Proposition: Model 1 operator

For every real $\lambda$, (eq:A1) is essentially self-adjoint on the total finite-particle domain. Its closure is defined on $\mathrm{Dom}\mathsf{N}$. It acts only on the gravitational oscillator and

in the scalar one-particle space, and is the identity on the orthogonal Fock factor.

Proof. The standard sector estimates are

Products of two such operators obey a constant multiple of $\Vert (\mathsf{N}+2)\Psi \Vert$. Expanding the two mixed terms in (eq:A1) gives a symmetric operator, changing number by at most two, with the estimates of Lemma lem:analytic. Its factors commute because the species differ. The lemma proves essential self-adjointness and the domain inclusion.

Fock factorization over $V_f\oplus {V_f}^{\perp }$ gives the active factor assertion first on finite-particle vectors and then for the unique self-adjoint closure.