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In this article we take a close look at three types of transformations usablein the Schwarzschild black hole perturbation theory: a standard (DT), a binary(BDT) and a generalized (GDT) Darboux transformations. In particular, wediscuss the absolutely crucial property of {\em isospectrality} of theaforementioned transformations which guarantees that the quasinormal mode (QNM)spectra of potentials, related via the transformation, completely coincide. Wedemonstrate that, while the first two types of the Darboux transformations (DTand BDT) are indeed isospectral, the situation is wildly different for the GDT:it violates the isospectrality requirement and is therefore only valid for thesolutions with just one fixed frequency. Furthermore, it is shown that althoughin this case the GDT does provide a relationship between two arbitrarypotentials (a short-ranged and a long-ranged potentials relationship being justa trivial example), this relationship ends up being completely formal. Finally,we consider frequency-dependent potentials. A new generalization of the Darbouxtransformation is constructed for them and it is proven (on a concrete example)that such transformations are also not isospectral. In short, we demonstratehow a little, almost incorporeal flaw may become a major problem for anotherwise perfectly admirable goal of mathematical generalization.

physics letters a

A look at the generalized Darboux transformations for the quasinormal  spectra in Schwarzschild black hole perturbation theory: just how general  should it be?