Spectral Statistics for one dimensional Anderson model with unbounded but decaying potential

Abstract

In this work, we study the spectral statistics for Anderson model on $\ell^2(\mathbb{N})$ with decaying randomness whose single site distribution has unbounded support. Here we consider the operator $H^ω$ given by $(H^ωu)_n=u_{n+1}+u_{n-1}+a_nω_n u_n$, $a_n\sim n^{-α}$ and $\{ω_n\}$ are real i.i.d random variables following symmetric distribution $μ$ with fat tail, i.e $μ((-R,R)^c)<\frac{C}{R^δ}$ for $R\gg 1$, for some constant $C$. In case of $α-\frac{1}δ>\frac{1}{2}$, we are able to show that the eigenvalue process in $(-2,2)$ is the clock process.