Some remarks on the magnetic field operators \nabla±iA and its applications

Abstract

In the present paper, we give some remarks on the magnetic field operators $\nabla \pm iA$. As its applications, we study the Schr\"{o}dinger equation with a magnetic field \begin{equation*} -\Delta u+|A(x)|^{2}u+iA(x)\cdot \nabla u=\mu u+|u|^{p}u,~x\in \mathbb{R}^{N}, \end{equation*} where $u$ is a complex-valued function and $\mu\in \mathbb{R}$. When $N>2$, for $2 p+2 \frac{2N}{N-2}$ or $N=2$, for $2 p+2 +\infty$, the existence and nonexistence of minimizers of the corresponding minimization problem are given via constrained variational methods. As a by-product, the above equation admits a normalized solution. We point out that the condition ${div}A(x)=0$ plays a crucial role in our study.