Lemma: Number bounds and essential self-adjointness

For every real $f\in \mathcal{S}(M)$, $F_f$ is Hilbert–Schmidt, $B_f$ is bounded, and

The operator $Q(f)$ is essentially self-adjoint on ${\mathcal{D}}_{\mathrm{fin}}$. Its restriction to ${\mathcal{D}}_E$ has the same self-adjoint closure.

Proof. Schwartz decay in the temporal Fourier variable gives

The radial measure is proportional to $p^0\mathrm{d}{p}^0$ on each shell, so the square of this bound is integrable for sufficiently large $M$. Thus $F_f$ is Hilbert–Schmidt.

For $B_f$, Schwartz decay gives, for large $M$,

The integral of this bound against $\mathrm{d}\mu (q)$ is uniformly bounded in $p$: on $|\mathbf{q}|<1$, the measure is integrable; on $|\mathbf{q}|\ge 1$, it is bounded by a constant times Lebesgue measure, and the translated integrable decay has constant integral. The corresponding column bound is identical. Schur's test proves boundedness.

On the $n$-particle sector, the pair creation and annihilation terms are bounded by constants times $\sqrt{(n+1)(n+2)}\Vert F_f\Vert$ and $\sqrt{n(n-1)}\Vert F_f\Vert$. The number-preserving term is bounded by $2n\Vert B_f\Vert$. This proves (eq:Qnumber).

Repeated application changes particle number by at most two. For a vector with maximal particle number $n$,

The analytic-vector series consequently converges for sufficiently small parameter. Nelson's analytic-vector theorem Nelson gives essential self-adjointness on ${\mathcal{D}}_{\mathrm{fin}}$.

On each fixed finite collection of particle sectors, (eq:Qnumber) bounds $Q(f)$ by a constant times the Hilbert norm. Energy-cutoff vectors in ${\mathcal{D}}_E$ approximate any vector of ${\mathcal{D}}_{\mathrm{fin}}$, both in Hilbert norm and after applying $Q(f)$. Thus ${\mathcal{D}}_E$ is a graph core for this particular closure.