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

TL;DR

This paper critically examines various Darboux transformations in Schwarzschild black hole perturbation theory, revealing that only some preserve the quasinormal mode spectra, and highlights limitations in the generalizations of these transformations.

Contribution

The study clarifies the spectral properties of standard, binary, and generalized Darboux transformations, showing that the generalized version often violates isospectrality, thus challenging their broad applicability.

Findings

DT and BDT are isospectral transformations.

GDT violates isospectrality and is only valid at fixed frequencies.

Frequency-dependent potentials also do not preserve spectra.

Abstract

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