Regularizing infinite sums of zeta-determinants

TL;DR

The paper introduces a novel multiparameter resolvent trace expansion for elliptic operators, enabling regularized sums of zeta-determinants, with applications to operators on surfaces of revolution and potential extensions to broader classes.

Contribution

It develops a new polyhomogeneous resolvent trace expansion for elliptic operators and applies it to compute zeta-determinants as regularized sums, including for infinite direct sums.

Findings

Derived a new resolvent trace expansion for elliptic operators.

Established a method to compute zeta-determinants via regularized sums.

Applied the method to the Laplace-Beltrami operator on surfaces of revolution.

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. 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 out method to extend to a much larger class of operators.