I have some questions about Dyson series in quantum theory. For perturbed hamiltonian $H=H_0+\lambda V(t)$, we have the following differential equations in interaction picture. $$ i \hbar \frac{d}{dt} |\psi_I(t)\rangle=\lambda V_I(t)|\psi_I(t) \rangle $$ Integrating iteratively gives the following dyson series. $$ |\psi_I(t)\big>=|\psi_I(0)\big>+\frac{\lambda}{i\hbar}\int_0^tV_I(t')|\psi_I(0)\rangle dt'+(\frac{\lambda}{i\hbar})^2\int_0^tV_I(t') \int_0^{t'}V_I(t'')|\psi_I(0)\rangle dt''dt'+... $$ I have several questions about this equation.
- First, is it right to understand this integral in sense of Bochner integral? I think so because we are integrating 'vectors' in hilbert space.
- $V_I(t)$ is in general unbounded operator. Is it okay to interchange order of integration and operator $V_I(t)$? (That is, is $\int V_I=V_I\int$ valid?)
- I think it is possible for $V_I(t')|\psi_I(0)\rangle\notin D(V_I(t))$ or $V_I(t')V_I(t'')|\psi_I(0)\rangle\notin D(V_I(t))$ ... to happen. Here, $D(V_I(t))$ is a domain of $V_I(t)$. If this happens, the equation above does not make sense. Is there an additional reasonable hypothesis on $V_I(t)$ that makes this not happen?
Thank you for reading these questions.