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We consider the linear stability of $4$-dimensional hairy black holes withmixed boundary conditions in Anti-de Sitter spacetime. We focus on the mass ofscalar fields around the maximally supersymmetric vacuum of the gauged$\mathcal{N}=8$ supergravity in four dimensions, $m^{2}=-2l^{-2}$. It is shownthat 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, allowsfor non-trivial self-adjoint extensions and each of them correspons to a classof mixed boundary conditions in the gravitational theory. Discarding theself-adjoint extensions with a negative mode impose a restriction on theseboundary conditions. The restriction is given in terms of an integral of thepotential in the Schr\"{o}dinger operator resembling the estimate of Simon forSchr\"{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 dualeffective potential.

Hairy Black Hole Stability in AdS, Quantum Mechanics on the Half-Line and Holography