On the Completeness of the Quasinormal Modes of the Pöschl–Teller Potential

The completeness of the quasinormal modes of the wave equation with Pöschl–Teller potential is investigated. A main result is that after a large enough time t ₀, the solutions of this equation corresponding to &C ^∞-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.

Authors and Affiliations

1. Max Planck Institute for Gravitational Physics, Albert Einstein Institute, D-14476 Golm, Germany.¶E-mail: horst@aei-potsdam.mpg.de, , , , , , DE

   Horst R. Beyer

Received: 10 March 1998 / Accepted: 1 February 1999

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