Spectral Zeta Function & Casimir Energy

The spectral zeta function ζ_A(s) = Σ λ_n^(-s) encodes eigenvalue spectra. Analytic continuation to s=-1/2 yields zero-point energy — the mathematical heart of the Casimir effect, connecting quantum vacuum fluctuations to spectral geometry.

Spectrum

Geometry Max eigenvalue index N 40 Re(s) for ζ(s) 2.0

ζ_A(s) = Σ λ_n^(-s)

ζ_A(s): —

ζ_A(-½) (vac.E): —

Weyl counting: —

Spectral gap: —