Invariant-Component Confinement in the Vlasov–Poisson System with Quadratic Repulsion

Abstract We construct compact spherical equilibria for the three-dimensional gravitational Vlasov–Poisson system with a repulsive quadratic external field through an invariant-component confinement mechanism. A global positive-part energy ansatz has infinite mass because the effective potential tends to minus infinity at large radius; the distribution is therefore placed on the bounded connected component of an energy sublevel. Starting from a compact Newtonian equilibrium with a transverse first zero, we prove local continuation at fixed total mass and match the interior solution to the exterior Newtonian field. The boundary flux identity puts the material radius strictly below the attraction–repulsion balance radius and yields a nonempty Hill-type forbidden interval. For general finite-mass solutions, the centre of mass separates exactly from the internal dynamics and obeys an inverted-oscillator equation, with corresponding energy and virial splittings. For equilibria with additional cutoff regularity, the support-preserving linearization is a compact perturbation of the Stone transport generator. Off the transport spectrum, a Schur complement reduces the kinetic resolvent to an analytic compact field pencil. Rotational covariance separates multipole sectors, coherent translations generate the exact eigenvalues ±√λ, and prescribed distant tidal fields act first in the quadrupole sector. Thus the same bounded invariant component supports the nonlinear equilibrium, exact collective translation, and collisionless internal response. Preprint status This is the author’s preprint corresponding to a manuscript submitted to Annals of Physics on 22 July 2026. It has not undergone journal peer review. If a version of record is published, this Zenodo record will be updated with the journal citation and DOI.