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 Poincar\'e 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.