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We investigate the superradiant instability for a charged scalar field in a$D$-dimensional small Reissner-Nordstr\"om-Anti-de Sitter (RN-AdS) black hole.Firstly, we solve the charged Klein-Gordon equation analytically by a matchingmethod. We show that the general $D$-dimensional quasinormal frequencies dependon the relation between the angular momentum quantum number, $\ell$, and $D$.When $\ell$ is a (non-negative) integer multiple of $D-3$, i.e $\ell=p(D-3)$,we find an analytical quasinormal frequency formula adding a purely imaginarycorrection to the AdS normal frequencies. This is the case for all $\ell$ modesin $D=4$. For general $D$ there are two more cases: i) when $\ell$ obeys$\ell=(p+\frac{1}{2})(D-3)$, which can occur for odd $D$, we observe that thematching method fails, since the near and far region solutions have differentfunctional dependences; ii) for all other cases, the analytical quasinormalfrequency formula gives a complex correction to the AdS normal frequencies.Secondly, we perform a numerical calculation which confirms the analyticalformulas obtained with the matching method and allows us to explore the casewhere that method failed. In the latter case, as in the former, we verify thatall $\ell$ modes for all $D$ may become superradiant, which contradicts aprevious claim in the literature.

Superradiant instabilities in a $D$-dimensional small Reissner-Nordstr\"om-Anti-de Sitter black hole