Wannier–Stark Localization for Time Quasi-Periodic Hamiltonian Operator on \(\mathbb {Z}\)

In this paper, we consider the time (quasi)-periodic quantum Hamiltonian of the form \(\textrm{H}(t)=\textrm{H}_\gamma +\textrm{V}(\omega t)\), where \(\textrm{H}_\gamma \) is a power-law long-range lattice operator with uniform electric fields on \(\mathbb {Z}\), \(\textrm{V}(\omega t)\) is a time quasi-periodic perturbation. In particular, we can obtain the uniform power-law localization of the Floquet Hamiltonian operator \(-{\textbf{i}}\omega \cdot \partial _{\phi }+\textrm{H}(\phi )\), and the dynamical localization of the Hamiltonian operator \(\textrm{H}(t)\). No assumptions are made on the size of the perturbation; however, we require the time quasi-periodic perturbation is a “quasi-Töplitz” operator (close to a Töplitz operator).