Isospectrality of Darboux-related axial potentials for arbitrary Riccati constant in Schwarzschild

Establish a complete proof that, in Schwarzschild spacetime, mode equality (isospectrality) holds for all axial potentials obtained via Darboux transformations generated by solutions W of the Riccati equation W_{,σ} − W^2 + V = c for an arbitrary constant c, i.e., that for every c the transformed potential v = V + 2 W_{,σ} yields the same mode spectrum as the original potential V.

Background

The paper discusses Darboux covariance as a hidden symmetry in black hole perturbation theory, relating infinite families of master equations and potentials through Darboux transformations. In the Schwarzschild exterior, these transformations imply that the associated axial potentials are isospectral in the frequency domain. The Darboux generating function W satisfies a Riccati equation W_{,σ} − W^2 + V = c, where c is a constant, and the transformed potential is v = V + 2 W_{,σ}.

While specific families of solutions are known to be isospectral, the authors note that a general, comprehensive proof of mode equality for every choice of the constant c is lacking. Prior work has demonstrated the existence of infinitely many such solutions, and related invariants (e.g., Korteweg–de Vries integrals) link to greybody factors, but a full proof covering all c remains to be established.