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 to 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 wavefunction localized at one end would evolve to its mirror image at the opposite chain end.