Dynamical localization and delocalization for random Schrodinger operators with $\delta$-interactions in $\mathbb{R}^3$

Abstract

We prove that the random Schrodinger operators on $\mathbb{R}^3$ with independent, identically distributed random variables and single-site potentials given by $\delta$-functions on $\mathbb{Z}^3$, exhibit both dynamical localization and dynamical delocalization with probability one. That is, there are regions in the deterministic spectrum that exhibit dynamical localization, the nonspreading of wave packets, and regions in the deterministic spectrum where the models also exhibit nontrivial quantum transport, almost surely. These models are the first examples of ergodic, random Schrodinger operators exhibiting both dynamical localization and delocalization in dimension three or higher. The nontrivial transport is due to the presence of delocalized generalized eigenfunctions at positive energies $E > \pi^2$. The general idea of the proof follows [Hislop, Kirsch, Krishna (2024)] in which lower bounds on moments of the position operator are constructed using these generalized eigenfunctions. A new result of independent interest is a proof of the Combes-Thomas estimate on exponential decay of the Green's function for Schrodinger operators with infinitely-many $\delta$-potentials.