AI 中文总结
针对分段实解析切割曲线上诺伊曼跳跃算子的行列式，提出局部角因子猜想。三个模型计算支持该猜想，还将常曲率多边形狄利克雷行列式简化为封闭对偶的行列式。虽奇异异常公式适用于对偶，但非零曲率下显式求值需解决一致化问题，且讨论了与其他工作的联系。
AI 中文摘要
我们针对分段实解析切割曲线上的诺伊曼跳跃算子的行列式，提出了一个局部角因子猜想。对于具有内角\(\pi\alpha_1,\ldots,\pi\alpha_N\)的单连通测地多边形的镜像对偶，猜想的行列式为\[ \Det_{\angle}'\cN = \frac{\length(\partial P)}2 \prod_{j=1}^N\alpha_j^{-1/2} \]。这里\(\Det_{\angle}'\)表示一个仍有待构建的内在行列式，需满足布尔热莱亚 - 弗里德兰德 - 卡佩勒胶合公式；只有其与\(\length(\partial P)\)的商纯粹依赖于角度。三个模型计算支持该猜想：一个平坦多边形及其对偶给出\(\frac12\prod_j\alpha_j^{-1/2}\)；一个分裂为全等月牙形的球形纺锤体对于两个角\(\pi\alpha\)给出\(1/(2\alpha)\)；每个具有角度\((\pi/p,\pi/q,\pi/r)\)的球形考克斯特三角形给出\(\frac12\sqrt{pqr}\)，八分体给出\(\sqrt2\)。该猜想还将常曲率多边形的狄利克雷行列式简化为其封闭对偶的行列式。虽然奇异异常公式一般适用于对偶，但在非零曲率下的显式求值需要解决其一致化问题。我们讨论了与维格曼 - 扎布罗丁和王的工作中的诺伊曼跳跃行列式以及最近关于具有角的域上库仑气体的格伦斯基算子方法的联系。
英文摘要
We formulate a local corner-factor conjecture for the determinant of the Neumann jump operator on a piecewise real-analytic cutting curve. For the mirror double of a simply connected geodesic polygon with interior angles $πα_1,\ldots,πα_N$, the conjectural determinant is \[ \Det_{\angle}'\cN = \frac{\length(\partial P)}2 \prod_{j=1}^Nα_j^{-1/2}. \] Here $\Det_{\angle}'$ denotes an intrinsic determinant, still to be constructed, that is required to satisfy a Burghelea--Friedlander--Kappeler gluing formula; only its quotient by $\length(\partial P)$ is purely angle-dependent. Three model calculations support the conjecture: a flat polygon and its double give $\frac12\prod_jα_j^{-1/2}$; a spherical spindle split into congruent lunes gives $1/(2α)$ for two angles $πα$; and every spherical Coxeter triangle with angles $(π/p,π/q,π/r)$ gives $\frac12\sqrt{pqr}$, including $\sqrt2$ for the octant. The conjecture also reduces the Dirichlet determinant of a constant-curvature polygon to that of its closed double. Although the singular anomaly formula applies to the double in general, an explicit evaluation in nonzero curvature requires solving its uniformization problem. We discuss connections with the Neumann jump determinants in the work of Wiegmann--Zabrodin and Wang and with the recent Grunsky-operator approach to Coulomb gases on domains with corners.
