Dispersionless pulse transport in mass-spring chains: All possible perfect Newton's cradles

A pulse traveling on a uniform nondissipative chain of N masses connected by springs is soon destructured by dispersion. Here it is shown that a proper modulation of the masses and the elastic constants makes it possible to obtain a periodic dynamics and a perfect transmission of any kind of pulse between the chain ends, since the initial configuration evolves to its mirror image in the half period. This makes the chain behave as a Newton's cradle. By a known algorithm based on orthogonal polynomials one can numerically solve the general inverse problem leading from the spectrum to the dynamical matrix and then to the corresponding mass-spring sequence, so yielding all possible "perfect cradles." As quantum linear systems obey the same dynamics of their classical counterparts, these results also apply to the quantum case: For instance, a wave function localized at one end would evolve to its mirror image at the opposite chain end.