Finite-rank approximations of spectral zeta residues

TL;DR

This paper presents a method to express residues of spectral zeta functions using heat trace asymptotics, extending to localized spectral zeta functions via regularized spectral sums.

Contribution

It introduces a novel approach to compute spectral zeta residues through asymptotic heat trace expansions, applicable to localized spectral functions.

Findings

Residues are expressed as regularized sums over the spectrum.

Method extends to spectral zeta functions localized by bounded operators.

Provides a unified framework for spectral residue calculations.

Abstract

We use the asymptotic expansion of the heat trace to express all residues of spectral zeta functions as regularized sums over the spectrum. The method extends to those spectral zeta functions that are localized by a bounded operator.