Singularity-Rank Signatures in Generalized Dirac Oscillators near a BTZ Horizon

Abstract

We study singular generalized Dirac-oscillator couplings in the near-horizon region of a nonextremal BTZ black hole. An effective covariant construction adapted to the stationary horizon congruence relates the two singular profiles to the proper acceleration and its radial gradient, whose leading near-horizon behaviors are $1/\rho$ and $1/{\rho }^2$. This motivates the family ${\mathcal{L}}_p(\rho )={\lambda }_0+{\beta }_p/{\rho }^p$, with $p=1,2$. The $p=1$ system has a regular singular point and is Fuchsian, whereas $p=2$ is the first irregular case; more generally, a leading ${\rho }^{-p}$ interaction with $p>1$ has Poincaré rank $p-1$. For $p=1$, a constant spinor transformation reduces the matrix-Coulomb system exactly to the Whittaker equation, while elimination in the original basis gives an equivalent confluent-Heun representation with an apparent finite singularity. For $p=2$, the local branches contain the essential factors $e^{\pm {\beta }_2/\rho }{\rho }^{-1/2}$, and their coefficient expansions exhibit generic factorial late-order growth. The finite-radius response ${\mathcal{Z}}_p(\Omega ;{\rho }_0)={\psi }_{2,p}({\rho }_0)/{\psi }_{1,p}({\rho }_0)$ obeys an exact Riccati equation whose large-damping expansion first becomes sensitive to the radial gradient of the interaction at order ${\gamma }^{-3}$. Even when the two couplings are matched at ${\rho }_0$, this response retains information about their different pole orders. The response functions provide local spinorial data for subsequent matching to the complete BTZ exterior.