  The wave equation $\square_{g_{M,a}}\psi=0$ on subextremal Kerr spacetimes
$(\mathcal{M}_{M,a},g_{M,a})$, $0<|a|<M$, does not admit real mode solutions,
as was established by Shlapentokh-Rothman. In this paper, we show that the
absence of real modes does not persist under the addition of an arbitrary
short-range non-negative potential $V$ to the wave equation or under changes of
the metric $g_{M,a}$ in the far away region of $\mathcal{M}_{M,a}$ (retaining
the causality of $T$ there).
  In particular, we first establish, for any $0<|a|<M$, the existence of real
mode solutions $\psi$ to equation $\square_{g_{M,a}}\psi-V\psi=0$, for a
suitably chosen time-independent real potential $V$ with compact support in
space, satisfying $V\ge0$. Exponentially growing modes are also obtained after
perturbing the potential $V$. Then, as an application of the above result, we
construct a family of spacetimes $(\mathcal{M}_{M,a},g_{M,a}^{(def)})$ which
are compact in space perturbations of $(\mathcal{M}_{M,a},g_{M,a})$, have the
same symmetries as $(\mathcal{M}_{M,a},g_{M,a})$ and moreover admit real and
exponentially growing modes. These spacetimes contain stably trapped null
geodesics, but we also construct a more complicated family of spacetimes with
normally hyperbolic trapped set, admitting real and exponentially growing
modes, at the expense of having conic asymptotics.
  The above results are in contrast with the case of stationary asymptotically
flat (or conic) spacetimes $(\mathcal{M},g)$ with a globally timelike Killing
field $T$, where real modes for equation $\square_{g}\psi-V\psi=0$ are always
absent, giving a useful zero-frequency continuity criterion for showing
stability for a smooth family of equations $\square_{g}\psi-V_{\lambda}\psi=0$,
with $\lambda\in[0,1]$ and $V_{0}=0$. We show explicitly that this criterion
fails on Kerr spacetime.
