Modified Black Hole Potentials and Their Korteweg-de Vries Integrals: Instabilities and Beyond

Black Hole (BH) Quasi-Normal Modes (QNMs) and Greybody Factors (GBFs) are key signatures of BH dynamics that are crucial for testing fundamental physics via gravitational waves. Recent studies of the BH pseudospectrum have revealed instabilities in QNMs. Here, we introduce a new perspective using hidden symmetries in the BH dynamics, specifically the Korteweg-de Vries (KdV) integrals - an infinite series of conserved quantities. By analyzing modified BH potentials, we find strong evidence that KdV integrals are valuable indicators for detecting instabilities in QNMs and GBFs.