An analysis has ruled out every quasinormal mode with a nonzero real frequency in a specified class of Teukolsky-equation solutions: under the paper's conditions, the only allowed solution is zero. The work studies those solutions in the Kerr domain of outer communication, with prescribed behavior at the event horizon and at infinity. The result therefore concerns wave patterns selected by the equation and those two boundary requirements.
The central theorem applies to modes with nonzero real frequency. Every quasinormal mode meeting the stated Teukolsky conditions is the zero solution. Here, mode stability means that the boundary conditions permit no nonzero mode in the class being examined. The conclusion is exact within that setup.
A proof built around frequencies
The paper's main methodological move is to replace the Whiting transform with the standard Fourier transform and distribution theory. The authors present that combination as a simpler route to a real-frequency mode-stability proof for subextremal Kerr black holes. The proof stays within the Teukolsky equation and the prescribed asymptotic conditions.
To develop the argument, the authors use Fredholm theory to obtain dual solutions in dual function spaces, including distributions supported outside the event horizon. These dual solutions form the companion side of the calculation. The proof then follows their behavior near the horizon and in the asymptotic region, setting up a frequency-space boundary pairing.
Fourier-transform analysis distinguishes a leading local horizon singularity from a subleading conormal singularity in the dual solution. In ordinary language, the second is a structured singular behavior that appears at a lower order than the leading term. That distinction matters because the proof targets the coefficient of the leading singularity and seeks to show that it vanishes.
The decisive calculation is a frequency-space boundary pairing, a way of combining the relevant boundary behavior after the equation has been transformed into frequencies. For negative frequency, the pairing is computed as a sum of two absolute-value squares, one associated with the second singular point and the other with asymptotic behavior. Because such squares are nonnegative, the calculation gives the condition needed for the leading singularity coefficient to vanish. This is the bridge between the local horizon analysis and the final zero-solution result.
After that coefficient vanishes, microlocal regularity, which controls the remaining singular behavior, and an energy estimate at a regular singular point yield the zero solution throughout the analyzed radial region. Put plainly, the proof uses the disappearance of the leading singular term to carry the vanishing conclusion across the region under study. The authors use this unique-continuation step to complete the mode-stability argument.
Where the argument stops
The method has a clear boundary. The paper analyzes dual modes in Kerr–de Sitter, but it does not obtain a mode-stability application there because the stated argument breaks down in that setting. The central real-frequency conclusion is therefore limited to the Kerr problem and the asymptotic conditions described in the paper.
A related comparison with the Klein–Gordon equation exposes another limit. For the stated sufficiently massive modification, the frequency-space boundary-pairing argument does not go through when the absolute value of the mass parameter is at least as large as the absolute value of the real mode frequency, because that condition removes the regular singular points used by the argument. The pairing is therefore tied to particular equation and singularity structures.
A separate horizon construction
In a separate zero-spin wave-equation case, the paper considers extending a mode across the horizons. A quasinormal mode that is smooth across the future event horizon can extend through the past event horizon as a distributional solution. This special construction is explicitly tied to the zero-spin case and does not widen the central mode-stability theorem beyond its stated setting.
The paper's status
The work is a preprint identified in its front matter as arXiv version 2 dated 21 Sep 2026. The authors report support from Swedish and Wallenberg sources, the National Science Foundation and the Simons Foundation.
Paper data and sources
Original title: Dual modes in Kerr spacetimes and the Whiting transform: Mode stability revisited
Authors: Oliver Petersen, András Vasy
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
Original paper · Full text