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arXiv · math-ph/0610046

Spectral analysis and zeta determinant on the deformed spheres

M. Spreafico·S. Zerbini
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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 $Δ_{S^N_k}$, we study the associated zeta functions $ζ(s,Δ_{S^N_k})$. We introduce a general method to deal with some classes of simple and double abstract zeta functions, generalizing the ones appearing in $ζ(s,Δ_{S^N_k})$. An application of this method allows to obtain the main zeta invariants for these zeta functions in all dimensions, and in particular $ζ(0,Δ_{S^N_k})$ and $ζ'(0,Δ_{S^N_k})$. We give explicit formulas for the zeta regularized determinant in the low dimensional cases, $N=2,3$, thus generalizing a result of Dowker \cite{Dow1}, and we compute the first coefficients in the expansion of these determinants in powers of the deformation parameter $k$.

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M. Spreafico, S. Zerbini. 2006-10-19. Spectral analysis and zeta determinant on the deformed spheres. https://doi.org/10.1007/s00220-007-0229-z

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