Dynamical localization for polynomial long-range hopping random operators on ℤ^{𝕕}

Abstract

In this paper, we prove a power-law version dynamical localization for a random operator H ω \mathrm {H}_{\omega } on Z d \mathbb {Z}^d with long-range hopping. In brief, we show that the Sobolev norm of the discrete linear Schrödinger equation i ∂ t u = H ω u \mathrm {i}\partial _{t}u=\mathrm {H}_{\omega }u with well-localized initial state is bounded for any t ≥ 0 t\geq 0 .