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Hessians of spectral zeta functions. (English) Zbl 1057.58014

From the author’s abstract: For a general geometric differential operator of Laplace type [on a closed Riemannian manifold – the reviewer] with eigenvalues \(0\leq\lambda_1\leq\lambda_2\leq\cdots\), we consider the spectral zeta function \(Z(s)=\sum_{\lambda_j\neq0}\lambda_j^{-s}\). The modified zeta function \({\mathcal Z}(s)=\Gamma(s)Z(s)/\Gamma(s-n/2)\) is an entire function of \(s\). For a fixed value of \(s\), we calculate the Hessian of \({\mathcal Z}(s)\) with respect to the metric and show that it is given by a pseudodifferential operator \(T_s=U_s+V_s\) where \(U_s\) is polyhomogeneous of degree \(n-2s\) and \(V_s\) is polyhomogeneous of degree \(2\). The operators \(U_s/\Gamma(n/2+1-s)\) and \(V_s/\Gamma(n/2+1-s)\) are entire in \(s\). The symbol expansion of \(U_s\) is computable from the symbol of the Laplacian.

MSC:

58J50 Spectral problems; spectral geometry; scattering theory on manifolds
11M36 Selberg zeta functions and regularized determinants; applications to spectral theory, Dirichlet series, Eisenstein series, etc. (explicit formulas)
58J40 Pseudodifferential and Fourier integral operators on manifolds
58E11 Critical metrics
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References:

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