The Darboux transformation in black hole perturbation theory

The Darboux transformation between ordinary differential equations is a 19th century technique that has seen wide use in quantum theory for producing exactly solvable potentials for the Schr\"odinger equation with specific spectral properties. In this paper we show that the same transformation appears in black hole theory, relating, for instance, the Zerilli and Regge-Wheeler equations for axial and polar Schwarzschild perturbations. The transformation reveals these two equations to be isospectral, a well known result whose method has been repeatedly reintroduced under different names. We highlight the key role that the so-called algebraically special solutions play in the black hole Darboux theory and show that a similar relation exists between the Chandrasekhar-Detweiler equations for Kerr perturbations. Finally, we discuss the limitations of the method when dealing with long-range potentials and explore the possibilities offered by a generalised Darboux transformation.