In Witten's note https://arxiv.org/abs/1803.04993, during the proof of Reeh-Schlieder theorem, he made an arguement that considering a function $$g(u)=\langle\chi|\phi(x_1)\dots e^{\mathrm{i}Hu}\phi(x_n)|\Omega\rangle\,.$$ Because the hamiltonian $H$ is bounded below by $0$, the operator $\exp(\mathrm{i}Hu)$ is holomorphic for $u$ in the upper half plane so that the function $g(u)$ is also holomorphic. Why the holomorphy of this function is connected to the boundness of $H$ and the reigon of $u$? I suppose that the upper half plane limit of $u$ is to stop the term $\exp(-Im(u)H )$ diverge to infinity, yet I am still confused by its relation to the holomorphy of the function.