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Mathematische Annalen

, Volume 361, Issue 3–4, pp 835–862 | Cite as

Regularizing infinite sums of zeta-determinants

  • Matthias Lesch
  • Boris VertmanEmail author
Article

Abstract

We present a new multiparameter resolvent trace expansion for elliptic operators, polyhomogeneous in both the resolvent and auxiliary variables. For elliptic operators on closed manifolds the expansion is a simple consequence of the parameter dependent pseudodifferential calculus. As an additional nontrivial toy example we treat here Sturm–Liouville operators with separated boundary conditions. As an application we give a new formula, in terms of regularized sums, for the \(\zeta \)–determinant of an infinite direct sum of Sturm–Liouville operators. The Laplace–Beltrami operator on a surface of revolution decomposes into an infinite direct sum of Sturm–Louville operators, parametrized by the eigenvalues of the Laplacian on the cross-section \(\mathbb {S}^1\). We apply the polyhomogeneous expansion to equate the zeta-determinant of the Laplace-Beltrami operator as a regularized sum of zeta-determinants of the Sturm–Liouville operators plus a locally computable term from the polyhomogeneous resolvent trace asymptotics. This approach provides a completely new method for summing up zeta-functions of operators and computing the meromorphic extension of that infinite sum to \(s=0\). We expect our method to extend to a much larger class of operators.

Keywords

Zeta-determinant Resolvent expansion Regularized sums 

Mathematics Subject Classification

Primary 58J52 Secondary 34S05 34B24 58J32 

Notes

Acknowledgments

The authors gratefully acknowledge helpful discussions with Leonid Friedlander and Rafe Mazzeo. They also thank Benedikt Sauer for a careful reading of the manuscript. The authors would like to thank the anonymous referee for valuable comments and suggestions. Both authors were supported by the Hausdorff Center for Mathematics.

References

  1. 1.
    Bordag, M., Kirsten, K., Dowker, S.: Heat-kernels and functional determinants on the generalized cone. Commun. Math. Phys. 182(2), 371–393 (1996)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Chamseddine, A.H., Connes, A.: Spectral action for Robertson–Walker metrics. J. High Energy Phys. 10, 101 (2012)CrossRefMathSciNetGoogle Scholar
  3. 3.
    Egorov, Y.V., Schulze, B.-W.: Pseudo-Differential Operators, Singularities, Applications, Operator Theory: Advances and Applications, vol. 93. Birkhäuser, Basel (1997)CrossRefGoogle Scholar
  4. 4.
    Gilkey, P.B.: Invariance theory, the heat equation, and the Atiyah–Singer index theorem. Studies in Advanced Mathematics, 2nd edn. CRC Press, Boca Raton (1995)Google Scholar
  5. 5.
    Grubb, G.: Functional calculus of pseudodifferential boundary problems. In: Progress in Mathematics, 2nd edn, vol. 65. Birkhäuser Boston Inc, Boston (1996)Google Scholar
  6. 6.
    Guillemin, V.W., Sternberg, S., Weitsman, J.: The Ehrhart function for symbols. Surveys in differential geometry. Surv. Differ. Geom, vol. 10, pp. 31–41. Int. Press, Somerville (2006)Google Scholar
  7. 7.
    Lesch, M.: Operators of Fuchs type, conical singularities, and asymptotic methods, Teubner-Texte zur Mathematik [Teubner Texts in Mathematics], vol. 136, B.G. Teubner Verlagsgesellschaft mbH, Stuttgart (1997). arXiv:dg-ga/9607005v1
  8. 8.
    Lesch, M.: Determinants of regular singular Sturm–Liouville operators. Math. Nachr. 194, 139–170 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Lesch, M.: Pseudodifferential operators and regularized traces, motives, quantum field theory, and pseudodifferential operators. In: Clay Math. Proc., vol. 12, pp. 37–72. Amer. Math. Soc., Providence (2010). arXiv:0901.1689 [math.OA]
  10. 10.
    Mazzeo, R., Vertman, B.: Analytic torsion on manifolds with edges. arXiv:1103.0448v1 [math.SP]
  11. 11.
    Mazzeo, R.: Elliptic theory of differential edge operators. I. Commun. Partial Differ. Equ. 16(10), 1615–1664 (1991)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Melrose, R.B.: The Atiyah–Patodi–Singer index theorem. In: Research Notes in Mathematics, vol. 4. A K Peters Ltd., Wellesley (1993)Google Scholar
  13. 13.
    Mooers, E.A.: Heat kernel asymptotics on manifolds with conic singularities. J. Anal. Math. 78, 1–36 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Sauer, B.: On the Resolvent Trace of Multi-Parametric Sturm–Liouville Operators. Universität Bonn, Diplomarbeit (2013)Google Scholar
  15. 15.
    Seeley, R.: The resolvent of an elliptic boundary problem. Am. J. Math. 91, 889–920 (1969)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Shubin, M.A.: Pseudodifferential operators and spectral theory, 2nd edn. Springer, Berlin (2001). Translated from the 1978 Russian original by Stig I. AnderssonGoogle Scholar
  17. 17.
    Spreafico, M.: Zeta function and regularized determinant on a disc and on a cone. J. Geom. Phys. 54(3), 355–371 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Spreafico, M.: Zeta invariants for Dirichlet series. Pac. J. Math. 224(1), 185–200 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Vertman, B.: Analytic torsion of a bounded generalized cone. Commun. Math. Phys. 290(3), 813–860 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Vertman, B.: Multiparameter resolvent trace expansion for elliptic boundary problems. arXiv:1301.7293 [math.SP] pp. 13, 19

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Mathematisches InstitutUniversität BonnBonnGermany

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