Theorem: Autonomous self-adjoint dynamics

Under (eq:regularity), the restriction $V_n$ is bounded and Hermitian, with

Each

is self-adjoint. The direct sum

is self-adjoint on

It defines $U(t)=e^{-\mathrm{i}tH}$, and every excitation sector reduces this group.

Moreover,

In particular, $H\ge 0$ if ${\omega }_0\ge {\kappa }_g$.

On the one-excitation sector,

with domain $\mathbb{C}\oplus D(M_{\nu })$.

Proof. On the $(n-1)$-particle sector,

For excited-source components ${\psi }_a$,

The annihilation block is its adjoint. The Hermitian off-diagonal block has the stated norm. The bounded self-adjoint perturbation theorem gives $H_n$ Kato. The direct sum is self-adjoint and has the stated domain. Finite sector sums of sector-domain vectors form an operator core.

For the lower bound, use finite-particle energy-form vectors and complete the square:

Here $a_a(\nu )\Psi$ denotes the usual annihilation-density map, interpreted in the energy-form integral. The condition $g/\sqrt{\nu }\in L^2$ makes the shifted integral and its cross terms well defined. The cross terms equal (eq:V), and ${\sum }_a{L}_a^{†}{L}_a=P_{\mathrm{ex}}$. The identity may first be checked on finite-particle vectors with one-particle tests supported away from zero and then extended in the form norm. It implies the claimed lower bound, which extends to the self-adjoint direct sum.

In ${\mathcal{H}}_1$, each source label couples only to the one-particle field of that label. The creation block sends the source vector to $g\otimes e_a$, giving (eq:friedrichs).