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We use Krein formula and the S-matrix formalism to give formulas for thezeta-regularized determinant of non-Friedrichs extensions of the Laplacian onEuclidean surfaces with Conical Singularities. This formula involves S(0) andwe show that the latter can be expressed using the Bergman projectiveconnection on the underlying Riemann surface.

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