Dynamical versions of Morgan's Uncertainty Principle and Electromagnetic Schrödinger Evolutions

This paper investigates the unique continuation properties of solutions of the electromagnetic Schrödinger equation $$i{\mathrm{\partial }}_{t}u(x,t)+(\mathrm{\nabla }-iA{)}^{2}u(x,t)=V(x,t)u(x,t)in{\mathbb{R}}^n\times [0,1],$$ where $A$ represents a time-independent magnetic vector potential and $V$ is a bounded, complex valued time-dependent potential. Given $1<p<2$ and $1\mathrm{/}p+1\mathrm{/}q=1$, we prove that if \begin{equation*} \int_{$\mathbb{R}$^{n}}|u(x,0)|^{2}e^{2α^{p}|x|^p/p}\ d x +\int_{$\mathbb{R}$^{n}}|u(x,1)|^{2}e^{2β^{q}|x|^q/q}\ d x <\infty, \end{equation*} for some $\alpha ,\beta >0$ and there exists $N_p>0$ such that \begin{equation*} αβ>N_p, \end{equation*} then $u\equiv 0$. These results can be interpreted as dynamical versions of the uncertainty principle of Morgan’s type. Furthermore, as an application, our results extend to a large class of semi-linear Schrödinger equations.