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

TL;DR

This paper establishes unique continuation properties for Schrödinger equations with gradient and potential terms using Carleman estimates, extending to magnetic Schrödinger equations, which is significant for understanding solution behaviors.

Contribution

It introduces novel Carleman estimates to prove unique continuation for Schrödinger equations with gradient and potential terms, including magnetic cases.

Findings

Proved unique continuation for Schrödinger equations with gradient terms.

Extended results to magnetic Schrödinger equations.

Developed new $L^2$ Carleman estimates for differential inequalities.

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.