Finite-difference zeta readout of one-loop operator spectra

Abstract

For a positive elliptic operator $A$, the logarithmic zeta determinant $\ln\det_ζA=-ζ_{A}'(0)$ combines UV information encoded by local heat-kernel coefficients with finite contributions determined by the full spectrum. We introduce a finite-difference zeta readout based on $ζ_{A}(0)$ and $ζ_{A}(q-1)$, defining a one-parameter meromorphic family whose node-coalescence limit $q\to 1$ recovers the standard logarithmic zeta determinant. The parameter $q$ fixes the Mellin evaluation point $s=q-1$, organising genuine poles, regular local special values, generic full-spectrum values, and the logarithmic determinant limit along a common coordinate, while simultaneously determining the spectral weight $λ^{-q}$ in the $q$-dependent variational response. In relative spectral problems, this coordinate distinguishes systems retaining a leading local hierarchy from those in which the entire local power-law hierarchy cancels, illustrated respectively by a reflectionless soliton and a twisted circle. In four dimensions, the framework recovers the standard local scale response at $q=1$, governed by the heat-kernel coefficient $a_{4}$, whereas at generic regular values of $q$ it retains finite mass-sensitive information beyond the local hierarchy. The construction thereby provides a unified analytic framework for comparing local UV structure, finite full-spectrum information, variational response, and relative spectral behaviour within fixed operator spectra.