An arXiv preprint examining spinor equations near a BTZ black-hole horizon finds that matching two singular interactions at an outer radius does not make them locally indistinguishable. A finite-radius spinor response still retains information about whether the model uses an inverse-radial (p = 1) or inverse-square (p = 2) coupling.
Two idealized near-horizon interactions
This is a deterministic mathematical comparison, not an empirical study. It asks how singular generalized-oscillator couplings change the analytical class and local response of separated one-body spinor equations. The BTZ geometry and tetrad are held fixed, while the couplings are treated as leading singular terms in an effective interaction.
The two profiles are motivated by different near-horizon geometric behavior: proper acceleration has a leading 1/ρ form, while its radial gradient has a leading 1/ρ² form.
Two mathematical regimes
In the p = 1 case, the radial system belongs to the regular-singular, or Fuchsian, class—a standard form of local singular behavior. The equations reduce exactly to a Whittaker spinor, with an equivalent confluent-Heun representation. The original component basis also contains an apparent finite singularity.
The p = 2 inverse-square case is the first irregular member. Its local solutions contain essential branches, with behavior of the form e^{±β₂/ρ}ρ⁻¹ᐟ². Its generic formal expansion has factorial late-order growth and is asymptotic rather than convergent.
More broadly, a leading ρ⁻ᵖ interaction with p > 1 has Poincaré rank p − 1, an index of the singularity’s irregularity, and essential behavior of exponential form. The result ties pole order to the analytical class of the local system.
The local response keeps the distinction
To compare the cases at a finite radius, the study follows a projective response: the ratio of the two spinor components. This ratio obeys an exact Riccati equation. In a formal large-damping expansion, the interaction value first enters at order γ⁻², while its radial gradient first enters at order γ⁻³.
That hierarchy supplies a formal local discriminator. Even after the outer values of the two interactions are matched, the next response datum retains direct information about pole order. In the calculation’s scaling, the absolute splitting of the projected responses at their intersection is linear in |η|, while the relative splitting is quadratic.
Where the result stops
The finding is local, not a global spectrum. A finite-radius spinor ratio is near-horizon matching data, not by itself a quasinormal-mode condition or a complete BTZ spectral condition. Turning it into global spectral information would require matching to the full exterior and admissible boundary data.
The calculation also does not establish that the locally bounded essential branch is the physical infalling branch. The branch is selected for finite-radius response data rather than imposed as a global quasinormal-mode or infalling boundary condition.
No empirical sample is described, and no data were created or analyzed. The generic inverse-square series is asymptotic rather than convergent, so the preprint should be read as a local analytic classification and matching result, not as a global BTZ spectrum.
Paper data and sources
Original title: Singularity-Rank Signatures in Generalized Dirac Oscillators near a BTZ Horizon
Authors: João V. R. Alencar, Allan R. P. Moreira, João B. R. Silva, Abdullah Guvendi
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-20
DOI: Not available
Original paper · Full text