Uniform asymptotic expansions for the infinite harmonic chain

Abstract

We study the dispersive behavior of waves in linear oscillator chains. We show that for general general dispersions it is possible to construct an expansion such that the remainder can be estimated by 1/t uniformly in space. In particalur we give precise asymptotics for the transition from the 1/t1/2 decay of nondegenerate wave numbers to the generate 1/t1/3 decay of generate wave numbers. This involves a careful description of the oscillatory integral involving the Airy function.