Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Albeverio, S., Gesztesy, F., Hoegh-Krohn, R., Holden, H.: Solvable Models in Quantum Mechanics. AMS Chelsea Publishing, Providence (2005)
Aurell, E., Salomonson, P.: On functional determinants of Laplacians in polygons and simplicial complexes. Commun. Math. Phys. 165(2), 233–259 (1994)
Bordag, M., Dowker, S., Kirsten, K.: Heat-kernels and functional determinants on the generalized cone. Commun. Math. Phys. 182(2), 371–393 (1996)
Birman, M.Sh., Solomjak, M.Z.: Spectral Theory of Selfadjoint Operators in Hilbert Space. Mathematics and Its Applications (Soviet Series). Reidel, Dordrecht (1987). Translated from the 1980 Russian original by S. Khrushchëv and V. Peller
Burghelea, D., Friedlander, L., Kappeler, T.: Meyer-Vietoris type formula for determinants of elliptic differential operators. J. Funct. Anal. 107(1), 34–65 (1992)
Carron, G.: Déterminant relatif et la fonction Xi. Am. J. Math. 124(2), 307–352 (2002)
Cheeger, J.: Spectral geometry of singular Riemannian spaces. J. Differ. Geom. 18(4), 575–657 (1983)
Cheeger, J., Taylor, M.: On the diffraction of waves by conical singularities. I. Commun. Pure Appl. Math. 35(3), 275–331 (1982)
Fay, J.: Kernel Functions, Analytic Torsion and Moduli Space. Memoirs of the AMS, vol. 92 (1992)
Forman, R.: Functional determinants and geometry. Invent. Math. 88, 447–493 (1987)
Forni, G.: Sobolev regularity of solutions of the cohomological equation (2007). arXiv:0707.0940v2
Gil, J., Krainer, T., Mendoza, G.: Dynamics on Grassmannians and resolvents of cone operators (2009). arXiv:0907.0023v1
Gesztesy, F., Simon, B.: The xi function. Acta Math. 176, 49–71 (1996)
Grubb, G.: Distributions and Operators. Graduate Texts in Mathematics, vol. 252. Springer, New York (2009)
Hassell, A., Zelditch, S.: Determinants of Laplacians in exterior domains. Int. Math. Res. Not. 18, 971–1004 (1999)
Hillairet, L.: Contribution of periodic diffractive geodesics. J. Funct. Anal. 226(1), 48–89 (2005)
Kirsten, K., Loya, P., Park, J.: Functional determinants for general self-adjoint extensions of Laplace-type operators resulting from the generalized cone. Manuscr. Math. 125, 95–126 (2008)
Kokotov, A.: Polyhedral surfaces and determinants of Laplacians. Proc. AMS (2012, to appear)
Kokotov, A., Korotkin, D.: Tau-functions on spaces of abelian differentials and higher genus generalizations of Ray-Singer formula. J. Differ. Geom. 82(1), 35–100 (2009)
Kontsevich, M., Zorich, A.: Connected components of the moduli spaces of Abelian differentials with prescribed singularities. Invent. Math. 153, 631–678 (2003)
Kostrykin, V., Schrader, R.: Laplacians on metric graphs: eigenvalues, resolvents and semigroups. In: Quantum graphs and their applications. Contemp. Math., vol. 415, pp. 201–225. Amer. Math. Soc., Providence (2006)
Loya, P., McDonald, P., Park, J.: Zeta regularized determinants for conic manifolds. J. Funct. Anal. 242, 195–229 (2007)
Mooers, E.: Heat kernel asymptotics on manifolds with conic singularities. J. Anal. Math. 78, 1–36 (1999)
Nazarov, S., Plamenevsky, B.: Elliptic Problems in Domains with Piecewise Smooth Boundaries. de Gruyter Expositions in Mathematics, vol. 13. de Gruyter, Berlin (1994)
Olver, F.J.: Asymptotics and Special Functions. Academic Press, New York (1997)
Reed, M., Simon, B.: Methods of Modern Mathematical Physics. II. Fourier Analysis, Self-adjointness. Academic Press/Harcourt Brace, New York (1975)
Troyanov, M.: Les surfaces euclidiennes à singularités coniques. Enseign. Math. 32(1–2), 79–94 (1986)
Yafaev, D.: Mathematical Scattering Theory: General Theory. Translations of Mathematical Monographs. AMS, Providence (1992)
Zorich, A.: Flat surfaces. In: Frontiers in Number Theory, Physics, and Geometry. I, pp. 437–583. Springer, Berlin (2006)
Acknowledgements
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Michael Taylor.
Rights and permissions
About this article
Cite this article
Hillairet, L., Kokotov, A. Krein Formula and S-Matrix for Euclidean Surfaces with Conical Singularities. J Geom Anal 23, 1498–1529 (2013). https://doi.org/10.1007/s12220-012-9295-3
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12220-012-9295-3