Unique continuation for the Schr\"odinger equation with gradient term

Abstract

We obtain a unique continuation result for the differential inequality $| (i\partial_t +\Delta)u | \leq |Vu| + | W\cdot\nabla u |$ by establishing $L^2$ Carleman estimates. Here, $V$ is a scalar function and $W$ is a vector function, which may be time-dependent or time-independent. As a consequence, we give a similar result for the magnetic Schr\"odinger equation.