Price's law on Minkowski space in the presence of an inverse square potential

TL;DR

This paper establishes Price's law decay rates for wave solutions in Minkowski space with an inverse square potential, revealing two distinct decay regimes depending on the spatial momentum, and provides a detailed analysis at their interface.

Contribution

It extends Price's law to singular models with inverse square potentials, identifying two decay regimes and analyzing their interface in Minkowski space.

Findings

Solutions exhibit two different decay rates at timelike infinity.

The decay rates depend on whether the spatial momentum is zero or non-zero.

The analysis precisely describes solutions at the interface of the two regimes.

Abstract

We consider the pointwise decay of solutions to wave-type equations in two model singular settings. Our main result is a form of Price's law for solutions of the massless Dirac-Coulomb system in (3+1)-dimensions. Using identical techniques, we prove a similar theorem for the wave equation on Minkowski space with an inverse square potential. One novel feature of these singular models is that solutions exhibit two different leading decay rates at timelike infinity in two regimes, distinguished by whether the spatial momentum along a curve which approaches timelike infinity is zero or non-zero. An important feature of our analysis is that it yields a precise description of solutions at the interface of these two regions which comprise the whole of timelike infinity.