# Green function and self-adjoint Laplacians on polyhedral surfaces

## TL;DR

This paper explicitly constructs a basis for the kernel of the adjoint Laplacian on polyhedral surfaces, computes the S-matrix at zero spectral parameter, and studies self-adjoint extensions, revealing geometry-dependent behaviors.

## Contribution

It introduces explicit constructions and computations of the Green function and S-matrix on polyhedral surfaces, advancing understanding of self-adjoint Laplacians in geometric analysis.

## Key findings

- The basis in the kernel of the adjoint Laplacian is explicitly constructed.
- The S-matrix at zero spectral parameter is computed for polyhedral surfaces.
- Behavior of the S-matrix at zero is sensitive to the polyhedron's geometry.

## Abstract

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.