In this talk, I will explain how to apply some standard integrability tools like Thermodynamic Bethe Ansatz (TBA) or Quantum Wronskian to realistic black holes models. In particular, the ringdown phase of black holes merging can be described by linear perturbation theory, especially the quasinormal modes. The perturbations obey relatively simple Ordinary Differential Equations (ODEs), which we closely analyse through the so-called ODE/IM correspondence with 2D Integrable Models (IMs). These ODEs are also shared by 4D N=2 SU(2) gauge theories, hence our treatment explains the use of the latter for the characterization and computation of quasinormal modes. A promising new method for the computation of these is also the Thermodynamic Bethe Ansatz. Based on arXiv:1908.08030, arXiv:2112.11434, arXiv:2208.14031 and arXiv:23**.***** .

https://indico.math.cnrs.fr/event/10623/