The Klein-Gordon equation on the toric adS-Schwarzschild black hole

We consider the Klein-Gordon equation on the exterior of the toric anti de-Sitter Schwarzschild black hole with Dirichlet, Neumann and Robin boundary conditions at $\mathcal{I}$. We define a non-degenerate energy for the equation which controls the renormalised $H^1$ norm of the field. We then establish both decay and integrated decay of this energy through vector field methods. Finally we demonstrate the necessity of `losing a derivative' in the integrated energy estimate through the construction of a Gaussian beam staying in the exterior of the event horizon for arbitrary long co-ordinate time.