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Spectral Analysis and Zeta Determinant on the Deformed Spheres

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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.

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Authors and Affiliations

  1. ICMC-Universidade de São Paulo, São Carlos, Brazil

    M. Spreafico

  2. Dipartimento di Fisica, Universitá di Trento, Gruppo Collegato di Trento, Sezione INFN di Padova, Padova, Italy

    S. Zerbini

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  1. M. Spreafico
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  2. S. Zerbini
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Correspondence to M. Spreafico.

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Communicated by G.W. Gibbons

Partially supported by FAPESP: 2005/04363-4

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Spreafico, M., Zerbini, S. Spectral Analysis and Zeta Determinant on the Deformed Spheres. Commun. Math. Phys. 273, 677–704 (2007). https://doi.org/10.1007/s00220-007-0229-z

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