Green function and self-adjoint Laplacians on polyhedral surfaces

Using Roelcke formula for the Green function, we explicitly construct a basis in the kernel of the adjoint Laplacian on a compact polyhedral surface \(X\) and compute the \(S\)-matrix of \(X\) at the zero value of the spectral parameter. We apply these results to study various self-adjoint extensions of a symmetric Laplacian on a compact polyhedral surface of genus two with a single conical point. It turns out that the behaviour of the \(S\)-matrix at the zero value of the spectral parameter is sensitive to the geometry of the polyhedron.