Stability of asymptotically Schrödinger RN black hole and superconductivity

We perform a perturbative (near-critical) analysis of the stability of an asymptotically Schrödinger Reissner-Nordström black hole with respect to generation of charged scalar hair. We find that apart from the expected instability at low temperatures typical of holographic models of superconductivity, in the presence of certain operators a similar instability appears as well at high temperatures. We propose that the reason for such a phase diagram could be due to peculiar features of the dual gauge theory or the failure of the model to provide a consistent holographic dual of a non-relativistic superconductor.