Superradiant stability of five and six-dimensional extremal Reissner-Nordstrom black holes

We revisit the superradiant stability of five and six-dimensional extremal Reissner-Nordstrom black holes under charged massive scalar perturbation with a new analytical method. In each case, it is analytically proved that the effective potential experienced by the scalar perturbation has only one maximum outside the black hole horizon and no potential well exists for the superradiance modes. So the five and six-dimensional extremal Reissner-Nordstrom black holes are superradiantly stable. The new method we developed is based on the Descartes' rule of signs for the polynomial equations. Our result provides a complementary support of previous studies on the stability of higher dimensional extremal Reissner-Nordstrom black holes based on numerical methods.