A model problem for quasinormal ringdown on asymptotically flat or extremal black holes

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 $\mathbb{C}$ which includes the region $\{$Re$(s) >-b$, $|$Im $(s)|> K\}$ for any $b>0$ and some $K=K(b) \gg 1$. As a corollary we establish the meromorphicity of the scattering resolvent in a sector $|$arg$(s)| <\varphi_0$ for some $\varphi_0 > \frac{2\pi}{3}$, and show that the poles occur only at quasinormal frequencies according to our definition. This result applies in situations where the method of complex scaling cannot be directly applied, as our potentials need not be analytic. Finally, we show that QNFs computed by the continued fraction method of Leaver are necessarily QNFs according to our new definition. This paper is a companion to [D. Gajic and C. Warnick, Quasinormal modes in extremal Reissner-Nordstr\"om spacetimes, preprint (2019)], which deals with the QNFs of the wave equation on the extremal Reissner-Nordstr\"om black hole.