We consider the linear stability of $4$-dimensional hairy black holes with mixed boundary conditions in Anti-de Sitter spacetime. We focus on the mass of scalar fields around the maximally supersymmetric vacuum of the gauged $mathcal{N}=8$ supergravity in four dimensions, $m^{2}=-2l^{-2}$. It is shown that the Schr{o}dinger operator on the half-line, governing the $S^{2}$, $H^{2}$ or $mathbb{R}^{2}$ invariant mode around the hairy black hole, allows for non-trivial self-adjoint extensions and each of them correspons to a class of mixed boundary conditions in the gravitational theory. Discarding the self-adjoint extensions with a negative mode impose a restriction on these boundary conditions. The restriction is given in terms of an integral of the potential in the Schr{o}dinger operator resembling the estimate of Simon for Schr{o}dinger operators on the real line. In the context of AdS/CFT duality, our result has a natural interpretation in terms of the field theory dual effective potential.
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