Repeated potentials in one dimension and Lorentz transformations

It is shown that Lorentz transformations in two spatial dimensions provide a representation for the group of transference matrices of one-dimensional short-range potentials. This gives rise to the possibility of obtaining the quantum-mechanical properties of a chain of localized potentials through the behavior of the product of the corresponding Lorentz transformations. As an example, it is shown that to the energy gaps of a periodic potential correspond an infinite sequence of Lorentz transformations whose final velocity tend ultimately to the velocity of light.