  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.
