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

Understand

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.