On pointwise decay of waves

TL;DR

This paper introduces the linear decay properties of waves, emphasizing spectral and scattering techniques, and surveys 15 years of results on wave decay in various backgrounds with a focus on zero energy effects.

Contribution

It provides an overview of recent advances in understanding wave decay, highlighting the role of spectral theory and zero energy phenomena.

Findings

Decay rates depend on spectral properties

Zero energy plays a crucial role in decay behavior

Survey of 15 years of research on wave decay

Abstract

This article serves as an introduction to the linear aspects of the recent submission by Krieger, Miao, and the author, see arXiv:2009.08843. It also surveys some of the results obtained on the decay of linear waves on various backgrounds or in the presence of a potential over the past 15 years. Particular emphasis is placed on the role of 0 energy. The techniques are based on spectral and scattering theory. Note that this is not a systematic review, as this would require a book.