An Explicit Model for Conic Laplacians on \(\mb P^1\)

We study an explicit model of a conic Laplacian on $\mb P^1$ with cone angle $2\pi n$ 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.