Wave Propagation in Nontrivial Backgrounds

TL;DR

This paper provides a systematic analysis of wave tails in general scattering potentials, revealing that late-time tails are primarily influenced by derivatives of the potential, a fact confirmed through analytical and numerical methods.

Contribution

It introduces a general framework for understanding wave tails in arbitrary potentials, emphasizing the importance of potential derivatives, which was not widely recognized before.

Findings

Late-time tails are determined by derivatives of the potential.

Analytical results are confirmed by numerical calculations.

The role of potential derivatives in tail formation is highlighted.

Abstract

It is well known that waves propagating in a nontrivial medium develop ``tails''. However, the exact form of the late-time tail has so far been determined only for a narrow class of models. We present a systematic analysis of the tail phenomenon for waves propagating under the influence of a {\it general} scattering potential $V(x)$. It is shown that, generically, the late-time tail is determined by spatial {\it derivatives} of the potential. The central role played by derivatives of the scattering potential appears not to be widely recognized. The analytical results are confirmed by numerical calculations.