Computing resonances of perturbed Schr\"odinger equations: Application to Reissner-Norsdtr\"om-de Sitter black holes

We present a numerical method for computing resonances of one-dimensional Schr\"odinger equations perturbed by a compactly supported potential, via finding zeros of the Wronskian associated with Jost solutions of the reference equation, computed through the resolution of Cauchy problems. All resonances located in a given domain are found efficiently using a defeated Newton algorithm. A key ingredient of the method is the choice of reference potential for which Jost solutions are known, which removes spurious resonances often encountered numerically. We test this method on three types of reference potentials and perturbations thereof: P\"oschl-Teller potentials, exponentially decaying potentials, and potentials associated with Reissner-Nordstr\"omde Sitter black holes. In particular we study the impact of perturbations on the resonances, and the stability of small resonances under perturbation. As an illustration, we use the method to numerically study the strong cosmic censorship hypothesis.