Dynamical versions of Morgan's Uncertainty Principle and Electromagnetic Schr\"{o}dinger Evolutions

Abstract

This paper investigates the unique continuation properties of solutions of the electromagnetic Schr\"{o}dinger equation $$ i\partial_{t}u(x,t)+(\nabla-i A)^{2}u(x,t)=V(x,t)u(x,t)\,\,\,\, \mbox{in} \,\,\,\mathbb{R}^{n}\times [0,1], $$ where $A$ represents a time-independent magnetic vector potential and $V$ is a bounded, complex valued time-dependent potential. Given $1<p<2$ and $1/p+1/q=1$, we prove that if \begin{equation*} \int_{\mathbb{R}^{n}}|u(x,0)|^{2}e^{2\alpha^{p}|x|^p/p}\ d x +\int_{\mathbb{R}^{n}}|u(x,1)|^{2}e^{2\beta^{q}|x|^q/q}\ d x <\infty, \end{equation*} for some $\alpha,\beta>0$ and there exists $N_{p}>0$ such that \begin{equation*} \alpha\beta>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 Schr\"{o}dinger equations.