Incompatibility of Frequency Splitting and Spatial Localization: A Quantitative Analysis of Hegerfeldt's Theorem

TL;DR

This paper provides a quantitative analysis demonstrating that solutions to the wave equation with spatially compact support and predominantly positive frequencies must contain significant high-frequency components, highlighting a fundamental incompatibility.

Contribution

It introduces quantitative versions of Hegerfeldt's theorem for 1+1 and 3+1-dimensional wave equations, linking spatial support and frequency content.

Findings

Solutions with compact support and positive frequencies have unavoidable high-frequency components.

The results extend Hegerfeldt's theorem to quantitative bounds in multiple dimensions.

The analysis clarifies the relationship between spatial localization and frequency spectrum in wave equations.

Abstract

We prove quantitative versions of the following statement: If a solution of the 1+1-dimensional wave equation has spatially compact support and consists mainly of positive frequencies, then it must have a significant high-frequency component. Similar results are proven for the 3+1-dimensional wave equation.