Dynamical localization and eigenvalue asymptotics: long-range hopping lattice operators with electric field

We prove that for polynomial long-range hopping lattice operators with uniform electric field under any bounded perturbation, the semi-uniform polynomial decay of the eigenfunctions is determined by the asymptotic behavior of the eigenvalues, and conversely. Consequently, we recover and refine recent results on power-law dynamical localization for this model. In this paper, to prove these results, we develop new arguments to deal with singularities in obtaining the semi-uniform polynomial decay of eigenfunctions, prove an independent result on the asymptotic behavior of eigenvalues for discrete operators, revisit a notion of Power-Law SULE, and utilize a known perturbation result on localization, proved through the KAM method. Unlike existing results in the literature, our approach does not rely on the specific form of the eigenfunctions, but rather the asymptotic behavior of the eigenvalues and the potential. It is worth underlining that our general results can be applied to other models such as Maryland-type potentials.