Singularity-Rank Signatures in Generalized Dirac Oscillators near a BTZ Horizon
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 $1/ρ$ and $1/ρ^2$. This motivates the family $\mathcal{L}_p(ρ)=λ_0+β_p/ρ^p$, 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 $ρ^{-p}$ 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 $e^{\pmβ_2/ρ}ρ^{-1/2}$, and their coefficient expansions exhibit generic factorial late-order growth. The finite-radius response $\mathcal{Z}_p(Ω;ρ_0)=ψ_{2,p}(ρ_0)/ψ_{1,p}(ρ_0)$ obeys an exact Riccati equation whose large-damping expansion first becomes sensitive to the radial gradient of the interaction at order $γ^{-3}$. Even when the two couplings are matched at $ρ_0$, this response retains information about their different pole orders. The response functions provide local spinorial data for subsequent matching to the complete BTZ exterior.