Assume $u\in C([0,1],L^{2})$ satisfies the following Schrodinger equation $$ \partial_t u=i(\Delta u+Vu), \text{in} ~\mathbb{R}^n\times[0,1],\\ u(0)=u_{0}. $$ with $V=V_1(x)+V_2(x,t)$, where $V_{1}$ is real valued and $\|V_{1}\|_{\infty}\leq M$, and $V_{2}$ satisfies that $\sup_{t\in[0,1]}\|e^{\frac{|x|^2}{\alpha t+\beta}}V(x,t)\|_{\infty}<\infty$.
If we denote $H=\Delta+V_{1}$, and $u_{\epsilon}(t)=e^{\epsilon t H}u(t)$, where $\epsilon, t\in [0,1]$. Then how to prove that $$ u_{\epsilon}\in L^{\infty}([0,1],L^{2}(\mathbb{R}^n))\cap L^{2}([0,1],H^{1}(\mathbb{R}^n)) ? $$ I came across this problem when reading the paper "Hardy's uncertainty principle, convexity and schrodinger evolutions" by Escauriaza, Kenig, Ponce and Vega. Since they stated it directly, I don't know if this is some standard results in PDE.
We know that $u_{\epsilon}(t)$ satisfies a parabolic type equation, I think the energy method may be applied to prove $u_{\epsilon}\in L^{2}([0,1],H^{1}(\mathbb{R}^n))$, but I don't know how to deal with the terms involved $V_{2}(x,t)$. Can some recommend me some references or show me how to prove the statement above?
