Krein Formula and S-Matrix for Euclidean Surfaces with Conical Singularities
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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.
KeywordsFlat Laplacian Determinant Conical singularities Complex structure
Mathematics Subject Classification (2000)58J52 (30F30)
The research of LH was partly supported by the ANR programs NONaa and Teichmüller.
The research of AK was supported by NSERC. AK thanks Hausdorff Research Institute for Mathematics (Bonn) and Laboratoire de Mathématiques Jean Leray (Nantes) for hospitality. AK also thanks the MATPYL program for supporting his travel and stay in Nantes, where this research began.
We acknowledge useful conversations with G. Carron and with D. Korotkin, whose advice in particular helped us simplify some constructions from Sect. 4.2.
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