AI 中文总结
研究\(\mb P^1\)上特定圆锥拉普拉斯算子模型，通过傅里叶分解简化方程，利用勒让德连接公式诱导胶合映射，计算出弗里德里希斯谱等，可从边界连接映射显式恢复外尔函数及\(S\)-矩阵。
AI 中文摘要
我们研究了\(\mb P^1\)上圆锥拉普拉斯算子的显式模型，在\(0\)和\(\infty\)处的锥角为\(2\pi n\)。利用傅里叶分解，将特征值方程简化为一族关联勒让德方程，并描述了相应的渐近边界数据。经典勒让德连接公式在两个圆锥点的边界数据之间诱导了一个显式胶合映射。我们计算了弗里德里希斯谱和特征函数，并表明可以从这个边界连接映射中显式恢复外尔函数以及相关的\(S\)-矩阵。
英文摘要
We study an explicit model of a conic Laplacian on \(\mb P^1\) with cone angle \(2πn\) at \(0\) and \(\infty\). Using Fourier decomposition, we reduce the eigenvalue equation to a family of associated Legendre equations and describe the corresponding asymptotic boundary data. The classical Legendre connection formulas induce an explicit gluing map between the boundary data at the two conic points. We compute the Friedrichs spectrum and eigenfunctions and show that the Weyl function, and hence the associated \(S\)-matrix, can be recovered explicitly from this boundary connection map.
