Dynamical versions of Morgan's Uncertainty Principle and Electromagnetic Schrödinger Evolutions
Shanlin Huang, Zhenqiang Wang
TL;DR
This work proves a dynamical Morgan-type uncertainty principle for the electromagnetic Schrödinger equation with a magnetic potential, establishing that decay of a solution at times $0$ and $1$ under compatible exponential weights implies triviality when the product of weights exceeds a threshold. The authors develop a robust analytic toolkit—gauge normalization via the Cronström gauge, a pseudo-conformal Appell transform to normalize decays, linear exponential decay estimates, and a Carleman inequality for the magnetic operator—to obtain quantitative unique continuation results. The results hold for $1<p<2$ with $1/p+1/q=1$, and extend to a broad class of nonlinear Schrödinger equations, under a structural Condition A on the potentials. This work deepens the connection between dynamical uncertainty principles and magnetic Schrödinger evolutions, providing a method to address nonlinear EM Schrödinger models and offering insight into how magnetic fields influence unique continuation properties in PDEs.
Abstract
This paper investigates the unique continuation properties of solutions of the electromagnetic Schrödinger 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^{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 $α,β>0$ 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.