AI 中文总结
该研究为朗道规范的格里博夫视界构建局部谱框架，证明相关二次型指标定理，通过临界投影算子等确定谱流跳跃等，还在\(SU(2)\)刺猬模型中获稳定区间，用狄利克雷盒谱作有限体积比较，不涉数值拟合得出连续结果。
AI 中文摘要
我们为朗道规范的格里博夫视界开发了一个局部谱框架，该框架区分规范轨道投影和规范固定黑塞矩阵的简并性。在平坦环面上，空间常数鬼场形成一个残余全局色核；去除该核后，约化的法捷耶夫 - 波波夫算子是轨道范数泛函的正规莫尔斯 - 博特黑塞矩阵，而协变拉普拉斯算子定义正交联络。对于相关的仿射焦铅笔，我们证明了一个带有莫尔斯 - 博特端点的二次型指标定理。在正则孤立交叉点，临界投影算子\(P\)和可逆压缩导数\(\Gamma = P\dot{\mathcal{M}}P\)确定谱流跳跃和源鬼场预解式的主导洛朗系数；对于简单零点，它们还给出壁共法向。从正区域沿仿射线到达的交叉点具有负定的\(\Gamma\)，包括对称保护多重态，而第二次舒尔约化确定沿切向路径的极点阶数。一个投影单谐波模型检验投影的费曼 - 海尔曼关系。对于\(\mathbb{R}^3\)上的\(SU(2)\)刺猬，我们在每个耦合自旋 - 轨道通道中构造一个可归一化的阈值态，并在整个塔上最小化以获得精确的稳定区间\(-2 < g < 1\)。狄利克雷盒谱接近这些阈值并用作有限体积比较；连续结果不涉及数值拟合。
英文摘要
We develop a local spectral framework for the Landau-gauge Gribov horizon that distinguishes gauge-orbit projection from degeneracy of the gauge-fixing Hessian. On a flat torus, spatially constant ghosts form a residual global-color kernel; after removal of this kernel, the reduced Faddeev-Popov operator is the normal Morse-Bott Hessian of the orbit-norm functional, whereas the covariant Laplacian defines the orthogonal connection. For the associated affine focal pencil, we prove a quadratic-form index theorem with a Morse-Bott endpoint. At a regular isolated crossing, the critical projector $P$ and invertible compressed derivative $Γ=P\dot{\mathcal{M}}P$ determine the spectral-flow jump and leading Laurent coefficient of the sourced ghost resolvent; for a simple zero, they also give the wall conormal. Crossings reached along affine rays from the positive region have negative-definite $Γ$, including symmetry-protected multiplets, while a second Schur reduction determines pole orders along tangential paths. A projected single-harmonic model checks the projected Feynman-Hellmann relation. For an $SU(2)$ hedgehog on $\mathbb R^3$, we construct a normalizable threshold state in every coupled spin-orbit channel and minimize over the full tower to obtain the exact stability interval $-2<g<1$. Dirichlet-box spectra approach these thresholds and serve as finite-volume comparisons; no numerical fit enters the continuum result.
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