  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.
