Zeno product formula revisited (Q877852)

Let \(H\) be a non-negative self-adjoint operator in a Hilbert space \(\mathcal{H}\) and \(\mathcal{K}\) be a closed subspace of \(\mathcal{H}\) such that \(P:\mathcal{H}\longrightarrow\mathcal{K}\) is the orthogonal projection from \(\mathcal{H}\) onto \(\mathcal{K}\). Suppose that \(\text{dom}(\sqrt{H})\cap\mathcal{K}\) is dense in \(\mathcal{K}\) and denote by \(K\) the non-negative self-adjoint operator associated with the closed sesquilinear form \(l(f,g):=(\sqrt{H}f,\sqrt{H} g)\), \(f,g\in\text{dom}(l)=\text{dom}(\sqrt{H})\cap\mathcal{K}\). A Borel measurable function \(\phi(\cdot):[0,\infty)\longrightarrow\mathbb{C}\) is called admissible if the conditions \(|\phi(x)|=1\) \((x \in [0,\infty))\), \(\phi (0) = 1\), and \(\phi^{'}(+0)=-i\), are satisfied. The authors prove that if \(\phi\) is an admissible function which obeys \(\text{Im}(\phi(x))\leq 0\) for \(x\in[0,\infty)\), then for any \(t_{0}>0\), one has \[ \text{s-}\lim_{n \longrightarrow\infty}(P\phi(tH/n)P)^{n}=e^{-itK} \] uniformly in \(t\in[0,t_{0}]\), where the strong convergence is meant on \(\mathcal{K}\). It is also shown that if \(\mathcal{K}\subset\text{dom}(\sqrt{H})\), then the condition \(\text{Im}(\phi(x))\leq 0\) \((x \in [0,\infty))\) is not required.