Spectral Geometry of Fourth Order Scalar Fields

A classical scalar field theory governed by the fourth-order operator ∇⁴ − α∇² + m² with a quartic nonlinearity, analyzed through the single spectral polynomial P_E(λ) = λ² + αλ + m². The paper establishes when the quadratic energy is nonnegative and when the linearized gradient flow is stable, including a Swift–Hohenberg-type finite-wavevector instability at k₀ = √(|α|/2) for α < 0. Spectral factorization gives the Green's function in two and three dimensions: double-Yukawa for α > 0, α² > 4m², and oscillatory-screened for α² < 4m², with the boundary identified as the Fisher–Widom line for α > 0. For prescribed sources, the interaction energy in the oscillatory regime has force-zero separations about π/q apart, alternately stable and unstable; the binding is weak, and in the tested configurations the cubic nonlinearity shifts the separations by less than 0.1. In the phase-separated regime the same kernels govern interface and droplet tails, and in one dimension free interfaces lock at separations whose stability is confirmed by converged Hessian spectra. For a Gaussian-chain polymer blend, the random-phase approximation gives the continuum coefficients, and a census of the poles of the full response shows that the slow pole of the k⁴ truncation is physical while the fast pole is an artifact of the truncation. Every computed number is reproducible from the accompanying scripts and logs.