Positivity Improvement and Ground-State Simplicity for a Renormalized Moving-Mirror Hamiltonian on a Ring

A self-adjoint Hamiltonian can possess a ground-state eigenvector without having aunique one. For a moving mirror coupled to a harmonic oscillator and an unboundedbosonic field, this distinction persists after ultraviolet renormalization. This articleexamines the periodic-ring version of that problem. We formulate the operator on thefull bosonic Fock space, rotate the oscillator into a real position coupling, and exploitthe local multiplication structure of every symmetric finite-mode interaction. A support-locality identity survives the ultraviolet limit whenever the companion quadratic-formconstruction gives convergence on one common free-energy form domain. The identitytransfers any hypothetical invariant measurable support of the interacting semigroup tothe free semigroup. The latter is positivity improving, including on its infinite Gaussianfield-coordinate space. We therefore prove, under the explicitly stated ultraviolet formhypotheses, positivity improvement at every fixed real coupling. Compact resolvent thengives a unique strictly positive ground eigenvector and a strictly positive, generally non-quantitative spectral gap. We additionally derive evenness and continuity in the coupling,a positive minimum of the gap on every compact coupling interval, and convergenceof ground-state projections for admissible hard and smooth regulators. These resultsconcern the specified periodic model; they do not prove an infinite-line ground state or auniform positive gap as the ring length diverges.