Lemma: Field domains and the number estimate

For $v\in \mathfrak{h}$, $\Phi (v)$ is essentially self-adjoint on ${\mathcal{D}}_{\mathrm{fin}}$. Its closure satisfies, as a quadratic-form inequality on $D(N^{1/2})$,

Consequently, whenever $n(\rho )<\infty$,

Proof. The creation and annihilation estimates are

Apply $\text{norm}{x+y}^2\le 2\text{norm}{x}^2+2\text{norm}{y}^2$.

If $\psi$ has particle support at most $m$, repeated application of these estimates bounds

The corresponding analytic-vector series converges for every fixed argument. Thus ${\mathcal{D}}_{\mathrm{fin}}$ is a dense invariant domain of analytic vectors, and the analytic-vector theorem establishes essential self-adjointness.

The form estimate extends to $D(N^{1/2})$ by graph-norm approximation. Expand a positive trace-class $\rho$ in an orthonormal eigenbasis and sum the form estimate. Finite $n(\rho )$ ensures that the relevant eigenvectors with positive eigenvalue lie in $D(N^{1/2})$. The resulting bound is (eq:number-bound).