Anabalon, A., Astefanesei, D., & Oliva, J. (2015). Hairy black hole stability in AdS, quantum mechanics on the halfline and holography. J. High Energy Phys., (10), 15 pp.
Abstract: We consider the linear stability of 4dimensional hairy black holes with mixed boundary conditions in Antide Sitter spacetinie. We focus on the mass of scalar fields around the maximally supersymmetric vacuum of the gauged N = 8 supergravity in four dimensions, m(2) = 2l(2). It is shown that the Schrodinger operator on the halfline, governing the S2, H2 or R2 invariant mode around the hairy black hole, allows for nontrivial selfadjoint extensions and each of them corresponds to a class of mixed boundary conditions in the gravitational theory. Discarding the selfadjoint 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 Schrodinger operator resembling the estimate of Simon for Schrodinger 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.
