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


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.


Zeta Function Deformation Parameter Elliptic Function Spectral Type Dirichlet Series 
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© 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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