Green Function and Self-adjoint Laplacians on Polyhedral Surfaces

Abstract

AbstractUsing Roelcke’s 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.