Krein Formula and S-Matrix for Euclidean Surfaces with Conical Singularities

Using the Krein formula for the difference of the resolvents of two self-adjoint extensions of a symmetric operator with finite deficiency indices, we establish a comparison formula for ζ-regularized determinants of two self-adjoint extensions of the Laplace operator on a Euclidean surface with conical singularities (E.s.c.s.). The ratio of two determinants is expressed through the value S(0) of the S-matrix, S(λ), of the surface. We study the asymptotic behavior of the S-matrix, give an explicit expression for S(0) relating it to the Bergman projective connection on the underlying compact Riemann surface, and derive variational formulas for S(λ) with respect to coordinates on the moduli space of E.s.c.s. with trivial holonomy.