Proposition: Self-adjointness and active factors

The operator (eq:A1) is essentially self-adjoint on the total finite-particle domain. Its closure contains $\mathrm{Dom}\mathsf{N}$, and

Only the gravitational oscillator and ${\mathrm{Span}}_{\mathbb{C}}\{u_0,u_e,u_r\}$ are active.

Proof. Creation and annihilation satisfy

Products of two such operators obey the stated number bound. Different species commute, so the polynomial is symmetric and changes total number by at most two.

If $\Psi$ has at most $n$ particles, iteration gives

Thus every finite-particle vector is analytic for $A$. Nelson's analytic-vector theorem Nelson proves essential self-adjointness. Approximation by number truncations proves the domain inclusion and bound. Fock factorization over the displayed one-particle span and its orthogonal complement proves the final claim.