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The deep reason why the equations describing axial and polar perturbations ofSchwarzschild black holes have the same spectrum is far from trivial. In thisarticle, we revisit the original proof and try to make it clearer. Stillfocusing on uncharged and non-rotating black holes, we extend the results tospacetimes including a cosmological constant, which have so far mostly beeninvestigated numerically from this perspective.

Analytical proof of the isospectrality of quasinormal modes for Schwarzschild-de Sitter and Schwarzschild-Anti de Sitter spacetimes