Phase retrieval for Schrödinger evolutions

Consider the one-dimensional Schrödinger evolution $$i{\mathrm{\partial }}_{t}w+{\mathrm{\partial }}_x^{2}w=Vw,(t,x)\in [0,T]\times \mathbb{R},$$ where $V$ is a real-valued time-dependent potential belonging to $L_t^1{L}_x^{\mathrm{\infty }}+L_{t,x}^2$. We prove that an $L^2$ solution to the above equation is determined by its mass density $\mathrm{\mid }w{\mathrm{\mid }}^2$ modulo a global unimodular phase factor. This answers a question recently posed by Jaming for a large class of potentials, and recovers his earlier result for the free Schrödinger evolution. Our proof is simple and robust. It combines the basic local smoothing properties of the above Schrödinger evolutions, Strichartz estimates, and a stripwise version of a unique continuation theorem of Ionescu and Kenig to show that the interaction function $$F(t,x,y)=u(t,x)v(t,y)-u(t,y)v(t,x)$$ vanishes identically on $[0,T]\times {\mathbb{R}}^2$ whenever $u$ and $v$ are $L^2$ solutions to a Schröinder euqation with potential $V$ with the same mass density. The method immediately extends to several important nonlinear models such as the cubic NLS equation, which is globally well-posed in $L^2(\mathbb{R})$ with $V=\mathrm{\mid }u{\mathrm{\mid }}^2$. Interestingly, we also show that the above phase recovery property fails in dimensions two and higher, even for the free equation with Schwartz initial data.