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Communications in Mathematical Physics

, Volume 273, Issue 3, pp 677–704 | Cite as

Spectral Analysis and Zeta Determinant on the Deformed Spheres

  • M. SpreaficoEmail author
  • S. Zerbini
Article

Abstract

We consider a class of singular Riemannian manifolds, the deformed spheres \({S^{N}_{k}}\) , defined as the classical spheres with a one parameter family g[k] of singular Riemannian structures, that reduces for k = 1 to the classical metric. After giving explicit formulas for the eigenvalues and eigenfunctions of the metric Laplacian \({\Delta_{{S^{N}_{k}}}}\) , we study the associated zeta functions \({\zeta(s, \Delta_{{S^{N}_{k}}})}\) . We introduce a general method to deal with some classes of simple and double abstract zeta functions, generalizing the ones appearing in \({\zeta(s,\Delta_{{S^{N}_{k}}})}\) . An application of this method allows to obtain the main zeta invariants for these zeta functions in all dimensions, and in particular \({\zeta(0,\Delta_{{S^{N}_{k}}})}\) and \({\zeta'(0,\Delta_{{S^{N}_{k}}})}\) . We give explicit formulas for the zeta regularized determinant in the low dimensional cases, N = 2,3, thus generalizing a result of Dowker [25], and we compute the first coefficients in the expansion of these determinants in powers of the deformation parameter k.

Keywords

Zeta Function Deformation Parameter Elliptic Function Spectral Type Dirichlet Series 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.ICMC-Universidade de São PauloSão CarlosBrazil
  2. 2.Dipartimento di FisicaUniversitá di Trento, Gruppo Collegato di TrentoPadovaItaly

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