Theorem: Mixed bound and real polynomial closure

For compact real $g,f$, let

Then ${\mathcal{R}}_{g,f}$ is symmetric and preserves ${\mathcal{D}}_{\mathrm{mix}}$, and

It extends continuously from $\mathrm{Dom}(N)$, with its graph norm, to $\mathcal{H}$, and is essentially self-adjoint on ${\mathcal{D}}_{\mathrm{mix}}$.

For every real $\lambda$, the real polynomial

is also essentially self-adjoint on ${\mathcal{D}}_{\mathrm{mix}}$. This is a statement about the displayed polynomial, not an all-orders interacting field.

Proof. The oscillator estimates and scalar estimates are

They act on commuting factors. Using

proves (eq:mixedbound). Schwartz $v_e,v_r$ and finite oscillator shifts prove invariance. Reality gives symmetry.

The domain is a core for $N$: truncate particle number and approximate each remaining scalar wavefunction in Schwartz space. Completion gives the graph extension.

Let $A$ be either operator in the theorem. There is $C_A<\infty$ such that $\Vert A\Psi \Vert \le C_A\Vert (N+2)\Psi \Vert$, and $A$ raises total particle number by at most two. If $\Psi$ has sectors at most $M$, induction gives

Thus every vector of ${\mathcal{D}}_{\mathrm{mix}}$ is analytic for $A$. Nelson's analytic-vector criterion Nelson proves essential self-adjointness.