Exact solvable family of discrete Schrödinger operators with long-range hoppings

It is shown that one-dimensional discrete Schrödinger operators, with a uniform electric field and long-range hoppings, form an isospectral family with discrete and simple spectrum (and isospectral operators with eigenfunctions having different decay properties). Explicit relations between hopping decay rates and eigenvectors decay rates are obtained. Small perturbations of exponentially decay hopping operators preserve dynamical localization.