Mathematische Annalen

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

Regularizing infinite sums of zeta-determinants

  • Matthias Lesch
  • Boris VertmanEmail author


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.


Zeta-determinant Resolvent expansion Regularized sums 

Mathematics Subject Classification

Primary 58J52 Secondary 34S05 34B24 58J32 



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.


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© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Mathematisches InstitutUniversität BonnBonnGermany

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