AI 中文总结
本文研究非极端BTZ黑洞近视界区域的奇异广义狄拉克振子耦合，构造含p=1、2的ℒₚ族系，分析其奇点性质、响应函数并为匹配BTZ外域提供局域旋量数据。
AI 中文摘要
我们研究非极端BTZ黑洞近视界区域内奇异广义狄拉克振子耦合项。针对稳态视界同余的有效协变构造将两种奇异轮廓与固有加速度及其径向梯度关联，其近视界主导行为为1/ρ和1/ρ²，由此引出族系ℒₚ(ρ)=λ₀+βₚ/ρᵖ，其中p=1,2。p=1的系统具有正则奇点且为富克斯型，而p=2是首个不规则情形；更一般地，p>1的主导ρ⁻ᵖ相互作用具有庞加莱秩p-1。对于p=1，常数旋量变换将矩阵库仑系统精确约化为惠特克方程，而在原基矢中消元得到等价的合流黑恩（confluent-Heun）表示，带有表观有限奇点。对于p=2，局部分支包含本质因子e^(±β₂/ρ)ρ^(-1/2)，其系数展开呈现典型的阶乘型晚期增长。有限半径响应ℤₚ(Ω;ρ₀)=ψ₂,ₚ(ρ₀)/ψ₁,ₚ(ρ₀)满足精确的里卡蒂方程，其大阻尼展开在γ⁻³阶首次对相互作用的径向梯度敏感；即使两个耦合项在ρ₀处匹配，该响应仍保留其不同极点阶的信息。这些响应函数为后续匹配到完整BTZ外域提供局域旋量数据。
英文摘要
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.
