# Hamiltonian: Commutator

Given a Hilbert space $\mathcal{H}$.

Consider a Hamiltonian: $$H:\mathcal{D}(H)\to\mathcal{H}:\quad H=H^*$$

and an operator: $$A:\mathcal{D}(A)\to\mathcal{H}:\quad A=A^*$$

Denote for shorthand: $$\mathcal{D}_0\subseteq\mathcal{D}:=\mathcal{D}(H)\cap\mathcal{D}(A)$$

with invariant core: $$e^{i\alpha A}\mathcal{D}_0\subseteq\mathcal{D}_0:\quad\overline{H_0}=H$$

Introduce the form: $$s(\varphi,\psi):=i\langle A\varphi,H\psi\rangle-i\langle H\varphi,A\psi\rangle\quad(\varphi,\psi\in\mathcal{D})$$

bounded-below on: $$s_0(\varphi,\varphi)\geq\sigma\|\varphi\|^2\quad(\varphi\in\mathcal{D}_0)$$

Regard its operator: $$\hat{s_0}(\varphi,\psi)=\langle i[H,A]_0\varphi,\psi\rangle$$

Then it extends: $$\mathcal{D}(i[H,A]_0)\supseteq\mathcal{D}(H)\implies s(\varphi,\psi)=\langle i[H,A]_0\varphi,\psi\rangle$$

Giving rise to: $$\hat{s}(\varphi,\psi)=\langle i[H,A]\varphi,\psi\rangle$$

How can I prove this from scratch?