相结合，对操作员的光谱和动力学特性产生了重大影响，从而导致“对角线”和“垂直”模型的对比行为。此外，“垂直”模型在其（纯粹）绝对连续的频谱中表现出急剧的相变。这是由Avron和Simon引入的绝对连续频谱的瞬时和复发组件的概念所捕获的。
We study spectral and dynamical properties of random Schrödinger operators $H_{\mathrm{Vert}}=-A_{\mathbb{G}_{\mathrm{Vert}}}+V_ω$ and $H_{\mathrm{Diag}}=-A_{\mathbb{G}_{\mathrm{Diag}}}+V_ω$ on certain two dimensional graphs ${\mathbb{G}_{\mathrm{Vert}}}$ and ${\mathbb{G}_{\mathrm{Diag}}}$. Differently from the standard Anderson model, the random potentials are not independent but, instead, are constant along any vertical line, i.e $V_ω(n)=ω(n_1)$, for $n=(n_1,n_2)$. In particular, the potentials studied here exhibit long range correlations. We present examples where geometric changes to the underlying graph, combined with high disorder, have a significant impact on the spectral and dynamical properties of the operators, leading to contrasting behaviors for the "diagonal" and "vertical" models. Moreover, the "vertical" model exhibits a sharp phase transition within its (purely) absolutely continuous spectrum. This is captured by the notions of transient and recurrent components of the absolutely continuous spectrum, introduced by Avron and Simon.
