On the Completeness of the Quasinormal Modes of the Poeschl-Teller Potential

TL;DR

This paper investigates the completeness of quasinormal modes for the wave equation with Poeschl-Teller potential, showing conditions under which solutions can be expanded in these modes and highlighting cases of non-absolute convergence.

Contribution

It provides explicit conditions and estimates for the convergence of quasinormal mode expansions, demonstrating both convergence and non-convergence scenarios.

Findings

Solutions can be expanded in quasinormal modes after a certain time

Explicit estimates for the time threshold $t_0$ are provided

Non-absolute convergence occurs at early times with zero distance between support and observation

Abstract

The completeness of the quasinormal modes of the wave equation with Poeschl-Teller potential is investigated. A main result is that after a large enough time $t_0$, the solutions of this equation corresponding to $C^{\infty}$-data with compact support can be expanded uniformly in time with respect to the quasinormal modes, thereby leading to absolutely convergent series. Explicit estimates for $t_0$ depending on both the support of the data and the point of observation are given. For the particular case of an ``early'' time and zero distance between the support of the data and observational point, it is shown that the corresponding series is not absolutely convergent, and hence that there is no associated sum which is independent of the order of summation.