doi: 10.17706/ijapm.2026.16.2.52-61
Spectral Analysis of the Inverse-square Model Operator Associated with the Cosmological Horizon Blow-up Index: Bessel Eigenvalues, Heat Trace, and Zeta Function
Craig S. Wright
- University of Exeter, Exeter, United Kingdom.
- Manuscript received March 25, 2026
- accepted August 7, 2026
- published October 27, 2026
Abstract
In FLRW spacetimes with finite conformal-time budget, the cosmological time-dilation factor (1+z) diverges at the last-emission surface; a previous companion paper classifies this divergence by the Karamata blow-up index β =p/(p-1) for scale factors α∈RVp. We study the model operator Lv=-d2/dx2+(v2-1/4)x-2 on L2((0,L]) with Dirichlet condition at x=L and limit-point condition at x=0, where ν=β+1/2. Proved: (1) the Dirichlet realisation is self-adjoint with purely discrete spectrum for ν≥1; (2) eigenvalues λn=jv,n2/L2; (3) a Weyl law with β-dependent sub-leading coefficient; (4) the diagonal resolvent vanishes linearly at the singular endpoint; (5) two-sided Sturm comparison bounds for the cosmological operator; (6) exact heat-trace constants a1=-1/2 at ν=1/2 and a1=-1 at ν=3/2; (7) ζ1/2(0)=-1/2 and detL1/2=2L. Conditional on standard meromorphic continuation: a formal Hadamard-product representation for the zeta-regularised determinant; the general-ν value of ζv(0) is not proved in this paper. Numerically supported: for ν=2,5/2, direct summation over 104 Bessel zeros is consistent with the conjectural a1=-v /2-1/4. For general ν, we give evidence that the Dirichlet-realisation zeta function avoids the previously reported branch-point pathology, but do not carry out the full contour-integral proof.
Keywords
- Sturm–Liouville operator
- spectral zeta function
- Bessel zeros
- regular variation
- functional determinant
- cosmological event horizon
How to Cite
Craig S. Wright, "Spectral Analysis of the Inverse-square Model Operator Associated with the Cosmological Horizon Blow-up Index: Bessel Eigenvalues, Heat Trace, and Zeta Function," International Journal of Applied Physics and Mathematics, vol. 16, no. 2, pp. 52-61, 2026. https://doi.org/ 10.17706/ijapm.2026.16.2.52-61
Copyright & License
Copyright © 2026 by the authors. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).