Dispersive and long-time behavior of oscillations in lattices

We study the long-time dynamics of oscillations in lattices of infinitely many particles interacting via harmonic springs and adumbrate how decay rates for the displacements are proved analytically applying methods for oscillatory integrals. After recapitulating the results of an infinite chain with nearest-neighbour interaction, we highlight the differences to the case of next-nearest-neighbour interaction. Finally we give an outlook on the two dimensional lattice with nearest-neighbour interaction where more complicated singularities occur. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)