On the stability of pure point spectrum of discrete operators with long-range hopping

Abstract

Let T be a multiplication operator on ${\ell }^2\left(Z^d\right)$ with pure-point and simple spectrum. In this paper, we study perturbation of T by some off-diagonal Toeplitz operator V with its matrix elements $V_{m,n}$ satisfying $\left|V_{m,n}\right|\le e^{-r{\log }^t(1+|m-n\left|\right)}$ $(t>1,r>0,m,n\in Z^d)$. Under some explicit conditions on the eigenvalues of T and for small enough ε, we prove via a KAM diagonalization method that the operator $T+\epsilon V$ has pure-point spectrum with eigenfunctions ${\left\{u_n\right(k\left)\right\}}_{n\in Z^d}$ obeying $\left|u_n\right(k\left)\right|\le 2e^{-\frac{r}{2}{\log }^t(1+|k-n\left|\right)}$ for all $n,k\in Z^d$.