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Description
We investigate quantum vacuum effects for a massless scalar field propagating on a one-parameter family of isotropic ultrastatic spacetimes containing two exact solutions of the $(n+1)$-dimensional Einstein field equations: a scalar-flat ultrastatic vacuum and a constant-curvature geometry. Using spectral zeta-function regularization, we evaluate the one-loop effective action associated with the corresponding Klein–Gordon operator. We show that the constant-curvature solution constitutes a distinguished spectral critical point, where the Liouville transformation becomes logarithmic and the radial eigenfunctions undergo a transition from Bessel modes to plane waves in the transformed coordinate. This transition produces a logarithmic non-analyticity in the renormalized one-loop effective action and Casimir energy, whereas the scalar-flat branch exhibits a qualitatively different infrared behavior. These results establish a direct connection between the scaling properties of the background geometry and the spectral structure of quantum vacuum fluctuations, providing a new example of geometry-induced spectral criticality in semiclassical gravity.