2026/04/30 by Keisuke Okamura
Physics and Astronomy · Mathematics · #math-ph #math.MP
18 pages
arxiv created 2026/08/01 · arxiv updated 2026/08/04
For a positive elliptic operator A, the logarithmic zeta determinant lndetζA=-ζA'(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 ζA(0) and ζA(q-1), defining a one-parameter meromorphic family whose node-coalescence limit q→ 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 a4, 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.