Consider the linear Schrödinger equation, $$ (LS) \begin{cases} \partial_{t}u= i\Delta u, t\in \mathbb R,\\ u(x,0)=u_{0}(x), \end{cases} $$ $x\in \mathbb R^{n}.$ Taking the Fourier transform with respect to the space variable $x$ in $(LS)$, we obtain, $$ \begin{cases} \widehat{\partial_{t}u}(\xi,t)=\partial_{t}\hat{u}(\xi,t)=\widehat{i\Delta}u(\xi,t)=-4\pi^{2}i|\xi|^{2}\hat{u}(\xi,t),\\ \hat{u}(\xi, 0)=\hat{u_{0}}(\xi). \end{cases} $$ The solution of this family of ordinary differential equations, with parameter $\xi,$ can be written as, $\hat{u}(\xi,t)= e^{-4\pi^{2}it|\xi|^{2}}\hat{u_{0}}(\xi);$ and then taking inverse Fourier transform, we have, $$u(x,t)=(e^{-4\pi^{2} it|\xi|^{2}}\hat{u_{0}}(\xi) )^\vee:= e^{it\Delta}u_{0}(x).$$ Next, we consider, inhomogeneous problem, $$ (NLS) \begin{cases} \partial_{t}u= i\Delta u +F(x,t), t\in \mathbb R,\\ u(x,0)=u_{0}(x), \end{cases} $$ $x\in \mathbb R^{n}, F\in C(\mathbb R:\mathcal{S}(\mathbb R)).$
My Question is:
How to show that the solution of $(NLS)$ is given by the formula, $$u(x,t)=e^{it\Delta}u_{0}+\int_{0}^{t}e^{i(t-t')\Delta} F(\cdot,t') dt'.$$
Thanks,