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We study a real, massive Klein-Gordon field in the Poincar\'e fundamentaldomain of the $(d+1)$-dimensional anti-de Sitter (AdS) spacetime, subject to aparticular choice of dynamical boundary conditions of generalized Wentzelltype, whereby the boundary data solves a non-homogeneous, boundary Klein-Gordonequation, with the source term fixed by the normal derivative of the scalarfield at the boundary. This naturally defines a field in the conformal boundaryof the Poincar\'e fundamental domain of AdS. We completely solve the equationsfor the bulk and boundary fields and investigate the existence of bound statesolutions, motivated by the analogous problem with Robin boundary conditions,which are recovered as a limiting case. Finally, we argue that both Robin andgeneralized Wentzell boundary conditions are distinguished in the sense thatthey are invariant under the action of the isometry group of the AdS conformalboundary, a condition which ensures in addition that the total flux of energyacross the boundary vanishes.

Mode solutions for a Klein-Gordon field in anti-de Sitter spacetime with dynamical boundary conditions of Wentzell type