Quasinormal mode expansion and the exact solution of the Cauchy problem for wave equations

TL;DR

This paper introduces a Laplace-transform method to solve wave equations with effective potentials, clarifies the role of quasinormal modes in bounded regions, and resolves divergence issues in their expansions, supported by numerical simulations.

Contribution

It provides an exact solution framework for wave equations using quasinormal modes that avoids divergence and clarifies their spatial validity, with numerical validation.

Findings

Quasinormal modes appear in solutions only in a spatially truncated form.

Coefficients of quasinormal modes are uniquely determined by initial data.

Numerical simulations support the theoretical predictions.

Abstract

Solutions for a class of wave equations with effective potentials are obtained by a method of a Laplace-transform. Quasinormal modes appear naturally in the solutions only in a spatially truncated form; their coefficients are uniquely determined by the initial data and are constant only in some region of spacetime -- in contrast to normal modes. This solves the problem of divergence of the usual expansion into spatially unbounded quasinormal modes and a contradiction with the causal propagation of signals. It also partially answers the question about the region of validity of the expansion. Results of numerical simulations are presented. They fully support the theoretical predictions.