A Paley-Wiener type uniqueness result for the electromagnetic Schr\"odinger equation

Abstract

In this paper, we establish a Paley-Wiener type uncertainty principle for Schr\"odinger equations with bounded electric and magnetic potentials, \begin{align*} i\partial_tu+\Delta_Au+V(t,x)u=0,\,\,u(0,x)=u_0(x), \end{align*} where $\Delta_A=(\nabla-iA)^2$ denotes the magnetic Schr\"odinger operator. Specifically, under suitable assumptions on $A$ and $V$, we show that if a solution $u$ exhibits linear exponential decay and support property in one spatial direction at times $t=0$ and $t=1$ respectively, then $u$ must vanish identically. This result extends the theorem of Kenig-Ponce-Vega [Ann. Sci. \'Ec. Norm. Sup\'er. (4) 47 (2014), 539-557] to the case $A\neq0$. We overcome the difficulty brought by the magnetic potential which breaks the translation invariance in the leading term of Hamiltonian $H=\Delta_A+V$. As a direct consequence, we also obtain a uniqueness result for a class of semi-linear Schr\"odinger equation with electromagnetic potentials.