Abstract
We consider a wave equation with a potential on the half-line as a model problem for wave propagation close to an extremal horizon or the asymptotically flat end of a black hole spacetime. We propose a definition of quasinormal frequencies (QNFs) as eigenvalues of the generator of time translations for a null foliation, acting on an appropriate (Gevrey based) Hilbert space. We show that this QNF spectrum is discrete in a subset of C, which includes the region {Rs>−b,Is>K} for any b > 0 and some K = K(b) ≫ 1. As a corollary, we establish the meromorphicity of the scattering resolvent in a sector args<φ0 for some φ0>2π3 and show that the poles occur only at quasinormal frequencies according to our definition. Finally, we show that QNFs computed by the continued fraction method of Leaver are necessarily QNFs according to our new definition. A companion to our paper (arXiv:1910.08481), which deals with the QNFs of the wave equation on the extremal Reissner–Nördstrom black hole.
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