标题： Lifshitz 尾部对于 Laurent 矩阵的随机对角扰动 显示英文标题

标题： Lifshitz tails for random diagonal perturbations of Laurent matrices

摘要： 我们研究了一维随机算子在$\ell^2(\mathbb Z)$上的积分态密度，其形式为$T + V_\omega$，其中$T$是一个 Laurent（也称为双无限 Toeplitz）矩阵，$V_\omega$是由独立同分布随机变量生成的 Anderson 势。 我们假设算子$T$与一个有界、Hölder 连续的符号$f$相关联，该符号在其最小值点处取得最小值。 我们允许$f$在代数意义上取得其最小值。 得到的算子$T$是长程的，具有弱（代数）的非对角衰减。 我们证明了该算子在谱的下边缘表现出 Lifshitz 尾部，其指数由$T$在下谱边缘的态密度给出。 该证明依赖于将 Dirichlet-Neumann 夹逼方法推广到长程情形，以及将 Temple 不等式推广到退化基态能量的情形。 显示英文摘要

摘要： We study the Integrated Density of States of one-dimensional random operators acting on $\ell^2(\mathbb Z)$ of the form $T + V_\omega$ where $T$ is a Laurent (also called bi-infinite Toeplitz) matrix and $V_\omega$ is an Anderson potential generated by i.i.d. random variables. We assume that the operator $T$ is associated to a bounded, H\"older-continuous symbol $f$, that attains its minimum at a finite number of points. We allow for $f$ to attain its minima algebraically. The resulting operator $T$ is long-range with weak (algebraic) off-diagonal decay. We prove that this operator exhibits Lifshitz tails at the lower edge of the spectrum with an exponent given by the Integrated Density of States of $T$ at the lower spectral edge. The proof relies on generalizations of Dirichlet-Neumann bracketing to the long-range setting and a generalization of Temple's inequality to degenerate ground state energies.