A bstract We study the scalar perturbations of asymptotically flat extremal Reissner-Nordström black holes via the quantum Seiberg-Witten geometry of N N = 2 SU (2) gauge theory with N f = 2 flavors. The radial master equation, governed by a double confluent Heun equation, is exactly mapped to the quantum Seiberg-Witten curve, providing an exact quantization condition derived from the non-perturbative Nekrasov-Shatashvili free energy. Analytically, this exact dictionary unveils precise gauge-theoretic interpretations for critical physical thresholds, demonstrating that the superradiance and mass decoupling limits naturally reduce the master equation to the Whittaker equation and the reduced doubly confluent Heun equation (the latter corresponds to the SW geometry of the N N = 2 SU (2) gauge theory with N f = 1), respectively. At the strict extremal limit, the coalescence of horizons induces a topological singularity that complicates the spectral analysis. By accommodating this irregular singularity, our geometric framework resolves the singularity coalescence and enables the extraction of the discrete global quasinormal mode. As our main contribution, we provide non-perturbative evaluation of the quasinormal modes spectrum for simultaneously charged and massive scalar fields directly at strict extremity. Furthermore, our analytical results reproduce numerical benchmarks for both neutral and charged massless probes, and naturally capture quasi-resonance behaviors.
Wang et al. (Mon,) studied this question.