Uniform Asymptotic Expansions for the Fundamental Solution of Infinite Harmonic Chains

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 particlar we give precise asymptotics for the cross-over from the $t^{-1/2}$ decay of nondegenerate wave numbers to the degenerate $t^{-1/3}$ decay of degenerate wave numbers. This involves a careful description of the oscillatory integral involving the Airy function.