Phase retrieval for Schrödinger evolutions

Consider the one-dimensional Schrödinger evolution $$ i\partial_tw+\partial_x^2w=Vw,\hspace{5mm}(t,x)\in [0,T]\times\mathbb{R},$$ where $V$ is a real-valued time-dependent potential belonging to $L^1_tL^\infty_x+L^2_{t,x}$. We prove that an $L^2$ solution to the above equation is determined by its m