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

TL;DR

This paper establishes dynamical uncertainty principles for electromagnetic Schrödinger equations, showing that under certain exponential decay conditions at two different times, solutions must be trivial, extending Morgan's uncertainty principle.

Contribution

It introduces dynamical versions of Morgan's uncertainty principle for electromagnetic Schrödinger equations with magnetic potentials and extends results to semi-linear cases.

Findings

Solutions with certain exponential decay at two times are identically zero.

Results apply to a broad class of semi-linear Schrödinger equations.

Provides conditions under which unique continuation holds.

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.