Regularizing infinite sums of zeta-determinants

Abstract

We present a new multiparameter resolvent trace expansion for elliptic operators, polyhomogeneous in both the resolvent and auxiliary variables. For elliptic operators on closed manifolds the expansion is a simple consequence of the parameter dependent pseudodifferential calculus. As an additional nontrivial toy example we treat here Sturm–Liouville operators with separated boundary conditions. As an application we give a new formula, in terms of regularized sums, for the $\zeta$–determinant of an infinite direct sum of Sturm–Liouville operators. The Laplace–Beltrami operator on a surface of revolution decomposes into an infinite direct sum of Sturm–Louville operators, parametrized by the eigenvalues of the Laplacian on the cross-section $\mathbb {S}^1$. We apply the polyhomogeneous expansion to equate the zeta-determinant of the Laplace-Beltrami operator as a regularized sum of zeta-determinants of the Sturm–Liouville operators plus a locally computable term from the polyhomogeneous resolvent trace asymptotics. This approach provides a completely new method for summing up zeta-functions of operators and computing the meromorphic extension of that infinite sum to $s=0$. We expect our method to extend to a much larger class of operators.