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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 theabsence of real modes does not persist under the addition of an arbitraryshort-range non-negative potential $V$ to the wave equation or under changes ofthe metric $g_{M,a}$ in the far away region of $\mathcal{M}_{M,a}$ (retainingthe causality of $T$ there).  In particular, we first establish, for any $0<|a|<M$, the existence of realmode solutions $\psi$ to equation $\square_{g_{M,a}}\psi-V\psi=0$, for asuitably chosen time-independent real potential $V$ with compact support inspace, satisfying $V\ge0$. Exponentially growing modes are also obtained afterperturbing the potential $V$. Then, as an application of the above result, weconstruct a family of spacetimes $(\mathcal{M}_{M,a},g_{M,a}^{(def)})$ whichare compact in space perturbations of $(\mathcal{M}_{M,a},g_{M,a})$, have thesame symmetries as $(\mathcal{M}_{M,a},g_{M,a})$ and moreover admit real andexponentially growing modes. These spacetimes contain stably trapped nullgeodesics, but we also construct a more complicated family of spacetimes withnormally hyperbolic trapped set, admitting real and exponentially growingmodes, at the expense of having conic asymptotics.  The above results are in contrast with the case of stationary asymptoticallyflat (or conic) spacetimes $(\mathcal{M},g)$ with a globally timelike Killingfield $T$, where real modes for equation $\square_{g}\psi-V\psi=0$ are alwaysabsent, giving a useful zero-frequency continuity criterion for showingstability 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 criterionfails on Kerr spacetime.

Superradiant instabilities for short-range non-negative potentials on Kerr spacetimes and applications