The Darboux transformation in black hole perturbation theory

In an extreme mass ratio inspiral (EMRI), we consider a solar mass black hole as a perturbation on a supermassive black hole's (SMBH) spacetime. Both axial and polar perturbations lead to the same equation with different potentials. In the 1970s, Chandrasekhar found by brute force calculation that these potentials are isospectral. In the 90s, when one of us visited to speak with Chandra, he mentioned his dissappointment that people didn't consider this an important avenue of research and instead only used the transformation for convenience. What has become known as the Chandrasekhar transformation has been known for a century to mathematicians as the Darboux transformation and plays an important role in scattering theory and supersymmetry. Here we discuss using the classical and generalized Darboux transformations to find algebraically special solutions and relate potentials in black hole perturbation theory, and more broadly how they might be used to find new potentials that are shorter ranged and therefore better for computation.